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Spatial stochastic models of HSV-2 lesion dynamics and their link with HIV-1 acquisition Byrne, Catherine Margaret McCombe 2016

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Spatial Stochastic Models of HSV-2Lesion Dynamics and Their Link withHIV-1 AcquisitionbyCatherine Margaret McCombe ByrneB.Sc. (Hons), Queen’s University, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2016c© Catherine Margaret McCombe Byrne 2016AbstractPatients with Herpes Simplex Virus-2 (HSV-2) infection face a significantlyhigher risk of contracting HIV-1. This marked increase is thought to bedue not only to herpetic lesions serving as an entry point for the HIV-1 virus, but also to the increase in CD4+ T cells in the human genitalmucosa during HSV-2 lesional events. By creating a stochastic, spatial,mathematical model describing the behaviour of the HSV-2 infection andimmune response in the genital mucosa, I first capture the dynamics thatoccur during the development of an HSV-2 lesion. I then use this model toquantify the risk of acquiring HIV-1 in HSV-2 positive patients upon sexualexposure, and determine whether antivirals meant to control HSV-2 candecrease HIV-1 infectivity. While theory predicts that HSV-2 treatmentshould lower HIV-1 infection probability, my results show that this maynot be the case unless a critical dosage of HSV-2 treatment is given to thepatient. These results help to explain the conflicting data on HIV-1 infectionprobability in HSV-2 patients and allow for further insight into the type oftreatment HSV-2 positive patients should receive to prevent HIV-1 infection.iiPrefaceAll work presented in this dissertation is original, remains unpublished, andhas been independently conducted by the author, Catherine Byrne.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Biology of a Genital HSV-2 Infection . . . . . . . . . . . 11.2 The Link Between HSV-2 Lesional Events and HIV-1 Con-traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Effects of HSV-2 Antiviral Drugs on HIV-1 Prevention . . . 31.4 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A Basic Model of HSV-2 Dynamics . . . . . . . . . . . . . . 52.1 Designing a Model of HSV-2 Infection in the Genital Mucosa 52.1.1 Adding Spatial Resolution to the Model SimulationRegion . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Defining Infection-Dependant Rates of Immune CellProliferation . . . . . . . . . . . . . . . . . . . . . . . 92.2 Parameters of the Model . . . . . . . . . . . . . . . . . . . . 102.3 Simulations of HSV-2 Infection . . . . . . . . . . . . . . . . . 122.3.1 Results of Model I for Immune Cell Proliferation . . . 132.3.2 Results of Model II for Immune Cell Proliferation . . 20ivTable of Contents3 The Establishment of HSV-2 - HIV-1 Coinfection . . . . . 263.1 A Description of HIV-1 Infection Dynamics . . . . . . . . . . 263.1.1 Defining HIV-1 Entry . . . . . . . . . . . . . . . . . . 273.1.2 Parameters for HIV-1 . . . . . . . . . . . . . . . . . . 293.1.3 The Dynamics of HIV-1 Infection . . . . . . . . . . . 293.2 Determining HIV-1 Infection Probability . . . . . . . . . . . 334 Effects of HSV-2 Antivirals on HIV-1 Infection Probability 424.1 Impact of Antivirals on HSV-2 Infection and Lesion Develop-ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Pritelivir’s Impact on HIV-1 Infection Probability . . . . . . 464.2.1 HIV-1 Infection Probability for the Entire Genital Re-gion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57AppendixA Main Python Code for HSV-2 Infection Simulations . . . 63vList of Tables2.1 Model parameter values for HSV-2 infection . . . . . . . . . . 123.1 Model parameter values for HIV-1 infection . . . . . . . . . . 293.2 Simulation scenarios used to find HIV-1 infection probabilities 353.3 Fold increases in per-coital HIV-1 risk when comparing expo-sure to chronic versus acute HIV-1 viral loads . . . . . . . . . 40viList of Figures2.1 Description of the model . . . . . . . . . . . . . . . . . . . . . 82.2 HSV-2 infection dynamics with no spatial resolution . . . . . 152.3 Summed HSV-2 infection dynamics on a 5× 5 grid . . . . . . 162.4 Spatial HSV-2 infection dynamics on a 5× 5 grid . . . . . . . 172.5 Summed HSV-2 infection dynamics on a 15× 15 grid . . . . . 182.6 Spatial HSV-2 infection dynamics on a 15× 15 grid . . . . . . 192.7 Summed HSV-2 infection dynamics on a 15×15 grid with theinclusion of cytokine effects . . . . . . . . . . . . . . . . . . . 222.8 Spatial HSV-2 infection dynamics on a 15× 15 grid with theinclusion of cytokine effects . . . . . . . . . . . . . . . . . . . 232.9 Ten simulations of HSV-2 infection dynamics . . . . . . . . . 253.1 Dynamics of HIV-1 infection establishment in an HSV-2 pos-itive patient upon exposure to an acute dose of HIV-1 in thesemen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Spatial dynamics of HIV-1 infection establishment . . . . . . 323.3 Per-coital act probability of HIV-1 infection in one simulationregion upon exposure to HIV-1 from a chronically infectedpartner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Per-coital act probability of HIV-1 infection in one simulationregion upon exposure to HIV-1 from an acutely infected partner 373.5 Probability of HIV-1 infection as dictated by CD4+ T cellcount and damaged tissue amount . . . . . . . . . . . . . . . 394.1 The effect of antivirals on lesion duration during HSV-2 in-fection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 The effect of antivirals on lesional tissue damage during HSV-2 infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 The effect of antivirals on viral shedding during HSV-2 infection 454.4 The effect of antivirals on median HSV-2 infection traits . . . 474.5 The effect of antivirals on median risk of HIV-1 contraction . 484.6 Changing risk of HIV-1 over time for different antiviral doses 49viiList of Figures4.7 Probability of HIV-1 infection when considering all vaginaltissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52viiiAcknowledgementsThank-you to Dr. Daniel Coombs for his guidance and support throughoutthe process of this thesis and to Rebeca Cardim-Falca˜o and Cole Zmurchokfor their initial help in formulating the ideas of this research.ixDedicationTo my family and to David who now knows more about herpes virus than Ibelieve he ever wanted to.xChapter 1IntroductionHerpes Simplex Virus-2 (HSV-2) is one of the most common sexually trans-mitted infections (STIs) with an estimated 11.3% of the world’s populationbeing infected as of 2012 [26]. While the overall weakening of the immunesystem caused by some STIs often means that patients with STIs are coin-fected with other viruses, shockingly common is the establishment of HIV-1in HSV-2 infected individuals. An estimated 38-60% of new HIV-1 infectionsin women and 8-49% of new HIV-1 infections in men may be attributed topreviously established HSV-2 infections due to the optimal conditions a her-petic genital lesion presents for the entry and establishment of an invadingHIV-1 virus [14, 20, 48]. As genital tissue becomes infected by HSV-2, largelesions can appear, damaging the natural barrier of the skin and providingentry points for the HIV-1 virus. In addition, the tissue surrounding a her-petic lesion is often rich in CD4+ immune cells, the main target cell for theHIV-1 virus. These conditions can cause a two to three-fold increase in theprobability of HIV-1 virus infection establishment and drastically add to thespread of the HIV-1 epidemic [14, 30]. An increasing amount of research isbeing done to understand the relationship between HSV-2 and HIV-1 in-fections and to find ways to decrease the risk of HSV-2 positive patientsacquiring an HIV-1 infection.1.1 The Biology of a Genital HSV-2 InfectionHSV-2, the virus most commonly responsible for genital herpes, is thoughtto infect approximately 20 million people worldwide every year [26]. Thespread of HSV-2 usually occurs through skin-to-skin genital contact wherethe virus enters the host through abrasions in the genital tract epithelium,infects an epithelial cell, and replicates. Following this initial infection, thevirus spreads to nearby neurons where it establishes latency in the dorsalroots of the neural ganglions [40]. With the nervous system having verylimited immune presence, this reservoir of HSV-2 in the neurons means alife-long infection for the host. A slow drip of virus from these neurons isreleased back into the genital tract where it may spark a new infection and11.2. The Link Between HSV-2 Lesional Events and HIV-1 Contractioncause viral shedding [34].The establishment and development of a herpetic lesion in the epithelialtissue is largely dependent on the immune presence at that site [34]. Despitethe high number of shedding episodes that occur in HSV-2 positive patients,the immune system rapidly responds, clearing minor infection sites in twoto twelve hours [15]. The two types of immune cells thought to be mostimportant in HSV-2 infection control are the CD4+ and CD8+ T cells. Oncethe lesion is resolved, these immune cells also work to prevent re-infection,remaining at previous sites of infection for up to twenty weeks [36].CD8+ T cells, which serve as cytotoxic cells for the immune system, areoften thought of as the main force responsible for the control of HSV-2, bothin neurons and in the infected epithelium [38]. As such, previous mathemat-ical models of HSV-2 have included CD8+ T cells as the only immune cellpresent during HSV-2 infections [11, 34–37]. However, CD4+ T cells havemore recently been shown to be important in the control of HSV. In experi-ments where CD8+ T cell deficient mice were infected with HSV-1, the virusand lesions could still be cleared at genital and neural sites. However, in thealternative situation where CD4+ T cell deficient mice were infected withHSV-1, the infection could not be cleared [23]. CD4+ T cells are thoughtto be the first cells to the site of infection and appear within the first 48hours of HSV-2 infection in the epithelium [41]. Once CD4+ T cells arrive,they release interferon gamma (IFN-γ) and other cytokines causing CD8+T cell recruitment to the infection site [28]. Activated CD8+ T cells thenkill infected cells by delivering perforin and activating apoptotic pathways inthe infected cell [9]. Further research on the role of CD4+ T cells in HSV-2infections is essential to get a full understanding of HSV-2’s pathobiology;similarly, CD4+ T cell dynamics should be included in mathematical modelsof the infection if we wish to obtain a full representation of the system.1.2 The Link Between HSV-2 Lesional Eventsand HIV-1 ContractionDuring an HSV-2 lesion, the presence of HSV-2 virus and infected epithelialcells cause stimulation and proliferation of immune cells which migrate tothe lesion site. This influx of immune cells creates a favourable environmentfor an invading HIV-1 virus. Not only does the HIV-1 virus have a greaterprobability of successfully entering the body through the damaged tissue atthe lesion site, but the immune response creates an environment dense inCD4+ T cells, the target cell of the HIV-1 virus [14, 43]. These CD4+ T21.3. Effects of HSV-2 Antiviral Drugs on HIV-1 Preventioncells are also often enriched with chemokine receptor type 5 (CCR5), a co-receptor which HIV-1 commonly uses to enter its target cell [48]. This allowsfor easier and more frequent infection. The effect of HSV-2 infection onHIV-1 acquisition probability has also been shown through ex-vivo studies.When cervical tissue cultures infected with HSV-2 were exposed to HIV-1virions, the virions bound more frequently to sites containing HSV-2 infectedepithelial cells than to sites of healthy cells, directly showing that areas ofHSV-2 infection are preferential sites for HIV-1 infection establishment [20].HSV-2 coinfection with HIV-1 can also lead to higher transmission ofHIV-1, with genital ulcers or microlesions shedding both HIV-1 and HSV-2virus in coinfection scenarios [33]. With 80% of HSV-2 viral shedding eventsnot coinciding with visible lesions, but instead to only small microlesions,this creates a potentially dangerous transmission scenario for both viruses[15]. Without proper knowledge about the state of their infection, infectedindividuals may remain asymptomatic and unknowingly pass on the HSV-2or HIV-1 virus to sexual partners. Reciprocally, an asymptomatic HSV-2positive individual may unknowingly be at an increased risk to invadingviruses, including HIV-1, as they enter through the herpetic microlesionsthat are present. The synergy of these two viruses makes the scenario ofcoinfection dangerous and underscores the importance of preventing the es-tablishment of the coinfection. An important avenue being explored for thecontrol of these infections is the use of antiviral drugs.1.3 Effects of HSV-2 Antiviral Drugs on HIV-1PreventionWhile no drug or vaccine has been developed that is capable of completelyclearing or preventing HSV-2 infection, antivirals designed to decrease in-fection severity and outbreak frequency have long been available. Acyclovirand other related compounds work by inhibiting HSV-specific DNA poly-merases [1]. By preventing