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A simple model for behaviour change in epidemics Brauer, Fred Feb 25, 2011

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RESEARCH Open AccessA simple model for behaviour change in epidemicsFred BrauerAbstractBackground: People change their behaviour during an epidemic. Infectious members of a population may reducethe number of contacts they make with other people because of the physical effects of their illness and possiblybecause of public health announcements asking them to do so in order to decrease the number of newinfections, while susceptible members of the population may reduce the number of contacts they make in orderto try to avoid becoming infected.Methods: We consider a simple epidemic model in which susceptible and infectious members respond to adisease outbreak by reducing contacts by different fractions and analyze the effect of such contact reductions onthe size of the epidemic. We assume constant fractional reductions, without attempting to consider the way inwhich susceptible members might respond to information about the epidemic.Results: We are able to derive upper and lower bounds for the final size of an epidemic, both for simple andstaged progression models.Conclusions: The responses of uninfected and infected individuals in a disease outbreak are different, and thisdifference affects estimates of epidemic size.IntroductionDuring the course of an epidemic, there are changes inbehaviour which have an effect on the transmission ofinfection. Individuals who are infected may make fewercontacts with others because the debilitating effects oftheir illness or because of advice by public health orga-nizations to stay home in order to avoid infectingothers. Individuals who have not been infected maytake hygienic measures to reduce the risk of beinginfected and may take other steps such as avoidance oflarge public gatherings. There is evidence that suchmeasures had substantial effects during the 1918 influ-enza pandemic [1].The question of what factors influence people tochange their behaviour is a difficult one, probably morein the areas of psychology and sociology than epide-miology and public health. In this study, we avoid thisquestion, and assume only reduction of contacts suffi-cient to transmit infection by members of the popula-tion. Since the factors affecting such behaviour changesare different for those who are infected and those whowish to avoid becoming infected, it is necessary toassume different fractional reductions in these twogroups. This implies that, even in a model in whichmixing is assumed homogeneous without behaviouralchange, it is necessary to recognize that the mixingbecomes heterogeneous, and this may affect the beha-viour of the model.In this note, our purpose is to estimate the effect thatgiven reductions in contacts have on the final size of anepidemic, without trying to model the factors that mightcause such reductions. It would be more realistic toassume that the rate or amount of behavioural change,at least for uninfected members of the population, isdependent on some information about the extent of theepidemic, perhaps the number of infectious people orthe total number of reported disease deaths. Study ofsuch questions is one of the most important gaps inscientific knowledge of the spread of communicable dis-eases. One contribution in the direction of studying thisarea is [4]A simple SIR epidemic modelThe simplest special case of the Kermack-McKendrickepidemic model isCorrespondence: brauer@math.ubc.caDepartment of Mathematics, University of British Columbia, Vancouver, BC,V6T 1Z2, CanadaBrauer BMC Public Health 2011, 11(Suppl 1):S3http://www.biomedcentral.com/1471-2458-11-S1-S3© 2011 Brauer; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.     S SII SI IR I  ,(1)in which it is assumed that contact between individualssatisfies a mass action law with members making a con-stant number bN contacts in unit time, that there is anexponential distribution of infected periods with meanlength 1/a, and that there are no disease deaths (so thatthe total population size N remains constant) [5]. Weassume that initially the population consists of S0 suscep-tible members and a (presumably small) number I0 ofinfectious members, with S0 + I0 = N. For the model (1)it is known that the basic reproduction number is0  N,that the number of susceptibles decreases to a positivelimit S∞and the number of infectious membersdecreases to zero as t ® ∞, and that the attack rate, theextent of the epidemic which is defined asASN  1satisfies the final size relationln .SSSN00 1  (2)This was originally derived in [5], although the basicreproduction number was not given explicitly. Morerecent derivations may be found in [2,6].Since (1) is a two-dimensional autonomous system ofdifferential equations, the natural approach would be tofind equilibria and linearize about each equilibrium todetermine its stability. However, since every point withI = 0 is an equilibrium, the system (1) has a line of equi-libria and this approach is not applicable (the lineariza-tion matrix at each equilibrium has a zero eigenvalue).It is possible to analyze the system in the phase plane(the (S, I) plane) obtaining a phase portrait and also thefinal size relation. Although this derivation of the finalsize relation is simple and has a useful geometric inter-pretation, it does not generalize readily to more compli-cated compartmental models. For this reason, we givealso an analytic argument which does generalize.Because S is a decreasing non-negative function, it has alimit S∞≥ 0 as t®∞. The sum of the two equations of (1) is(S + I)′ = –aI.Thus S + I is a non-negative smooth decreasing funtionand therefore tends to a limit as t® ∞. Also, it is not dif-ficult to prove that the derivative of a smooth decreasingfunction must tend to zero, and this shows thatI I tt  lim ( ) .0Thus S + I has limit S∞.Integration of the sum of the two equations of (1)from 0 to ∞ gives I t dt S t I t dt S I S N S( ) [ ( ) ( )] .          