HSV-2 replication, fewer virions are present toinfect epithelial cells and initiate the development of herpetic lesions. Acy-clovir has been shown to reduce the occurrence of genital lesions by 47-75%and the rate of viral shedding by 80% [8, 17].These reductions in HSV-2 infection severity have promising implicationsfor decreasing the rate of HIV-1 contraction. With fewer herpetic lesions,not only should there be less damaged tissue for the virus to enter through,but also fewer immune cells for the HIV-1 virus to infect. Despite thistheory, studies on HIV-1 contraction risks present conflicting results. In31.4. Goalsclinical studies observing HSV-2 positive individuals, the incidence of HIV-1 infection remained the same regardless of whether participants were orwere not receiving acyclovir treatment [8, 18, 46]. While some have arguedthat antiviral doses may simply have not been high enough to observe areduction in HIV-1 infection rates, the reason for this discrepancy is largelyunknown [18]. Determining whether these antivirals can in fact decreaseHIV-1 infection rates at a higher dosage would have important implicationsfor the control of HIV-1 spread. The use of mathematical models can bringa better understanding of HSV-2 infection dynamics in patients receivingantivirals, and may help to determine correct dosage amounts.1.4 GoalsWhile a considerable amount of mathematical modelling has been done tostudy HIV-1 or HSV-2 infections as they occur alone, [10, 11, 29, 31, 34–37, 39], none have analyzed the establishment of HIV-1 coinfection in indi-viduals with chronic HSV-2 from a mechanistic and immunological perspec-tive. Further, mathematical models have yet to be utilized to understandhow HSV-2 antivirals may decrease the risk of HIV-1 contraction. Using aspatial stochastic model to describe the dynamics of a genital herpetic le-sion caused by HSV-2, I quantify the risk of HIV-1 acquisition based on anindividual’s current state of HSV-2 infection. Further, I analyze the effectHSV-2 antiviral drugs may play in controlling HSV-2 outbreaks and inhibit-ing the establishment of HIV-1 infection, predicting dosage amounts neededfor significant reductions in HIV-1 infection probability. This informationhas the potential to help both clinicians and patients understand when risksof HIV-1 acquisition are at their highest, and help guide doctors in choos-ing appropriate doses of HSV-2 antivirals to protect their patients againstcontracting HIV-1.4Chapter 2A Basic Model of HSV-2Dynamics2.1 Designing a Model of HSV-2 Infection in theGenital MucosaTo address questions related to viral dynamics at the site of herpetic lesions,I began by creating a mathematical model describing the basic dynamicsoccurring in the genital mucosa of an individual with a chronic HSV-2 infec-tion. These dynamics, written as a system of ordinary differential equations(ODEs), are largely based on previous mathematical models of HSV-2 in-fections [11, 34–37], and are described in equations 2.1-2.5, tracking thenumber of healthy epithelial cells (H), infected epithelial cells (I), CD8+ Tcells (E), CD4+ T cells (T ), and HSV-2 virus (V ) within the model.dHdt= g(H0 −H)− βHV (2.1)dIdt= βHV − fIE − aI (2.2)dEdt= X1E − δE (2.3)dVdt= φ+ pI − βHV − cV (2.4)dTdt= λ+X2T − dT (2.5)The model is intended to describe a 2 cm × 2 cm patch of epithelial cellswithin the genital epidermis of a chronically HSV-2 infected person. Withherpetic lesions rarely reaching diameters greater than 6 mm, the 4 cm2surface area is meant to be large enough to contain any herpetic lesion thatmay occur at this site [34]. The simulation region has an assumed depth of74 µm, representing the average thickness of infectible epithelial tissue asmeasured by previous histological studies [34].52.1. Designing a Model of HSV-2 Infection in the Genital MucosaH0 represents the number of healthy epithelial cells present at the modelsite in the absence of infection. Taking the diameter of a healthy epithelialcell to be 17 µm, and assuming a cuboidal cell shape, gives an H0 of approx-imately 6 × 106 cells inside the model region [34]. The number of healthyepithelial cells either grows or shrinks proportional to how far H has strayedfrom H0 at a rate constant g, capturing tissue repair mechanisms.HSV-2 infections are initiated by the release of virus from neurons in-nervating the genital mucosa, which occurs at a constant rate φ. Healthyepithelial cells become infected by HSV-2 virus following the law of massaction at a rate proportional to β, and new virus is produced by these in-fected cells at a per-capita rate p. Free HSV-2 virus decays at a per-capitarate c. The presence of infected epithelial cells stimulates the proliferationof CD8+ T cells and CD4+ T cells in the genital mucosa at per-capita ratesX1 and X2 respectively. X1 and X2 are assumed to be dependent on thestate of the infection. Throughout the modelling process, X1 and X2 willtake varying forms as described in subsection 2.1.2. Modelled CD8+ T cellsare taken to be HSV-2 specific and interact and kill infected epithelial cellsat a rate proportional to f , again following the law of mass action. Infectedcells also die at a per-capita rate a as they succumb to the infection. CD8+T cells leave the system at a per-capita rate δ, chosen to be low, representingthe long period these cells remain at, and protect, previous sites of infection[23, 48]. In the infection-free state, a population of CD4+ T cells remain atthe site, existing at an equilibrium number of λ/d. CD4+ T cells have notbeen included in previous models of HSV-2 infection [11, 34–37]. I includetheir dynamics due to their importance in the establishment of HIV-1 infec-tions, which I analyze later, and in order to get a better representation ofthe full dynamics occurring at herpes lesion sites.While this system of ODEs serves as a useful framework to write downthe dynamics that are occurring in a herpetic lesion, their constant rates donot provide a good representation of what is occurring biologically duringa lesion event. Solutions to a system of ODEs report smooth, average be-haviours over time; however, the dynamics of a herpes infection show littleresemblance to an average [34]. Instead, dynamics show fast, apparentlyrandom spikes of infection, followed often by equally fast clearance [34]. Assuch, I transferred the dynamics into a better suited stochastic framework,designing a chemical master equation to describe the system and the proba-bility of different states. Using this chemical master equation, I implementedthe Gillespie algorithm to simulate the progression of the system throughtime.In the traditional Gillespie algorithm, the system progresses through62.1. Designing a Model of HSV-2 Infection in the Genital Mucosasmall time intervals, randomly chosen, during which one reaction is allowedto occur. Both the size of the time interval and reaction choice are madedependent on the probability of different events occurring within the system[12]. While previous mathematical models of herpes infections have beenanalyzed in a stochastic framework, none to my knowledge have used the trueGillespie algorithm, instead progressing the system through small, constanttime steps and allowing reactions to occur with probabilities taken fromknown parameter distributions [11, 34–37]. While the Gillespie algorithmcan be computationally expensive if reaction rates are fast, it is exact andbetter describes the behaviour of a system.2.1.1 Adding Spatial Resolution to the Model SimulationRegionWhile knowing the overall dynamics occurring within the simulation regionis useful, it would be helpful to have a clear idea about how a single lesionspatially develops. To gain a more specific representation of where reactionevents take place, I added a spatial component to the Gillespie algorithm,placing the modelled region on a grid. The surface is divided into n × nequally sized sites, with each spanning the depth of the simulation region.This design allows the dynamics within each grid site to be tracked as themodel runs. While the system still progresses through the Gillespie algo-rithm in much the same fashion, I now assume reactions only occur betweencells and virions that exist within the same grid site. Since immune cells ac-tively move around the epithelium in search of infection and viruses diffusethrough their environment, I allow HSV-2 virus, CD4+ T cells, and CD8+T cells to diffuse horizontally through the tissue into neighbouring grid sites.I define the rate of diffusion for a specific diffusing body N ∈ {E, T, V } atgrid site (i, j), where i, j ∈ {1, 2, ..., n}, asDNi,j = ωNNi,j−1 +Ni−1,j − 4Ni,j +Ni+1,j +Ni,j+1h2. (2.6)Here, ωN represents the diffusion coefficient specific to the diffusing bodyand h2 represents the horizontal cross sectional area of each grid site.At the boundaries of the simulation region, virus, CD8+ T cells, andCD4+ T cells can diffuse out of the system; however, as surrounding tissuealso has immune presence, I allow CD8+ and CD4+ T cells to diffuse intothe system from the borders. By allowing Eave and Tave to represent theaverage CD8+ T cell concentrations expected to exist in chronic HSV-2infected tissue per simulation region, Eave/n2 and Tave/n2 represent the72.1. Designing a Model of HSV-2 Infection in the Genital Mucosaaverage CD8+ and CD4+ T cell concentrations per grid site on an n × ngrid. Diffusion of CD8+ T cells and CD4+ T cells therefore occurs at ratesωEEave/n2 and ωTTave/n2 respectively. Virus is not allowed to enter fromthe boundary as I assume no other lesions are in close proximity to themodelled patch. A summary of this new model system is described in figure2.1 while a description of the parameter values appears in Section 2.2.Figure 2.1: Description of the model. The ODE system presented in equa-tions 2.1-2.5 is put into a spatial stochastic framework to capture the randomevents that occur during a herpes infection. The system progresses throughsmall time steps (∆t) following an exponential distribution with rate Rtotequal to the sum of vector ~R which contains all reaction rates in the sys-tem. During each time step, the system updates with one event (∆S) beingallowed to occur, chosen from a multinomial distribution where the proba-bility of each event equals its rate of occurrence divided by Rtot. To gainfurther resolution on where reactions are occurring, the system is dividedinto n×n equally sized grid sites so that I can track the location of reactionsand the dynamics occurring at every grid site (i, j), where i, j ∈ {1, 2, ..., n}.In addition to the dynamics described in the system of ODEs, HSV-2 virus,CD4+ T cells and CD8+ T cells are allowed to diffuse into neighbouringsites while infected and healthy epithelial cells are assumed to remain sta-tionary. With n2 grid sites per simulation region, parameters dependent onspace were either divided or multiplied by n2 to allow for the correct unitconversion.82.1. Designing a Model of HSV-2 Infection in the Genital Mucosa2.1.2 Defining Infection-Dependant Rates of Immune CellProliferationModel I: Immune Stimulation by Infected CellsIn determining the best representation of immune stimulation, I began withthe simplest model, making X1 and X2, the terms describing CD8+ andCD4+ T cell stimulation respectively, saturating functions dependent onthe number of infected epithelial cells.X1 =Ii,jIi,j + r1/n2θ1 (2.7)X2 =Ii,jIi,j + r2/n2θ2 (2.8)Here, θ1,2 is the maximum per-capita proliferation rate of immune cells,and r1,2 is the number of infected cells needed to achieve half this prolif-eration rate within the entire simulation region. 1, 2 correspond to CD8+stimulation and CD4+ stimulation respectively. To scale parameters to bespecific to the size of a grid site, r1 and r2 are divided by n2.While this representation of immune stimulation is functional in repre-senting infection-dependent immune cell proliferation, it remains a simplifi-cation of the true biological system. Cytokines, small signalling moleculesproduced in response to infection, are the true components regulating andstimulating immune cell proliferation [28, 41]. As such, I develop a second,more complex model of the HSV-2 infection system, including the presenceof cytokines and their effects on immune cell proliferation.Model II: Including The Effects of Cytokines on ImmuneStimulationTo capture the effects of cytokines on the immune response to HSV-2 infec-tion within the genital mucosa, I define cytokine dynamics (C) at site (i, j)as follows:∆Ci,j = [bIi,j −mCi,j +DCi,j ]∆t, (2.9)whereDCi,j = ωCCi,j−1 + Ci−1,j − 4Ci,j + Ci+1,j + Ci,j+1h2. (2.10)92.2. Parameters of the ModelCytokines are assumed to be produced at a rate b, dependent on thenumber of infected epithelial cells in the same grid site, and decay from thesystem at a per-capita ratem. I also assume they may diffuse into neighbour-ing grid sites with diffusion coefficient ωC . These dynamics are transferredinto the chemical master equation describing the system as a whole andchanges in cytokine numbers occur through the Gillespie algorithm.Rates of immune cell proliferation at site (i, j) are then made to bedependent on the number of cytokines at that site.X1 =Ci,jCi,j + r3/n2θ1, (2.11)X2 =Ci,jCi,j + r4/n2θ2. (2.12)Again, I choose to use a saturating function to describe the rate at whichcytokines stimulate immune cell production, with r3 and r4 now representingthe number of cytokines needed to cause half the maximum proliferation rateof CD8+ and CD4+ T cells, respectively, in the entire simulation region.By including the dynamics of cytokines, in particular their ability to diffuseinto neighbouring grid sites, the immune response to infection may showvery different results to those predicted through the use of Model I.2.2 Parameters of the ModelDue to recent interest in mathematically analyzing HSV-2 infections in thegenital mucosa, many of the dynamics describing HSV-2 infection are wellparameterized [11, 34–37]. However, my inclusion of CD4+ T cell dynamicsat the lesion site has not previously been examined from a mathematicalstandpoint. I therefore determined new values for the parameters governingCD4+ T cell behaviour in the genital epithelium. One challenge surroundingthis task was the limitation in data reporting CD4+ T cell numbers in thegenital mucosa; however, as CD4+ T cells are the main target of the HIV-1virus, a representation of their dynamics is essential if we want to addressquestions related to HIV-1 infection. Fortunately, recent studies on theimmune presence in the genital mucosa during different stages of herpeticlesion development reported both CD4+ and CD8+ T cell numbers at thelesion sites [24, 48]. Using this data, I determined values for the undefinedparameters.A healthy individual without HSV-2 infection has approximately 68CD4+ T cells per mm2 circulating around the epidermal layer of the gen-ital epithelium [48]. Scaling this number to the 4 cm2 of my model region102.2. Parameters of the Modelindicates there should be 27200 CD4+ cells in the model when the patientis infection-free, directly corresponding to the fraction λ/d. Assuming thedeath rate, d, of CD4+ T cells is similar to that of CD8+ T cells, I setd = 0.07/day and λ = 1900/region-day to achieve the correct infection-freeequilibrium value. With these two parameter values chosen, the parame-ters found in the expression for X2, the rate of CD4+ T cell response toinfection, remained to define. One common way of fitting mathematicalmodels to data is to adjust the parameters of the model until the ODE so-lution curve matches the experimental data curve. However, this process isnot as straightforward in the case of HSV-2 lesional events. As previouslystated, the curves predicted by the ODE model are smooth averages of thesystem and have little resemblance to the stochastic nature of HSV-2 in-fection dynamics. Furthermore, lesion events within a patch of epitheliumare relatively rare. It therefore makes little sense to fit a curve representingthe average to data representing rare events. Instead, I used the closelylinked dynamics of CD4+ and CD8+ T cells to determine parameters val-ues governing X2. A striking feature appearing in the data is the relativelyunchanging ratio between CD4+ and CD8+ T cells [24, 48]. During the pro-gression of an HSV-2 genital lesion, the CD4+ to CD8+ T cell ratio remainsfairly constant, ranging from approximately 0.6 to 2.0 with a mean valueof 1.06 in healthy tissue and 1.24 in HSV-infected tissue [24, 48]. As pa-rameters describing CD8+ T cell dynamics are already well known, I simplyvaried the parameters describing X2 until I reached a state where the CD4+to CD8+ T cell ratio consistently fell within an appropriate range. In run-ning fifty, one-year simulations of the full model including cytokines, withparameters of r3 = 42/day, r4 = 38/day, θ1 = 1.70/day, and θ2 = 1.40/day,the CD4+ to CD8+ ratio had an average of 1.4, ranging from 0.7-3.1. Withthese ratios similar to those observed experimentally, these values were as-sumed acceptable for all remaining simulations.Due to the division of the simulation region into specific grid sites duringthe spatial analysis of the model, those parameters dependent on spaceneeded to be scaled accordingly. With n2 grid sites per simulation region,parameters dependent on space were either divided or multiplied by n2 toallow for the correct unit conversion. All parameter values of the model arerecorded in table Simulations of HSV-2 InfectionTable 2.1: List of parameter values used in the model. Most parametervalues were chosen to fall within the range of those found in the literature.Parameters without previously recorded values were estimated so that themodel showed the correct, expected dynamics. Parameter time units are interms of days, while those dependent on space are scaled to be in terms ofone 2 cm × 2 cm × 74 µm simulation region.Units Value Chosen Range in Literature Citationg /day 0.22 0.22 [35]β region/day 1.0× 10−7 2.7× 10−9 − 6.6× 10−7 [11, 34–37]a /day 1.20 1.20 − 1.33 [34, 35]f region/day 0.010 0.001 − 0.200 [34, 35]r1 /region 42 5 − 200 [11, 34–37]r2 /region 38 − −r3 /region 42 − −r4 /region 38 − −θ1 /day 1.70 0.98 − 7.20 [11, 34–37]θ2 /day 1.40 − −δ /day 0.05 6.64× 10−4 − 8.30× 10−2 [11, 34–37]p /day 7.05× 103 103 − 105 [11, 34–37]c /day 8.8 6.2 − 96.0 [11, 34–37, 42]φ /region-day 50 1 − 2000 [11, 34–37]λ /region-day 1900 − −d /day 0.07 − −ωV cm2/day 7.2× 10−4 2.8× 10−6 − 3.1× 10−3 [5, 25, 27]ωE , ωT cm2/day 7.2× 10−4 1.3× 10−4 − 1.4× 10−3 [2, 4]ωC cm2/day 2.45× 10−2 1.30× 10−2 − 8.64× 10−1 [16, 21]m /day 6.2 2.8 − 6.6 [21, 45]b /day 24.8 − −n − 5, 15 − −Eave /region 60,000 60, 000 [34]Tave /region 50,000 50, 000 [48]H0 /region 6× 106 6× 106 [34]With the model now fully developed, I move on to determine the effects ofdifferent spatial resolutions of the model and find which resolution presentsthe clearest representation of lesion development in the genital mucosa.2.3 Simulations of HSV-2 InfectionA simulation begins with the modelled epithelial patch being lesion andvirus-free, but still displaying the characteristics of chronic HSV-2 infection.The initial number of healthy epithelial cells is set to equal H0 and theHSV-2 and infected cell counts are both zero. As immune cells have a pro-longed presence at the sites of previous HSV-2 infection, the initial numbersof CD4+ and CD8+ T cells are not set to zero, but to numbers indicative122.3. Simulations of HSV-2 Infectionof recently healed tissue with E(0) = 60, 000 cells and T (0) = 50, 000 cells[34]. The 2 cm × 2 cm simulation region is assumed to be centered aroundan HSV-2 infected nerve, with virus being released into the central grid siteat rate φ. To explore the effects of different definitions for immune stimu-lation rates, I first allowed proliferation rates to be dependent on infectedcell counts, as described in Model I for immune cell proliferation, and laterexamined the effects of cytokine-dependent immune cell proliferation as de-scribed in Model II.2.3.1 Results of Model I for Immune Cell ProliferationModel simulations with X1 and X2 as described in equations 2.7 and 2.8, andwith spatial resolutions of 1, 25, and 225 grid sites per simulation region, arereported in figures 2.2-2.6. Figures 2.2, 2.3, and 2.5 present HSV-2, infectedcell, and immune cell counts summed across the entire simulation regionover two months of infection. Figures 2.4 and 2.6 present spatially-specificdata for model regions divided into 25 and 225 grid squares, showing HSV-2counts, CD8+ T cell counts, and amounts of damaged tissue at each gridsite during key times of lesion development. Here, tissue damage at site(i, j) is reported as the fraction of epithelial cells missing from a grid siteand contributing to the lesion. This value is defined asLi,j =H0/n2 −Hi,j − Ii,jH0/n2, (2.13)with H0/n2 being the expected number of healthy epithelial cells at a gridsite in the absence of infection. This formula can also be modified to findthe fraction of tissue damage within the entire simulation region:Ltot =H0 −∑∀i∑∀j Hi,j −∑∀i∑∀j Ii,jH0. (2.14)Regardless of the spatial resolution, each simulation captured thestochastic dynamics that appear during herpetic lesions. Viral peaks in thesimulations varied in size, corresponding to different severities of outbreaks,and matching with those recorded in the literature [34, 48].On examination of the spatially resolved information collected duringthe simulations, one can note a large difference in the visualization of lesiondevelopment when the simulation region is divided into 25 versus 225 gridsquares. While simulations with a 25 grid square resolution have difficultydisplaying the connected, round shape we expect from a herpetic lesion,132.3. Simulations of HSV-2 Infectionthese characteristics are displayed well at a 225 grid square resolution. Thisobservation indicates that a high number of grid squares allows for a moreaccurate picture of lesion dynamics.While the higher spatial resolution appears to produce a better repre-sentation of lesion development, further examination of these simulations,with X1 and X2 described as in equations 2.7 and 2.8, reveals a new issue.We see that lesions develop as an expanding ring, eventually surpassing theboundaries of the simulation region. This effect should not occur as the sizeof the simulation region was chosen to contain the largest of biologicallyrealistic herpetic lesions. These dynamics can be explained by the nature ofhow the immune response has been defined. With virus dripping into themodel region at the center grid site, infection most often begins here. Asmore epithelial cells become infected, CD8+ T cells are recruited to the siteand begin attacking and clearing the infection from the center. However,virus is concurrently spreading to the surrounding grid sites. Since immunecell proliferation is only stimulated at sites where infected cells exist, im-mune presence at these sites is low when the virus first enters, allowingthe virus to thrive. Immune cell response eventually catches up but, asthe simulations show, can only chase, rather than prevent, infection spread.This phenomenon leads to the expanding ring we observe. To truly stopthe infection, it would appear that increased immune cell presence needs tobe stimulated in surrounding, uninfected sites to prevent the infection fromspreading. This idea motivates a new definition for the terms X1 and X2 inthe model and how immune cell production should be stimulated.142.3. Simulations of HSV-2 InfectionFigure 2.2: HSV-2 infection dynamics with no spatial resolution. Graphsdepict counts of HSV-2 virus, infected epithelial cells, and CD4+ and CD8+immune cells in the simulation region during one, sixty day, model run. Here,immune cell stimulation terms, X1 and X2, are defined as in equations 2.7and 2.8. All cells and virions within the system are assumed to be wellmixed, with no spatial divisions. Infection dynamics show rapid infectiondevelopment and extinction, with viral and immune cell loads correspondingwith those found in the literature [34, 48]. Parameter values are the sameas those reported in table 2.1, except f = .003 region/day, δ = 0.07/day,λ = 2700/region-day, θ1 = 2.84/day and θ2 = 2.5/day.152.3. Simulations of HSV-2 InfectionFigure 2.3: HSV-2 infection dynamics within a model region divided intoa 5 × 5 grid. Immune cell stimulation terms, X1 and X2, are defined as inequations 2.7 and 2.8. Upper graphs show the combined counts of HSV-2 virus, infected epithelial cells, and CD4+ and CD8+ immune cells inthe entire simulation region during one model run. Lower graphs displayspatially-specific counts of HSV-2 virus and the percent of tissue damagein each grid site taken at the time of the simulation’s highest viral load.While HSV-2 viral loads reach values over 104 virions in some grid sites,tissue damage due to the developing lesion remains low, ranging from 0-6%in each grid site.162.3. Simulations of HSV-2 InfectionFigure 2.4: Stills taken from an HSV-2 infection simulation where the modelregion was divided into a 5×5 grid and X1 and X2 are defined as in equations2.7 and 2.8. Stills show spatially resolved log10 counts of HSV-2 virus (left),percent of damaged tissue (centre), and log10 counts of CD8+ T cells (right)at days 1.0, 4.0 and 7.0 of the simulation. Note that scales change overpanels. Infection begins at the center grid site, and spreads, leading to anincrease in damaged tissue and the recruitment of CD8+ T cells. Whilecapturing the spread of infection within the patch, the spatially resolvedimages present a poor representation of a rounded, connected lesion weexpect in a herpes infection.172.3. Simulations of HSV-2 InfectionFigure 2.5: HSV-2 infection dynamics within a model region divided intoa 15 × 15 grid. Immune cell stimulation terms, X1 and X2, are definedas in equations 2.7 and 2.8. Upper graphs show the combined counts ofHSV-2 virus, infected epithelial cells, and CD4+ and CD8+ immune cellsin the entire simulation region during one model run. Lower graphs displayspatially-specific counts of HSV-2 virus and the percent of tissue damagein each grid site taken at the time of the simulation’s highest viral load.With the simulation being divided into a 15 × 15 grid, I obtain a clearvisualization of a round, developing lesion; however due to the design of themodel’s immune response, infection develops as a spreading ring.182.3. Simulations of HSV-2 InfectionFigure 2.6: Stills taken from an HSV-2 infection simulation where the modelregion was divided into a 15 × 15 grid and X1 and X2 are defined as inequations 2.7 and 2.8. Stills show spatially resolved log10 counts of HSV-2virus (left), percent of damaged tissue (centre), and log10 counts of CD8+T cells (right) for the first seven days of the simulation. Note that scaleschange over panels. Infection begins at the center and grows quickly withviral counts reaching 104 and damaged tissue accounting for up to 30% ofsome grid sites. CD8+ T cells are recruited to the infection site; however,rather than being cleared, the infection develops into an expanding ring,eventually surpassing the boundaries of the simulation region.192.3. Simulations of HSV-2 InfectionFigure 2.6 cont’d2.3.2 Results of Model II for Immune Cell ProliferationWhile capturing some of the dynamics seen in HSV-2 infections, executionsof Model I failed to produce biologically accurate growth of a lesion. Byimplementing Model II as described in equations 2.9-2.12, I now account forthe effects of cytokines on the system and their role in stimulating immunecell proliferation. Because cytokines are allowed to diffuse, immune cellproliferation can be stimulated away from the site of infection, potentiallyallowing immune cell response to get ahead of, and stop, the spreading ringof infection.202.3. Simulations of HSV-2 InfectionTo further the effects of cytokines on the model, I also took into ac-count cytokines’ ability to create a chemical gradient within tissue whichexisting immune cells migrate towards [28]. To capture this idea, I madethe direction of immune cell diffusion dependent on the number of cytokinespresent in neighbouring grid squares. As the simulation ran, if at a giventime step immune cell diffusion was the event chosen to occur, the programfirst checked the four neighbouring grid squares for cytokine presence. Ifnone were present, then diffusion into any of the four neighbouring squaresoccurred with equal probability. However, if cytokines were present, theprobability of diffusion into each of the neighbouring square was given byProbi,j =Ci,jCtot, (2.15)with (i, j) taking on the indices of the four surrounding grid sites and Ctotbeing the total quantity of cytokines in these four sites. The full pythoncode used to execute this model can be found in Appendix A.Depictions of HSV-2 infection dynamics with the inclusion of cytokinesare shown in figures 2.7-2.9. Again, the magnitude of viral and tissue damagepeaks vary throughout a simulation, capturing the differences in lesion sizesand severities seen in patients. Stills of the simulation region throughoutlesion development are shown in figure 2.8. In the absence of large lesions,runs often show small amounts of virus existing in the system, matching theasymptomatic viral shedding that is well described in chronically patients[34]. Occasionally, more severe infections become established in the tissue,leading to a greater viral peak and subsequently a high amount of tissue dam-age. The beginning of a viral peak appears to always correspond to timesof low immune cell presence, suggesting a potential threshold in conditionsneeded for infection to break through. With the inclusion of cytokines in themodel, virus and developing lesions are now cleared within the model regionrather than surpassing its borders. These simulations present an excellentrepresentation of the development of HSV-2 lesions in the genital tract, al-lowing me to move on and examine how the state of these developing lesionsmay dictate the probability of contracting HIV-1 upon exposure.212.3. Simulations of HSV-2 InfectionFigure 2.7: Summed HSV-2 infection dynamics for a single 120 day simula-tion within a model region divided into a 15×15 grid and with the inclusionof cytokines. Infection dynamics remain consistent with those in the liter-ature [34, 48]. Graph a. shows how peaks in HSV-2 directly correspondwith peaks in infected cells. Slightly delayed are the corresponding peaks ofcytokines and immune cells shown in graphs b. and c. respectively. Whilethe infection is cleared rapidly, tissue heals at a slower rate as depicted ingraph d..222.3. Simulations of HSV-2 InfectionFigure 2.8: Stills of HSV-2 lesion development taken from a model simulationincluding the effects of cytokines. Log10 HSV-2 counts (left), percents oftissue damage due to lesion development (centre), and log10 CD8+ immunecell counts (right) are shown across various days. Note that scales changeacross panels. With the inclusion of cytokines, simulations no longer presentlesions developing as an expanding ring, but rather as a connected, growingpatch. The lesion grows to a biologically realistic maximum size within thesimulation area and then slowly heals. CD8+ immune cell presence becomesconcentrated in areas of high infection, and remains at high, protective,concentrations even once virus has been cleared from the system.232.3. Simulations of HSV-2 InfectionFigure 2.8 cont’d242.3. Simulations of HSV-2 InfectionFigure 2.9: The results of ten simulations of chronic HSV-2 infection in a4 cm2 patch of the genital mucosa. The simulation region was divided intoa 15 × 15 square grid and terms X1 and X2 of the model are defined as inequations 2.11 and 2.12. Graph a. shows the total quantity of HSV-2 overthe whole simulation region and graph b. shows the total percent of tissuedamage in the simulation region at all time points throughout year longsimulations calculated using equation 2.14. Large variation exists betweensimulations, with each showing varying sizes of viral peaks and amount oftissue damage over time. At times of severe lesions, tissue damage canaccount for up to 50% of the 4 cm2 patch of tissue, with HSV-2 countsreaching values close to 108 per simulation region.25Chapter 3The Establishment of HSV-2- HIV-1 Coinfection3.1 A Description of HIV-1 Infection DynamicsWhen HIV enters its host, it must infect a target cell in order for the infectionto become established. These target cells are most commonly CD4+ T cells;however other immune cells expressing the CD4+ cell surface receptor suchas tissue macrophages, and dendritic cells may also serve as a target. Once acell is infected, it migrates to nearby draining lymph nodes where the virusgains access to a reservoir of immune cells. The virus replicates and infectionpropagates as infected cells spread throughout the blood stream. Previousexperiments quantifying the immune presence in herpetic genital lesionsshow that the number of CD4+ cells greatly increases during lesional events[24, 48]. This result helps to explain why these patients show increased risksof HIV-1 contraction [14, 20]. By using the spatial stochastic model of thegenital mucosa, I examined the initial establishment of HIV-1 infection inthe tissue of patients with chronic HSV-2.In order to examine how HIV-1 infection becomes established in an HSV-2 infected individual, I introduced HIV-1 virus into the modelled genitalepithelial patch and tracked infection development. While the stochasticdynamics of the herpes infection remain as previously defined, I define newequations to describe the development of an HIV-1 infection as follows:∆Ti,j =[ λn2+X2Ti,j − kn2Ti,jPi,j − dTi,j]∆t, (3.1)∆Pi,j = [ψT2i,j − kn2Pi,jTi,j − `Pi,j +DPi,j ]∆t, (3.2)∆T1i,j = [kn2Ti,jPi,j − ηT1i,j − g1T1i,j ]∆t, (3.3)∆T2i,j = [ηT1i,j − g2T2i,j ]∆t. (3.4)CD4+ T cell dynamics now include loss due to infection by HIV-1 (P ).This is set to occur through the properties of mass action at a rate propor-263.1. A Description of HIV-1 Infection Dynamicstional to kn2 at each grid site, with n2 being the number of grid squares inthe simulation region. Once a CD4+ cell becomes infected, it moves intothe T1 class, representing latently infected cells known to be in the eclipsephase. These cells do not produce HIV-1 virus, but mature into activelyinfected cells (T2) at a per-capita rate η. These actively infected CD4+ cellsproduce HIV-1 virus at a rate ψ and are cleared from the system at a rateg2. T1 cells may be cleared from the system at a per-capita rate g1; however,I assume that this rate is 0. The model includes no specific immune responseagainst HIV-1-infected CD4+ cells as I assume no immunity against HIV-1infection yet exists. These dynamics are again put into the spatial Gillespieframework with diffusion of HIV-1 occurring at a per-capita rate ωP betweenneighbouring grid sites. The diffusion events for HIV-1 at grid site (i, j) aresummarized by the term DPi,j as described in equation 2.6 with N = P .3.1.1 Defining HIV-1 EntryHSV-2 infections are thought to increase HIV-1 infectivity in two ways:by providing an increased number of HIV-1 target cells, and causing tissuedamage at the lesion site which serves as viral entry points. I therefore allowboth of these factors to play a role in the probability of HIV-1 infectionestablishment.The act of an HIV-1 virion penetrating the epithelium is a multi-stepprocess. The virus must come in contact with the body, cross the genitalmucus, and invade through the cells of the genital tract. The ability ofvirus to penetrate through the mucosa is variable and specific to the tis-sue exposed. The majority of human outer skin is composed of keratinizedsquamous epithelial cells which serve as a tough barrier for pathogenic in-vasion; however large portions of the genitals are composed of more delicatetissue. Regions that have direct contact with a partner’s body fluid and arecomposed of a single layer or few layers of columnar epithelial cells, like therectum, endocervix in women, and urethra in men, are the most susceptibleto HIV-1 invasion [7, 19]. Areas consisting of squamous, non-keratinized, orpoorly keratinized cells, like the inner foreskin in men, and ectocervix andvagina in women, are more protected, but still common other sites of HIV-1invasion [7, 19]. HIV-1 transmission probabilities per sexual act range from1 in 3000 to 1 in 20 depending on the genital tissue exposed, with the rectumbeing the site of highest risk [19, 30].Recent ex-vivo experiments on initial HIV-1 infection in tissue samplestaken from healthy females were used to estimate the number of HIV-1 viri-ons able to penetrate the genital mucosa per sexual act [7]. When female273.1. A Description of HIV-1 Infection Dynamicsgenital tissue was exposed to a semen concentration of 5.0 × 103 HIV-1virions/mL, characteristic of chronically infected semen loads in males notreceiving standard antiretroviral therapy (ART), 18 virions were estimatedto penetrate the epithelium of the vaginal surface per coital act [7]. Assum-ing the vaginal surface area is 88 cm2, this is approximately 0.2 virions/cm2[7]. When exposed to semen viral concentrations of 4.0 × 106 virions/mL,characteristic of acute HIV-1 infection in men not receiving ART, approxi-mately 170 virions/cm2 were predicted to pass through vaginal mucosa [7].Using these values, I estimated the amount of HIV-1 virus that wouldpass through the epithelium and enter the model system per sexual act.To simulate a female with healthy genital epithelial tissue who engages inheterosexual intercourse with an HIV-1 positive partner not receiving ART,1 virion per sexual act enters into the model region if the partner has achronic HIV-1 infection, and 800 virions per sexual act enters the modelregion if the partner has an acute HIV-1 infection due to the 800 timeshigher viral load in the semen of acutely versus chronically infected men. Ithen modified these numbers based on the amount of lesional damage to theepithelium, allowing more virus to enter. In looking at the effects of weak celljunctions, Carias et al. 2013 found that 10 times as many viruses were ableto penetrate the epithelium if weak cell junctions were present compared totissue without weak cell junctions. Assuming the effects of lesion damage aresimilar, I used this estimate as a conservative representation of the increasedamount of virus that enters due to damaged lesional tissue.The amount of HIV-1 virus expected to enter the model region at gridsite (i, j) in an individual exposed to chronic or acute HIV-1 viral loadsrespectively is described as follows:Pchronici,j =1 + 9Li,jn2, (3.5)Pacutei,j =800 + 7200Li,jn2= 800Pchronici,j . (3.6)Li,j measures tissue damage as defined in equation (2.9). Summing overall grid sites, we can obtain the total amount of virus expected to enter thesystem:Pchronic,tot =∑∀i∑∀jPchronici,j , (3.7)Pacute,tot =∑∀i∑∀jPacutei,j . (3.8)283.1. A Description of HIV-1 Infection DynamicsIf no lesion is present in the model region (Li,j = 0 ∀(i, j)), then thepreviously mentioned estimates of 1 or 800 HIV-1 viruses per simulationregion are allowed to enter. These values increase linearly with tissue damageleading to 10 times more virus entering the tissue if it is entirely damaged(Li,j = 1 ∀(i, j)).3.1.2 Parameters for HIV-1Many mathematical models have examined initial HIV-1 infection, leadingto a rich supply of previously determined parameter values. However, fewexplicitly examined the dynamics occurring within the epithelium, the mostcommon site of HIV-1 infection establishment. As HIV-1 and immune cellcounts usually come from blood or plasma samples, the parameters of mostmodels are fit to these numbers [10, 31, 39]. While dynamics occurringwithin the blood and epithelium may be similar, they may not be occur-ring at the same rates. While I was able to choose some parameter valuesbased on those used in previous models, others were estimated based on ourknowledge of HSV-2 infection behaviour in the genital mucosa. This is withspecific reference to the estimates for parameters ψ and `. As only within-blood estimates have been recorded in the literature for these parameters,I chose values that corresponded with those fit for HSV-2 dynamics. Here,ψ matches with the rate of HSV-2 production and ` matches with the valuefor HSV-2 clearance. All HIV-1 parameters used in the model are listed intable 3.1.Table 3.1: List of parameter values used in the model to describe HIV-1infection dynamicsUnits Value Chosen Range in Literature Citationk region/day 1× 10−7 3.7× 10−8 − 7.4× 10−4 [10, 29]ψ /day 7.05× 103 2× 104 [10]` /day 8.8 20 − 23 [10, 29]ωP cm2/day 7.2× 10−4 2.8× 10−6 − 3.1× 10−3 [5, 25, 27]η /day 1 0.7 − 5 [10, 29]g1 /day 0 0 − 0.5 [10, 29]g2 /day 1.2 0.583 − 1 [10, 29]3.1.3 The Dynamics of HIV-1 InfectionTo represent exposure to the HIV-1 virus, I paused the simulation at vari-ous states of lesion development, introduced the calculated amount of HIV-1293.1. A Description of HIV-1 Infection Dynamicsvirus depending on the current lesion state, and then restarted the simula-tion. An example run showing HIV-1 infection establishment upon exposureto an acute load of HIV-1 is shown in figure 3.1. In this scenario, HIV-1virus was introduced during a peak lesion event. From this figure we can seethat the infection establishes quickly, with HIV-1 viral loads approaching106 in the tissue within four days of infection. Figure 3.2 shows the spa-tial dynamics occurring during this infection. Virus is introduced, and whilemost virus quickly dies off, infection becomes prominently established at onesite and quickly develops there. We can note that after less than four daysof HIV-1 infection, the number of target CD4+ cells at the infection sitehave been depleted. While this may imply that target cell abundance maylimit HIV-1 infection expansion, it is important to keep in mind that thismodel does not include the migration of infected cells to the lymph nodeswhere the virus can find another large reservoir of target cells. However, weclearly note the fast establishment and importance of target cell number inthe epithelium during HIV-1 infection.303.1. A Description of HIV-1 Infection DynamicsFigure 3.1: Dynamics of HIV-1 infection establishment in an HSV-2 positivepatient upon exposure to an acute dose of HIV-1 in the semen. Here, thevertical red line indicates the time of HIV-1 introduction into the simulationregion. Plots show the amount of tissue damage due to HSV-2 lesions, theamount of HSV-2, and the number of healthy CD4+ T cells before and afterHIV-1 introduction. Also shown are the changing HIV-1 counts, numberof infected CD4+ cells in the eclipse phase, and number of infected CD4+cells in the active phase once HIV-1 has been introduced. By day 89 of thesimulation, 4 days after the HIV-1 transmission event, HIV-1 counts andthe number of infected CD4+ cells are high enough for the HIV-1 infectionto be considered fully established with little chance of it being cleared fromthe system.313.1. A Description of HIV-1 Infection DynamicsFigure 3.2: Spatial dynamics of HIV-1 introduction for the same simulationregion as shown in Figure 3.1. HIV-1 counts (left), uninfected CD4+ T cellcounts (centre), and infected CD4+ T cell counts (left) are shown acrossthe initial days of infection. Note that scales change across panels. HIV-1enters on day 85.4 of an HSV-2 infection simulation and quickly becomesestablished.323.2. Determining HIV-1 Infection Probability3.2 Determining HIV-1 Infection ProbabilityWhile individual runs of the model give information on how the HIV-1 in-fection first becomes initialized, many repetitions of these simulations needto be performed in order to approximate the probability of HIV-1 infec-tion establishment. One issue with this approach is the great amount oftime required to complete each full simulation. Tracking both the HSV-2and HIV-1 dynamics is computationally expensive and therefore performingmany simulations that record all the dynamics is unrealistic. To minimizecomputational time but still gather the data necessary to calculate HIV-1infection probability, I tracked only the dynamics related to HIV-1 infectionestablishment once HIV-1 virus was introduced into the simulation region.By making this simplification, I assume that the changes in HSV-2 infectiondynamics have a minimal effect on CD4+ T cell count during the small timewindow during which HIV-1 infection establishment occurs.To calculate the probability of HIV-1 infection in patients with differentHSV-2 lesion scenarios, I used the following algorithm:1. Stop an HSV-2 infection simulation at a point where exposure to HIV-1 is assumed to occur.2. Calculate the fraction of tissue damage at each simulation grid siteusing equation 2.13.3. Calculate the total number of HIV-1 viruses expected to enter thesimulation region using equation 3.7 or 3.8, depending on the type ofexposure being examined, and round it to the closest whole virus.4. Randomly distribute these viruses among the grid sites of the simula-tion region following a multinomial distribution with the probabilityof a virus being distributed to site (i, j) given byProbchronici,j =Pchronici,jPchronic,tot(3.9)orProbacutei,j =Pacutei,jPacute,tot. (3.10)depending on the situation being simulated.5. Isolate and simplify the dynamics occurring within each grid site toonly include the main components involved in HIV-1 infection (dy-namics of T , T1, T2, P and C).333.2. Determining HIV-1 Infection Probability6. For each grid square, run the Gillespie algorithm for the simplifieddynamics, stopping the simulation once the infection has gone extinctor propagated enough to imply infection establishment. Here, I assumethat infection is established once the simulation region has at least 8infected cells [31].7. Repeat steps 4. to 6. 10,000 times.8. For every grid square (i, j), define the probability of HIV-1 infectionat that site, Probinfi,j , as the fraction of successful infections thatoccurred at that site over the 10,000 simulations.9. Combine the probabilities of infection at each grid site to achieve anoverall probability of HIV-1 infection for the entire simulation region.To do this, I look at the probability that infection becomes establishedin any of the grid squares of the simulation region using the formulaProbinf,tot = 1−∏∀i∏∀j(1− Probinfi,j ) (3.11)where Probinf,tot is the overall probability of HIV-1 infection over theentire simulation region.Examples of the results obtained from these simulations appear in fig-ures 3.3 and 3.4. In total, I examined 26 different simulated tissue samplesexperiencing various states of lesional damage. In particular, I focused onHIV-1 exposure to tissue that was healthy with no HSV-2 infection, tissueexposed one week before peak tissue damage, tissue exposed during peaktissue damage, tissue exposed during peak CD4+ T cell levels, and tissueexposed one, two, or four weeks after peak tissue damage. These initialconditions are summarized in table 3.2. Each tissue sample was exposed toHIV-1 counts representative of the amount within the semen of chronicallyand acutely infected males, leading to a total of 52 infection simulations.343.2. Determining HIV-1 Infection ProbabilityTable 3.2: Simulation scenarios used to find HIV-1 infection probabilities.Simulations were paused at different times of lesion development, creatinga sample scenario of when tissue may become exposed to HIV-1.Simulation InitialConditions# of SamplesAve CD4+ cellcount/mm2Across SamplesAve % TissueDamageAcross Sampleshealthy 1 40.00 01 week before peak lesion 4 83.75 0.000850peak lesion 5 448.6 18.3peak CD4+ cell count 2 872.5 15.51 week after peak lesion 5 605.0 4.522 weeks after peak lesion 5 392.8 0.9834 weeks after peak lesion 4 200.0 0.113Since the model assumes chronic HSV-2 infection, I had to make mod-ifications in order to simulate healthy tissue. The simulations representingHIV-1 infection in healthy tissue were started with 16000 CD4+ T cells ran-domly distributed throughout the simulation region, corresponding with theequilibrium concentration of CD4+ T cells in healthy epidermis as recordedin the literature [48]. All HSV-2 virus, HSV-2 specific CD8+ T cell, andHSV-2 infected cell dynamics were also removed. Since this system is com-paratively resistant to infection due to having no tissue damage and lowCD4+ T cell numbers, the above algorithm was modified slightly with thesimulation being repeated 50,000 times rather than 10,000 times so that thelow probability of infection in healthy tissue could be captured.353.2. Determining HIV-1 Infection ProbabilityFigure 3.3: Per-coital act probability of HIV-1 infection in one simulationregion upon exposure to HIV-1 from a chronically infected partner. Here,HIV-1 was entered into the simulation region during the peak of an HSV-2lesion. CD4+ T cell counts per mm2 and the amount of tissue damage inthe simulation region at the time of exposure are shown on the left. Thefigure on the right depicts the probability of an HIV-1 infection becomingestablished at each grid site. Combining these probabilities, as describedin equation 3.11, leads to a 0.65% risk of HIV-1 infection establishmentsomewhere within this model region.363.2. Determining HIV-1 Infection ProbabilityFigure 3.4: Per-coital act probability of HIV-1 infection in one simulationregion upon exposure to HIV-1 from an acutely infected partner. Here,HIV-1 was entered into the simulation region two weeks after the peak ofan HSV-2 lesion. CD4+ T cell counts per mm2 and the amount of tissuedamage in the simulation region at the time of exposure are shown on theleft. The figure on the right depicts the probability of an HIV-1 infectionbecoming established at each grid site. Combining these probabilities, asdescribed in equation 3.11, leads to a 73.67% risk of HIV-1 infection estab-lishment somewhere within this model region. This risk seems unrealisticallyhigh indicating an issue with our description of HIV-1 entry and infectiondevelopment in the acute scenario.The probability of HIV-1 infection for each of the 52 infection simula-tions, plotted against the number of CD4+ T cells and amount of tissuedamage at the time of HIV-1 introduction, appears in figure 3.5. By fit-ting linear planes to the data, the probability of HIV-1 infection can beexpressed as a function of CD4+ T cell number and tissue damage. The373.2. Determining HIV-1 Infection Probabilitylinear planes were forced to pass through the points representing HIV-1 in-fection probability in healthy tissue in each scenario in order to fully capturethe background risks of infection. These equations appear at the top of thegraphs in figure 3.5.383.2. Determining HIV-1 Infection ProbabilityFigure 3.5: Probability of HIV-1 infection in a 2 cm × 2 cm region exposedto an acute or chronic dose of HIV-1 in semen. Graphs a. and b. show datacollected during chronic and acute HIV-1 exposure simulations respectively.Here, risk of HIV-1 is plotted against the number of CD4+ T cells per mm2and the percent of herpetic lesional tissue damage taking up the simulationregion at the time of HIV-1 introduction. Through linear regression, I foundplanes of best fit, describing HIV-1 infection risk as a function of tissuedamage and CD4+ T cell count. Planes of best fit were made to passthrough the points describing HIV-1 infection in healthy tissue, shown inblack, in order to capture the background risk of infection. Equations fromthe linear regression of the simulated data appear above their respectivegraphs and show good fit with the R2 values being 0.9703 and 0.9204 forgraph a. and b. respectively.393.2. Determining HIV-1 Infection ProbabilityWhile the infection probabilities predicted by the model for chronic HIV-1 exposure seem plausible, those predicted for acute HIV-1 exposure seemunrealistically high. In healthy tissue, the probability of HIV-1 infection persimulation site was 0.005% per sexual act when exposed to chronic HIV-1viral loads and 2.7%, a 540 fold increase, per sexual act when exposed toacute HIV-1 viral loads. Similar comparisons of acute and chronic HIV-1risks appear in table 3.3. HIV-1 infection probability is known to increasewhen individuals are exposed to acute rather than chronic HIV-1 viral loads,however studies have reported this to only be a 4.3 fold increase [32]. Whilethis value is with respect to infection anywhere within the genital regionand the simulations here only report values with respect to infection risk inone 4 cm2 patch of genital epithelium, the increase seen from the results ofthe model appears to be too high.Table 3.3: Fold increases in per-coital HIV-1 risk when comparing exposureto chronic versus acute HIV-1 viral loads. For all tissue samples used to sim-ulate HIV-1 infection probabilities, risks of HIV-1 were significantly higherwhen tissue was exposed to acute doses of HIV-1 opposed to chronic dosesof HIV-1.Simulation InitialConditionsAve Risk of HIV-1(chronic exposure)Ave Risk of HIV-1(acute exposure)Fold Increasehealthy 0.005% 2.7% 540peak tissue damage/peak CD4+ cell number0.79% 98.94% 125mild to moderatetissue damage0.19% 59.97% 316There are a few reasons as to why the model may have predicted un-realistically high infection probabilities in patients exposed to acute HIV-1viral loads. The first may be due to how I define infection establishment.In the model, the infection is considered established once the HIV-1 infec-tion has managed to infect eight CD4+ cells. Even at this point there is asmall possibility that the infection could be cleared. Other models have hadthresholds of 32 infected CD4+ cells [31]. Having a higher threshold wouldlikely decrease the frequency of HIV-1 infection; however how this wouldaffect the relative increase in infection risk between chronic versus acuteHIV-1 viral load exposures remains unclear. Another explanation may berooted in the number of infectious HIV-1 virions assumed to be in acuteHIV-1 seminal loads. While I assume that all viruses that enter the systemare equally infectious and qualitatively the same in both chronic and acute403.2. Determining HIV-1 Infection Probabilitysemen loads, this may not be the case. If the HIV-1 virions produced duringthe acute phase of infection could be shown to be less infectious than thevirions produced later in infection, this could help explain why the resultshere on infection probabilities from exposure to acute HIV-1 viral loads aretoo high. Further, I may have overestimated the number of HIV-1 virionsable to penetrate the epithelium. While my calculations are based on theestimate that 170 virions/cm2 are able to penetrate the healthy genital mu-cosa when exposure to acute loads of HIV-1 occurs [7], this estimate doesnot consider how mucus on the surface of the tissue may hinder viral entry.Serving as a barrier to the epithelium, mucus is thought to protect againstinfection and may reduce the number of HIV-1 virions an individual is ex-posed to and decrease infection risk. Finally, the clearance rate of HIV-1was chosen to match with the clearance rate of HSV-2 in the genital epithe-lium since no within-epithelium estimates for this parameter are currentlyavailable in the literature. This parameter choice is approximately threetimes smaller than those estimates for this parameter in the blood [10, 29].Allowing for a rate of HIV-1 clearance similar to that seen in the bloodwould likely decrease the risk of HIV-1 acquisition; however it is difficult toknow which parameter value is best given the limitation of data on HIV-1dynamics in the genital epithelium. This point stresses the importance offurther study of HIV-1 initial infection in the epithelium.While the HIV-1 infection risks for exposure to acute HIV-1 viral loadsappear unrealistically high, those for exposure to chronic HIV-1 can helpinform both clinicians and HSV-2 infected individuals of the per-coital risksof HIV-1 contraction and when to take the most caution in sexual activitieswith HIV-1 positive partners. I next use the relationships between HSV-2infection state and HIV-1 risk to determine how doses of HSV-2 antiviralsmay reduce these infection probabilities. I however continue my analysissolely for HIV-1 infection upon exposure to chronic HIV-1 viral loads.41Chapter 4Effects of HSV-2 Antiviralson HIV-1 InfectionProbabilityWhile there is currently no vaccine or cure for HSV-2 infection, many HSV-2patients receive antiviral treatment to help control their lesion outbreaks.The most commonly prescribed drug is acyclovir and variants thereof. Thesedrugs work by inhibiting the virus’s DNA polymerase, preventing replication[1]. While not suppressing the HSV-2 virus entirely, acyclovir and its vari-ants decrease HSV-2 shedding and lesion development. Theoretically, witha decrease in the amount of virus and number of lesions, patients shouldexperience a decreased risk of contracting HIV-1; however, clinical studieshave had difficultly detecting this effect [8, 18, 46]. Studies have shownequal probability of HIV-1 contraction for HSV-2 positive patients indepen-dent of whether they are receiving acyclovir [8, 18, 46]. Here, I work tounderstand why this anomaly occurs and form a mathematical prediction ofHSV-2 antiviral drug doses needed to significantly decrease an individual’srisk of acquiring HIV-1.4.1 Impact of Antivirals on HSV-2 Infection andLesion DevelopmentTo account for the effects of HSV-2 antiviral drugs in the mathematicalmodel designed to describe chronic HSV-2 infection in the genital mucosa,I include a new parameter ζ to describe the effectiveness of anti-virals insuppressing HSV-2 replication. Here, ζ can take values in [0, 1] with 0 rep-resenting an antiviral dosage that has no effect on HSV-2 replication and 1representing complete suppression of replication. I assume this suppressiondecreases both the amount of virus that drips out of the neurons and theamount of virus an infected epithelial cell produces. The modified HSV-2viral dynamics are described as follows:424.1. Impact of Antivirals on HSV-2 Infection and Lesion Development∆Vi,j = [φ(1− ζ) + p(1− ζ)Ii,j − βn2Hi,jVi,j − cVi,j +DVi,j ]δt. (4.1)I analyzed four potential values for ζ (ζ = 0.15, ζ = 0.5, ζ = 0.7,ζ = 0.85) and studied the resulting effects on lesion dynamics.One of the most recent antivirals for HSV-2 treatment, currently in clin-ical trial, is pritelivir. It reduces HSV viral loads by targeting the DNAhelicase of HSV and is considered a stronger drug option for HSV treatment[37]. Development of these new drugs is important as some resistance toprevious HSV-2 antivirals has been observed [22]. Recent clinical trial datahave been released on the effectiveness of pritelivir at different doses [44].Previous analysis of this data predicts the following relationship betweenviral shedding reduction, an empirical equivalent to ζ, and daily pritelivirdose [37]:% reduction = 30.1 ln(daily dose)− 51.5, (4.2)with daily dose being measured in milligrams. From this function, we caninversely infer thatdaily dose = eζ+0.5150.301 . (4.3)Using this formula and rounding the daily dose to the closest 5 mil-ligrams, the ζ values being used correspond to doses of 10, 30, 55, and 80mg a day. These amounts all fall within the range of pritelivir doses givento patients in recent drug trials [44]. While other models have includedthe pharmacokinetics and pharmacodynamics of the drug in the body [37],I assume the patient has a constant dosage within their body for simplic-ity. This assumption can be considered acceptable due to pritelivir’s long80-hour half-life which keeps drug conditions within the body relatively con-stant [37].Using the new definition for the dynamics of the HSV-2 virus that in-cludes antiviral effects, I ran the stochastic Gillespie algorithm as describedpreviously and examined the characteristics of the lesions that developedunder these four drug scenarios. Specifically, I compared fifty, one-year sim-ulations of the model for each of the four treated cases and the untreatedcase, counting the number of viral peaks and lesions that occurred in eachrun, and the duration of each lesion episode. I classified a viral peak as aspike of virus greater than 1000 virions with no higher viral counts withinthe five previous and following days. Similarly, lesion peaks were classified434.1. Impact of Antivirals on HSV-2 Infection and Lesion Developmentas tissue damage taking up at least 1% of the simulation region and with nohigher levels of tissue damage within the five previous and following days.To measure the duration of a lesion, the lesion was assumed to begin whentissue damage was greater than 1% of the simulation region and was consid-ered over when tissue damage dropped below 1% of the simulation region.If a lesion episode was to occur within five days of a previous one, it wasclassified as the same lesion event. These results are summarized in figures4.1-4.3. Results show that as pritelivir dose increases, the severity of vi-ral and lesion peaks, and the duration of lesions, decrease. The frequencyat which viral peaks and lesions occur also decrease with higher pritelivirdoses. When compared to simulations of HSV-2 positive patients receivingno treatment, lesions are two times less frequent and last half as long dueto their decreased severity.Figure 4.1: Pritelivir’s effect on lesion duration during HSV-2 infection.Here, the duration of HSV-2 lesions at varying doses of pritelivir was calcu-lated and categorized based on length. A lesion was defined to begin whentissue damage due to HSV-2 infection was greater than 1% of the simulationregion and considered healed once tissue damage dropped below 1%. Theduration of all lesions is reported in days. Numbers appearing at the top ofthe graph represent the average number of days lesions lasted for simulationsrun at each dose of pritelivir. Lesion duration decreases as pritelivir doseincreases due to smaller lesions that occur.444.1. Impact of Antivirals on HSV-2 Infection and Lesion DevelopmentFigure 4.2: Pritelivir’s effect on tissue damage during HSV-2 infection. Here,the percent of tissue damage at times of peak lesions for varying doses ofpritelivir was calculated and categorized based on its severity. Peak lesionswere defined as when the greatest fraction of epithelial cells were absentfrom the system due to infection-caused cell death. Numbers appearing atthe top of the graph represent the average number of lesions that occur in aone-year simulation for each dose of pritelivir. The frequency and severityof HSV-2 lesions decrease as pritelivir dose increases.Figure 4.3: Pritelivir’s effect on viral shedding during HSV-2 infection. Vi-ral peaks that occurred during model simulations run at varying doses ofpritelivir were counted and classified based on their magnitude. Numbersappearing at the top of the graph represent the average number of viralpeaks occurring in a one-year simulation for each dose of pritelivir. Whileviral peaks are slightly worse at a dose of 10 mg/day compared to 30 mg/day,the general trend shows that as pritelivir dose increases, both the frequencyand severity of HSV-2 viral peaks decrease.454.2. Pritelivir’s Impact on HIV-1 Infection Probability4.2 Pritelivir’s Impact on HIV-1 InfectionProbabilityWith an understanding and mathematical representation of how antiviralHSV-2 drugs decrease the severity of herpes lesions and infections, it wouldbe valuable to determine if and by how much this drug may decrease theprobability of contracting HIV-1 in patients who are HSV-2 positive andfacing HIV-1 exposure.As presented in Chapter 3, HIV-1 infection probability per simulationsite can estimated by the equationProbinf,tot ≈ −0.0177 + 0.000567T + 0.0275L (4.4)for each sexual act with a chronically infected individual. Here T representsthe per mm2 average number of CD4+ T cells and L represents the totalpercent of tissue damage in the simulation region upon the time of exposureto HIV-1. By observing how CD4+ T cell counts and lesion damage changein patients being treated with HSV-2 antivirals, I obtained a representationof how these drugs may decrease the probability of infection.Median HSV-2, CD4+ T cell, and tissue damage amounts obtained fromfifty one-year-long simulations, run for varying daily doses of pritelivir, areshown in figure 4.4. Median, rather than average, values are presented dueto the highly skewed nature of the data during lesion events. This skew iscaused by the large number of model reaction events that occur during thesmall time window of infection outbreaks. As pritelivir dosage increases,the median number of HSV-2 virions, CD4+ T cells, and amount of tissuedamage decreases accordingly.464.2. Pritelivir’s Impact on HIV-1 Infection ProbabilityFigure 4.4: Median HSV-2 counts (a.), median CD4+ T cell counts per mm2(b.), and median percent of tissue damage (c.) for simulations run at fivedaily doses of pritelivir. The red horizontal line in b. shows the numberof CD4+ T cells in healthy individuals (40 cells/mm2). All three valuesdecrease as pritelivir dose increases; however, CD4+ T cell count remainsthe most stable.In patients not receiving pritelivir, median HSV-2 viral loads ranged from7-9 virions, and median tissue damage ranged from 0.003-0.017%. At dosesof 80 mg/day, median HSV-2 counts in the simulation region were reduced7.4 fold to between 1 and 3 virions, and tissue damage in the region wasreduced 24.6 fold to approximately 0.0001% the simulation region. Theseresults are confirmation that the symptoms associated with the HSV-2 in-fection have been greatly reduced as a result of antiviral use. Specifically, ifwe consider equation 4.4, the amount of median tissue damage at a pritelivirdose of 80 mg/day contributes only an additional 0.00000275% to the riskof HIV-1 contraction. Considering the background risk of HIV-1 is 0.005%,this increase can be considered negligible and indicates that at high dosesof antiviral drugs CD4+ T cell counts become the greatest determinant ofHIV-1 infection probability.As depicted in figure 4.4, CD4+ T cell count appears the most stableinfection characteristic as pritelivir dose increases. Between doses of 0 and80 mg/day of pritelivir, median CD4+ T cell count only decreased by 1.5fold and remained more than double the count seen in healthy tissue [48].While the effects of lesions may become negligible, the maintenance of highCD4+ T cell counts may hinder the reduction of HIV-1 infection probability.These trends are also apparent in figure 4.5 where median percent risksof HIV-1 contraction per coital act in one simulation region are shown for474.2. Pritelivir’s Impact on HIV-1 Infection Probabilityvarying doses of pritelivir. These values were obtained by inserting themedian values of CD4+ T cells per mm2 and median percents of tissuedamage into equation 4.4. Similar to the 1.5 fold decrease seen across CD4+T cells, the risk of HIV-1 infection shows a 1.8 fold decrease between doses of0 and 80 mg/day of pritelivir. These results may help to explain why studieshave not detected a significant decrease in HIV-1 infection probability inpatients receiving HSV-2 antivirals [8, 46]. Even at a dosage of 80 mg/day,the median risk of HIV-1 for each of the 50 year long simulations remained5.7-7.6 times larger than the 0.005% background risk of HIV-1 infection per(uninfected) simulation site that was calculated earlier. This result indicatesthat none of the pritelivir doses examined here are able to return the HSV-2infected tissue to a fully healthy state.Figure 4.5: Median CD4+ T cell counts per mm2 and median percent oftissue damage from model simulations describing the effects of varying dosesof pritelivir were used to calculate the median risk of HIV-1 contraction ina simulation region per coital act. Risk of HIV-1 decreases with increases inpritelivir dose; however, risk remains well above that seen in healthy tissue,marked by the horizontal red line (0.005% per coital act).While median HIV-1 infection probability shows a general decline aspritelivir dose increases, high variability in HIV-1 infection probability oc-curs throughout the course of a simulation. Figure 4.6 shows the changingrisk of HIV-1 infection throughout the course of a year for varying doses ofpritelivir. Spikes in infection probability occur during times of lesion devel-opment, showing up to a 60 fold increase. These spikes appear to occur atevery dose. This observation serves as a clear indication that while patientson doses of pritelivir have a lower median risk, the current state of their484.2. Pritelivir’s Impact on HIV-1 Infection ProbabilityHSV-2 infection remains an important factor in their risk of contractingHIV-1.Figure 4.6: Changing risk of HIV-1 over time. For daily pritelivir doses of0 mg, 30 mg, and 80 mg, two randomly chosen year-long simulations of themodel were used to display the changes in HIV-1 infection risk over time.At every recorded time point, CD4+ T cell count and tissue damage wereused to calculate the risk of HIV-1 using equation 4.4. For all doses, HIV-1risk shows great variability over time.4.2.1 HIV-1 Infection Probability for the Entire GenitalRegionThe simulations performed up to this point give a prediction of the proba-bility of HIV-1 infection at one 4 cm2 patch of genital epithelium. It may bemore informative, however, to know the probability of HIV-1 infection persexual act for the entire genital mucosa, as these values could be more easilycompared to those recorded in the literature [6, 30, 32, 47]. This conversionof probabilities proves difficult as we cannot be sure at which stage of le-sional development every patch of skin is experiencing, whether all patchesof skin are chronically infected with HSV-2, or whether all patches of skinare exposed to the HIV-1 virus during a sexual act. To form some approxi-mation, I looked at different states the estimated 88 cm2 of the vagina maybe experiencing in a chronically HSV-2 infected individual. I assumed thatevery 4 cm2 patch, corresponding with one simulation region, could be ex-periencing a peak lesion event, some other chronic HSV-2 infected state,494.2. Pritelivir’s Impact on HIV-1 Infection Probabilityor be unaffected by the HSV-2 virus and be considered healthy. I calcu-lated the overall probability of HIV-1 infection for the entire vaginal region(Probinf,vag) using the following formula.Probinf,vag = 1− (1−max)a(1−med)b(1− healthy)c (4.5)Here, a is the number of 4 cm2 epithelial patches that are experiencing a peaklesion, b is the number of 4 cm2 epithelial patches that are experiencing someother state of lesion development, and c is the number of 4 cm2 epithelialpatches that are unaffected by the HSV-2 infection and can be consideredhealthy. Note that a+ b+ c = 22 as each patch is 4 cm2 and the total areaof the vagina is 88 cm2.For each dose of pritelivir, the probability of HIV-1 in a simulation siteexperiencing a peak lesion, termed max in equation 4.5, was defined as theaverage of the 95th percentile probabilities of infection for all fifty, one-year simulations run at that dose. Similarly, the average of the medianprobabilities of infection for each drug dose were used to represent the riskof HIV-1 infection in a region of HSV-2 infected tissue not experiencing apeak lesion event, termed med in equation 4.5. The probability of HIV-1infection in a simulation region assumed to be healthy, termed healthy inequation 4.5, was set to 0.00005, as found earlier in Chapter 3.This formula assumes that infection at any site is independent of infec-tion at other sites, and calculates the probability HIV-1 infection is estab-lished somewhere in the 22 patches. It also assumes all sites are exposed toHIV-1, and the concentration of HIV-1 virus is uniformly distributed acrossthe vaginal tissue.Results from these calculations for doses of 0 mg/day, 30 mg/day, and80 mg/day of pritelivir are shown in figure 4.7. For an individual with allhealthy tissue, the probability of HIV-1 infection is 0.11% per sexual act,matching with those values found in the literature [6, 30, 32, 47]. When anindividual is not receiving treatment, every peak lesion adds an additional0.8% to the risk of contracting HIV-1, and every 4 cm2 patch of epithe-lial tissue that is experiencing some other state of HSV-2 infection adds anadditional 0.05% to the risk of contracting HIV-1. If patients receive 30mg/day of pritelivir, the risk of contracting HIV-1 in any of the vaginal tis-sue exposed shows a 1.11-1.32 fold decrease compared to the no treatmentscenario depending on the tissue composition compared. This decrease mayor may not be large enough to be captured in clinical studies. More promis-ing, however, is the decrease in HIV-1 contraction risk with a dose of 80mg/day of priteliver, causing a 1.3-1.78 fold decrease in risk compared tothe no treatment scenario.504.2. Pritelivir’s Impact on HIV-1 Infection ProbabilityWhen examining these overall risks of HIV-1 infection, another impor-tant aspect to consider is the likelihood of each tissue profile presented infigure 4.7. Since most HSV-2 positive patients spend the majority of theirtime with no symptomatic lesions, the top rows of these heatmaps are likelythe best representation of the HIV-1 contraction risks most commonly facedby HSV-2 infected patients. Furthermore, HSV-2 lesions often re-occur insimilar tissue regions, indicating that not the entire genital region should beconsidered chronically infected, further limiting the realistic tissue profile tothe top left corner of these heatmaps. Other tissue profiles represent theper-coital HIV-1 risks that patients with more severe HSV-2 infections mayexperience. These more severe cases also become less likely as the dose ofpritelivir increases and the occurrence of multiple genital lesions at one timebecomes rare.In patients not receiving antivirals, studies have recorded HIV-1 infectionrisk to be 2-3 times higher in HSV-2 asymptomatic, compared to healthy,patients and 7 times higher if lesions are present [3, 14]. These risks appearto match with those presented here when considering the most commontissue profiles.514.2. Pritelivir’s Impact on HIV-1 Infection ProbabilityFigure 4.7: Probability of HIV-1 infection in vaginal tissue of an HSV-2 positive female, per coital act with a chronically HIV-1 infected male.With a vaginal surface area of 88 cm2, I assume the vaginal tissue can berepresented by twenty-two 4 cm2 simulation regions experiencing varyingseverities of HSV-2 infection. In a., the probability of HIV-1 infection inuntreated individuals is shown. The top left grid square of the matrix rep-resents the probability of HIV-1 infection given that all twenty-two tissuesimulation regions in the vagina are healthy. Moving down the vertical axis,infection probability increases as sites of healthy tissue are replaced withsevere lesions. Moving across the horizontal axis, healthy tissue becomesreplaced with tissue experiencing median HSV-2 infection behaviour, alsoleading to an increase in HIV-1 infection probability, but to a lesser degree.For each examined tissue profile, heatmaps b. and c. show by what foldHIV-1 infection risk is predicted to decrease in patients receiving 30 mg and80 mg of pritelivir a day respectively, compared to untreated patients.524.2. Pritelivir’s Impact on HIV-1 Infection ProbabilityFigure 4.7 cont’d53Chapter 5ConclusionAs the development of a trusted vaccination for HIV-1 may be far off, otherstrategies for controlling the spread of HIV-1 are important to develop andunderstand. With up to 60% of HIV-1 cases being attributed to previouslyestablished HSV-2 infections [14], it is important to see HSV-2 treatment asa way of controlling HIV-1 spread.By developing a spatial stochastic model to describe the dynamics ofchronic HSV-2 infections in the genital mucosa, I determined how HIV-1infection probability can be expressed as a function of an individual’s currentHSV-2 infection state. With a particular focus on the newly developedHSV-2 antiviral drug pritelivir, I showed that doses currently being used inclinical trials should lower the risk of contracting HIV-1 by decreasing thefrequency and severity of HSV-2 lesions that serve as entry points, and thenumber of CD4+ T cells in the genital mucosa that serve as target cellsfor HIV-1 virions. While the risk of contracting HIV-1 as a chronic HSV-2infected individual receiving pritelivir still remains well above the risk facedby healthy, uninfected individuals, here I estimate that receiving 80 mg/dayof pritelivir should cause a 1.3-1.78 fold decrease in the risk of contractingHIV-1 per coital act.While theory has long predicted that such a decrease should result fromHSV-2 antiviral drugs, clinical studies have failed to detect the effect ofHSV-2 antivirals on the reduction of HIV-1 infection probability [18, 46].Numerous reasons may explain this disconnect between theory and data,the first related to the CD4+ T cell counts in the studies’ participants. Inpatients receiving HSV-2 antivirals, the mathematical model presented herepredicts that CD4+ T cell counts in the genital mucosa become the great-est determinant in HIV-1 infection risk. Lowering CD4+ T cell count inthe genital mucosa is essential to HIV-1 risk reduction. HSV-2 infection,however, may not be the only factor controlling CD4+ T cell counts in thegenital mucosa. In clinical studies observing the effect of HSV-2 antivi-rals on HIV-1 infection probability, many participants were coinfected withother STIs which were not controlled for [18, 46]. Non-ulcerative STIs canincrease CD4+ T cell levels to twice those seen in healthy tissue [13]. Even if54Chapter 5. Conclusionparticipants were receiving HSV-2 antivirals, these extra CD4+ T cells dueto other STIs may have kept CD4+ T cell counts too high for significantreductions in HIV-1 infection risk to occur.Another potential reason why differences in HIV-1 risk were not capturedwith and without the use of HSV-2 antivirals in clinical studies may be dueto sample sizes in the studies being too small. From the analysis presentedhere, the per-coital risk of contracting HIV-1 in chronically infected HSV-2positive individuals remains very low, likely below 1.5%. While my analysisassumed partners were exposed to HIV-1 through unprotected sex, condomuse in the clinical studies was encouraged [18, 46], likely bringing risks evenlower and making them difficult to capture without a large sample size.Even more difficult may be capturing the difference in infection probabilitybetween the treated and untreated scenarios.Differences in when, and how often, individuals from the control andtreatment groups were having sex may have also helped mask HIV-1 riskdifferences in clinical studies. While individuals receiving antivirals generallyhave less severe lesions, they may have more small, undetectable microlesionswhich cause a flare in CD4+ T cell number and damaged skin that allowseasier entry for the HIV-1 virus. While individuals may avoid sex whenthey have major, visible lesions, less caution may be taken during theseother vulnerable times. This behaviour could potentially bring HIV-1 risksin the treated study participants closer to those seen in the untreated studyparticipants.Another more worrying reason of why antiviral drugs were not shown toreduce HIV-1 contraction risk in clinical studies may be due to short drughalf lives. In the model presented here, antiviral drug decay was not con-sidered. Previous mathematical models of HSV-2 infection, however, haveincluded the effects of antiviral drug decay and show that when drug con-centrations reach sub-therapeutic levels, rapid HSV-2 breakouts can occur,preventing full control of the infection [37]. The clinical studies aimed atdetecting the effects of HSV-2 antivirals on HIV-1 infection risk have usedacyclovir as their study drug which unfortunately has a short half-life of3-4 hours [18, 46]. Lack of HIV-1 risk reduction in patients on these drugsmay be explained by frequent times of sub-therapeutic drug levels. Fortu-nately, pritelivir’s half-life is estimated to be approximately 80 hours [37].This longer half-life should reduce the effects of drug decay on HSV-2 infec-tion breakouts and HIV-1 infection probability. Longer half-lives also allowmodelling the effects of antivirals as constant throughout time to be a saferassumption.While the model presented here did not include the effects of drug decay,55Chapter 5. Conclusionit did incorporate CD4+ T cell dynamics in the genital mucosa, somethingwhich is novel to mathematical models describing HSV-2 infection. Unfor-tunately, their influence on the clearance of HSV-2 remained passive as notenough information is currently available to accurately predict their directeffects. By showing how essential CD4+ T cell numbers are to HIV-1 in-fection probabilities, this hopefully sparks more interest in studying theirrole and dynamics in herpes infections. It also may lead the way for moremathematical modelling of the synergy between HSV-2 and HIV-1 from theimmunological perspective.In conclusion, these results bring further insight to the mechanistic be-haviour of herpes lesion development in the genital mucosa, the synergybetween HSV-2 and HIV-1 infections, and provide support for HSV-2 an-tivirals being an effective way of controlling HIV-1 infection risk in patientswith chronic HSV-2 infections. This study also helps inform clinicians indetermining appropriate drug doses to see significant results in HIV-1 riskreduction. While clinical studies have been performed to examine the ef-fects of pritelivir on HSV-2 infection alone, none have yet examined itseffect of HIV-1 infection spread. The results presented here predict posi-tive outcomes for such a study, and support further consideration of HSV-2antivirals serving as a viable way to decrease probabilities of HIV-1 contrac-tion. 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Nature Medicine,15(8):886–892, 2009.62Appendix AMain Python Code forHSV-2 Infection Simulations1 #LIBRARIES2 import time3 import re4 import math5 import numpy as np6 from numpy import array7 from numpy . random import uniform , multinomial , exponent ia l ,random8 from numpy import arange , array , empty , zeros , l og9 import time10 import mu l t i p r o c e s s i ng1112 #TECHNICAL INFORMATION13 #Si z e Dimensions14 Dim= 2.0 #cm15 x box = 1516 y box = 1517 h = Dim/x box1819 #Run Time Sp e c i f i c a t i o n s20 S max = 121 N max =122 t max = 12023 r e c I n t e r v a l = 0 .012425 #REACTIONS26 #r0 = c r e a t i on o f hea l thy e p i t h e l i a l c e l l s27 #r1 = i n f e c t i o n o f hea l thy c e l l by an e p i t h e l i a l v i r u s28 #r2 = i n f e c t i o n o f a hea l thy c e l l by a neuronal v i r u s29 #r3 = death o f i n f e c t e d e p i t h e l i a l c e l l s30 #r4 = c l e a r anc e o f i n f e c t e d e p i t h e l i a l c e l l s by CD831 #r5 = product ion o f more CD8s32 #r6 = death o f CD8s33 #r7 = product ion o f e p i t h e l i a l v i r u s34 #r8 = decay o f e p i t h e l i a l v i r u s35 #r9 = decay o f neuronal v i r u s63Appendix A. Main Python Code for HSV-2 Infection Simulations36 #r10 = r e l e a s e o f v i r u s from neuron − only occurs in the cent r ebox37 #r11 = d i f f u s i o n o f e p i t h e l i a l v i r u s38 #r12 = d i f f u s i o n o f neuronal v i r u s39 #r13 = d i f f u s i o n o f CD840 #r14 = d i f f u s i o n o f CD441 #r15 = d i f f u s i o n o f cy tok ine s42 #r16 = br ing ing in o f more CD8 from borders43 #r17 = br ing ing in o f more CD4 from borders44 #r18 = product ion o f more CD4s45 #r19 = death o f CD4s46 #r20 = f low in o f more CD4s47 #r21 = product ion o f cy tok ine s48 #r22 = decay o f cy tok ine s4950 tmat = array ([ [1 ,−1 ,−1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,51 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ] , #H52 [ 0 , 1 , 1 ,−1 ,−1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,53 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ] , #I54 [ 0 , 0 , 0 , 0 , 0 , 1 ,−1 , 0 , 0 , 0 , 0 , 0 ,55 0 ,−1 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 ] , #E56 [0 ,−1 , 0 , 0 , 0 , 0 , 0 , 1 ,−1 , 0 , 0 ,−1 ,57 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ] , #Ve58 [ 0 , 0 ,−1 , 0 , 0 , 0 , 0 , 0 , 0 ,−1 , 1 , 0 ,59 −1, 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ] , #Vn60 [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ,0 ,0 , 0 , 0 , 0 ,61 0 , 0 ,−1 , 0 , 0 , 1 , 1 ,−1 , 1 , 0 , 0 ] , #T62 [ 0 , 0 , 0 , 0 , 0 , 0 , 0 ,0 ,0 , 0 , 0 , 0 ,63 0 , 0 , 0 ,−1 , 0 , 0 , 0 , 0 , 0 , 1 ,−1] #C64 ] )6566 de f f cn ( var1 , var2 , var3 ) :67 H = H counter [ var1 , var2 ]68 I = I coun t e r [ var1 , var2 ]69 E = E counter [ var1 , var2 ]70 Ve = Ve counter [ var1 , var2 ]71 Vn = Vn counter [ var1 , var2 ]72 T = T counter [ var1 , var2 ]73 C = C counter [ var1 , var2 ]74 #determine how many exposed edges the chosen box has75 edges = sum ( [ var1==0, var1 == x box−1, var2==0,var2==y box−1])76 ra = ze ro s (23 , dtype=f l o a t )77 ra [ 0 ] = g ∗(H0−H)78 ra [ 1 ] = beta ∗H∗Ve79 ra [ 2 ] = beta ∗H∗Vn80 ra [ 3 ] = a∗ I81 ra [ 4 ] = f ∗ I ∗E82 ra [ 5 ] = C/(C+r3 ) ∗ theta1 ∗E64Appendix A. Main Python Code for HSV-2 Infection Simulations83 ra [ 6 ] = de l t a ∗E84 ra [ 7 ] = p∗ I85 ra [ 8 ] = c∗Ve86 ra [ 9 ] = c∗Vn87 i f var1==round ( x box /2 ,0) and var2==round ( x box /2 ,0) :88 ra [ 1 0 ] = phi89 e l s e :90 ra [ 1 0 ] = 091 ra [ 1 1 ] = d i f f 1 ∗Ve92 ra [ 1 2 ] = d i f f 1 ∗Vn93 ra [ 1 3 ] = d i f f 2 ∗E94 ra [ 1 4 ] = d i f f 2 ∗T95 ra [ 1 5 ] = d i f f 3 ∗C96 i f edges >0:97 ra [ 1 6 ] = edges ∗D E98 ra [ 1 7 ] = edges ∗D T99 e l s e :100 ra [ 1 6 ] = 0101 ra [ 1 7 ] = 0102 ra [ 1 8 ] = C/(C+r4 ) ∗ theta2 ∗T103 ra [ 1 9 ] = d∗T104 ra [ 2 0 ] = Lambda105 ra [ 2 1 ] = b∗ I106 ra [ 2 2 ] = m∗C107108 ra t e = ra . sum( )109 i f var3==1:110 i f rate >0:111 d e l t a t = −math . l og (np . random . uniform (0 , 1 , 1 ) ) / ra t e112 e l s e :113 d e l t a t = 1000000114 return d e l t a t115 e l i f var3==2:116 event = np . random . mult inomial (1 , ra / ra t e )117 return event118119 de f d i f f u s i o n ( counter ) :120 i f id ( counter )==id ( C counter ) or id ( counter ) ==id ( Ve counter )or id ( counter ) == id ( Vn counter ) :121 event2 = np . random . mult inomial ( 1 , [ 0 . 2 5 , 0 . 2 5 , 0 . 2 5 , 0 . 2 5 ] )122 e l s e :123 Cneigh = ze ro s (4 , dtype=f l o a t )124 i f X>0:125 Cneigh [ 0 ] = C counter [X−1,Y]126 i f X<x box−1:127 Cneigh [ 1 ] = C counter [X+1,Y]128 i f Y>0:129 Cneigh [ 2 ] = C counter [X,Y−1]130 i f Y<y box−1:65Appendix A. Main Python Code for HSV-2 Infection Simulations131 Cneigh [ 3 ] = C counter [X,Y+1]132 Ctot = sum(Cneigh )133 i f Ctot>0:134 event2 = np . random . mult inomial (1 , Cneigh/Ctot )135 e l s e :136 event2 = np . random . mult inomial ( 1 , [ 0 . 2 5 , 0 . 2 5 , 0 . 2 5 , 0 . 2 5 ] )137 i f event2 [0]==1 and X>0:138 counter [X−1,Y]+=1139 d e l t a t = fcn (X−1,Y, 1 )140 t [X−1,Y] = t counte r+d e l t a t141 e l i f event2 [1]==1 and X<x box−1:142 counter [X+1,Y]+=1143 d e l t a t = fcn (X+1,Y, 1 )144 t [X+1,Y] = t counte r+d e l t a t145 e l i f event2 [2]==1 and Y>0:146 counter [X,Y−1]+=1147 d e l t a t = fcn (X,Y−1 ,1)148 t [X,Y−1] = t count e r+d e l t a t149 e l i f event2 [3]==1 and Y<y box−1:150 counter [X,Y+1]+=1151 d e l t a t = fcn (X,Y+1 ,1)152 t [X,Y+1] = t counte r+d e l t a t153154155 #INITIAL CONDITIONS156 H = round (6024382/( x box∗y box ) ,0 )157 I = 0158 E = 60000159 C = 0160 Ve = 0161 Vn = 0162 T =50000163164 #MODEL SPECIFICATIONS165166 #Parameters167 g = 0.22168 H0 = H169 beta = 1∗x box∗y box∗pow(10 ,−7)170 a = 1 .2171 f = 0.01∗ x box∗y box172 r3 = 42 .0/ ( x box∗y box )173 theta1 = 1 .7174 de l t a = 0.05175 p = 7.05∗pow(10 ,3 )176 c = 8 .8177 phi = 50178 m = 6.2179 b = 24 .866Appendix A. Main Python Code for HSV-2 Infection Simulations180 d = 0.07181 theta2 = 1 .4182 r4 = 38 .0/ ( x box∗y box )183 Lambda = 1900 .0/( x box∗y box )184 omegav = 0.00072185 omegae = 0.00072186 omegac = 0.0245187 d i f f 1 = 4∗omegav/pow(h , 2 )188 d i f f 2 = 4∗omegae/pow(h , 2 )189 d i f f 3 = 4∗omegac/pow(h , 2 )190 #ra t e s o f d i f f u s i o n from borders191 D E = omegae∗E/(pow(h , 2 ) ∗( x box∗y box ) )192 D T = omegae∗T/(pow(h , 2 ) ∗( x box∗y box ) )193 #Counters194 H counter = np . z e r o s ( ( x box , y box ) )195 I coun t e r = np . z e r o s ( ( x box , y box ) )196 E counter = np . z e r o s ( ( x box , y box ) )197 C counter = np . z e r o s ( ( x box , y box ) )198 Ve counter = np . z e ro s ( ( x box , y box ) )199 Vn counter = np . z e ro s ( ( x box , y box ) )200 T counter = np . z e r o s ( ( x box , y box ) )201 t count e r = 0202 t = np . z e ro s ( ( x box , y box ) )203204 #OTHER FUNCTIONS205206207 de f p r i n t i n g ( f i leName , counter , savetype , format ) :208 with f i l e ( f i leName , savetype ) as f i l e s h o r t :209 np . save txt ( f i l e s h o r t , counter , fmt=format )210 f i l e s h o r t . wr i t e ( ’# New step \n ’ )211212 #MODEL213 N = 0214 whi le (N<=N max−1) :215 S = 0216 H counter [ : , : ] = H217 I coun t e r [ : , : ] = I218 XY l i s t = np . random . random integers ( x box , s i z e=(E , 2 . ) )−1219 f o r i in range (0 ,E) :220 E counter [ XY l i s t [ i , 0 ] , XY l i s t [ i , 1 ] ] = E counter [ XY l i s t [ i, 0 ] , XY l i s t [ i , 1 ] ]+1221 C counter [ : , : ] = C222 Ve counter [ : , : ] = Ve223 Vn counter [ : , : ] = Vn224 XY l i s t2 = np . random . random integers ( x box , s i z e=(T, 2 . ) )−1225 f o r i in range (0 ,T) :226 T counter [ XY l i s t2 [ i , 0 ] , XY l i s t2 [ i , 1 ] ] = T counter [ XY l i s t2 [i , 0 ] , XY l i s t2 [ i , 1 ] ]+167Appendix A. Main Python Code for HSV-2 Infection Simulations227 t count e r = 0228 t [ : , : ] = 0229230231 whi l e t counter<t max :232 S +=1233 i f S==1:234 #Set up i n t i a l time queue235 f o r i in range (0 , x box ) :236 f o r j in range (0 , y box ) :237 d e l t a t = fcn ( i , j , 1 )238 t [ i , j ] = d e l t a t239240 #Se l e c t box f o r r e a c t i on to occur and update the t count e r241 Hit= np . where ( t == np . min ( t ) )242 Hit = np . asmatr ix ( Hit )243 X = Hit [ 0 ]244 Y = Hit [ 1 ]245246 i f np . shape ( Hit ) !=(2 ,1) :247 X = Hit [ 0 , 0 ]248 Y = Hit [ 1 , 0 ]249 p r in t ’True ’250 t count e r = t [X,Y]251252253 #Choose an event to occur the s e l e c t e d box254 event = fcn (X,Y, 2 )255 changes = tmat [ : , event . nonzero ( ) [ 0 ] [ 0 ] ]256 #Update cent r e box257 H counter [X,Y] += changes [ 0 ]258 I c oun t e r [X,Y] += changes [ 1 ]259 E counter [X,Y] += changes [ 2 ]260 Ve counter [X,Y] += changes [ 3 ]261 Vn counter [X,Y] += changes [ 4 ]262 T counter [X,Y] += changes [ 5 ]263 C counter [X,Y] += changes [ 6 ]264265 #Update s i d e boxes and time queue266 i f sum( event [ 1 1 : 1 6 ] ) >0:267268 i f event [11]==1:269 d i f f u s i o n ( Ve counter )270 e l i f event [12]==1:271 d i f f u s i o n ( Vn counter )272 e l i f event [13]==1:273 d i f f u s i o n ( E counter )274 e l i f event [14]==1:275 d i f f u s i o n ( T counter )68Appendix A. Main Python Code for HSV-2 Infection Simulations276 e l i f event [15]==1:277 d i f f u s i o n ( C counter )278279 #Update time queue f o r c e n t r a l box280 d e l t a t = fcn (X,Y, 1 )281 t [X,Y] = t [X,Y]+ d e l t a t282283 i f S==1 and N == 0 :284 tpo in t = t counte r285 p r i n t i n g ( ’H mat . txt ’ , H counter , ’w’ , ’%1 .0 f ’ )286 p r i n t i n g ( ’ I mat . txt ’ , I counte r , ’w’ , ’%1 .0 f ’ )287 p r i n t i n g ( ’ E mat . txt ’ , E counter , ’w’ , ’%1 .0 f ’ )288 p r i n t i n g ( ’ C mat . txt ’ , C counter , ’w’ , ’%1 .0 f ’ )289 p r i n t i n g ( ’V mat . txt ’ , Ve counter+Vn counter , ’w’ , ’%1 .0 f ’ )290 p r i n t i n g ( ’ Tce l l mat . txt ’ , T counter , ’w’ , ’%1 .0 f ’ )291 p r i n t i n g ( ’ t mat . txt ’ , t counter , ’w’ , ’%.10 e ’ )292293 e l i f (S==1 and N!=0) or \294 t counter>=tpo in t+r e c I n t e r v a l :295 tpo in t = t counte r296 p r i n t i n g ( ’H mat . txt ’ , H counter , ’ a ’ , ’%1 .0 f ’ )297 p r i n t i n g ( ’ I mat . txt ’ , I counte r , ’ a ’ , ’%1 .0 f ’ )298 p r i n t i n g ( ’ E mat . txt ’ , E counter , ’ a ’ , ’%1 .0 f ’ )299 p r i n t i n g ( ’ C mat . txt ’ , C counter , ’ a ’ , ’%1 .0 f ’ )300 p r i n t i n g ( ’V mat . txt ’ , Ve counter+Vn counter , ’ a ’ , ’%1 .0 f ’ )301 p r i n t i n g ( ’ Tce l l mat . txt ’ , T counter , ’ a ’ , ’%1 .0 f ’ )302 p r i n t i n g ( ’ t mat . txt ’ , t counter , ’ a ’ , ’%.10 e ’ )303304 N+=169


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