0 000Division of the first equation of (1) by S and integra-tion from 0 to ∞ gives the final size relation (2). Wenow modify the model (1) by assuming that susceptiblemembers decrease their rate of contact by a fraction p,0 ≤ p ≤ 1 and that infectious members decrease theirrate of contact by a fraction q, 0 ≤ q ≤ 1. As differentsubgroups of the population now have different activitylevels, we must specify the mixing between groups.Since the population is assumed to mix homoge-neously in the absence of disease, we assume propor-tionate mixing. Thus we assume that the number ofcontacts in unit time made by susceptible members,infectious members, and removed members are,respectively,pbN, qbN, bN,and the fraction of contacts made by susceptiblemembers that are with infectious members isqIpS qI R  .It is convenient to defineT = pS + qI + R,so that the rate of new infections isp NqIT ,and the model is given by the pair of differential equations    S NpqTSII NpqTSI I  .(3)From the fact that (S + I)′ = –aI it follows that I® 0 ast® ∞, and from integration of this equation it follows thatN S I t dt  ( ) .0 (4)Then integration of the equation for S in (3)ln( )( ).SSNpqI tT tdt00  (5)Brauer BMC Public Health 2011, 11(Suppl 1):S3http://www.biomedcentral.com/1471-2458-11-S1-S3Page 2 of 5From min(p, q)N ≤ T ≤ N it follows thatpqNpqTpqp q N min( , ),and this, combined with (4), and (5), givesNpqSNSSN pqp qSN1 10   lnmin( , ). (6)It is easy to see thatpqp qp qmin( , )max( , ),and this, together with (6), gives a final size inequalityNpqSNSSNp qSN1 10   ln max( , ) . (7)For the model (3), the next generation matrix calcula-tion [7] shows that0  q N ,while the reproduction number if there were no beha-vioural change would be  N,so thatR0 = qR* ≤ R*.We define R1, R2 by    1 0 2     pq N p pq p q N p q, max( , ) max( , ) .Then (7) takes the form 1 0 21 1   SNSSSNln . (8)It is clear thatR1≤ R0, R1≤ R*, R0≤ R2≤ R*,.In addition,R0< R* (q < 1), R1< R* (pq < 1), R1< R0 (p < 1)R2< R* (p < 1, q < 1), R2 = R0 (p ≤ q).The final size equationA final size equation of the formlnSSSN0 1  (9)determines S∞as a function S∞(R) of R. It is easy toshow [2] that the final size relation (9) has a uniquesolution S∞withSS  0 .Using implicit differentiation of (9) and this estimate,it is easy to see that the function S∞(R) is strictlydecreasing. The final size inequalities (8) imply that forthe model (3) the final size S∞satisfies the inequalitiesS∞(R2) < S∞ < S∞(R1).The behavioural response in the model (3) decreasesthe final susceptible population size from S∞(R*) toS∞(R2) or less. If one ignores the heterogeneity in themodel, one might assume that the final susceptiblepopulation size is S∞(R0). If p > q, so that R2> R0, it ispossible that the final susceptible population size couldbe smaller than this. However, simulations suggest thatthe final susceptible population size is usually largerthan S∞(R0).For example we simulate the model (3) withparametersbN = 0.45, a = 0.25, p = 0.9, q = 0.8,so thatR0 = 1.44, R* = 1.8, R1 = 1.30, R2 = 1.62.A simulation gives S∞= 483.3. The reproductionnumber corresponding to this value of S∞, which wemay call the effective reproduction number, denoted byRE, is 1.405. Further simulations suggest that the effec-tive reproduction number is likely to be close to andsomewhat smaller than R0. In fact, since R2 = R0 if p ≤q, (7) may be replaced by 1 0 01 1    SNSSSNp qln ( ). (10)Even if p >q the upper bound in (10) may be valid.Simulations suggest that the upper bound is valid exceptpossibly when p is very close to 1, q is very close tozero, and R0 is well below 1. It is possible to find exam-ples for which RE > R0, such as p = 0.9, q = 0.2, whichgivesR0 = 0.36, R1 = 0.324, R2 = 1.62, RE = 0.375.This indicates that behavioural response of the typeassumed here usually reduces the size of the epidemic alittle more than might be expected by a naive approach.Staged progression epidemic modelsIt is not feasible to extend the results of the previoussection to general age of infection epidemic models, butwe can analyze the special case of staged progressionmodels. We consider a SI1I2 ... In model in which theBrauer BMC Public Health 2011, 11(Suppl 1):S3http://www.biomedcentral.com/1471-2458-11-S1-S3Page 3 of 5relative infectivity in stage j is εj, and the rate of transferto the next stage is aj. This leads to the model    S S II S I II I Ij jjnj jjn    11 1 112 1 1 2 2  .........I I In n n n n    1 1 .(11)It is known [3] that, for this model,01 N jjjnWe now add to the model (11) the assumption thatsusceptibles reduce contacts by a factor p, infectiousmembers in stage j reduce their contacts by a factor qj,and that the mixing between groups is proportionate.Then the rate of contacts in unit time by susceptibles ispbN, and the fraction of these contacts that are withinfectious members in stage j is qj/T, whereT pS q I Ri iin   .1The resulting model is    Sp NTS q IIp NTS q I II I Ii i iini i iin    1111 12 1 1 2 2  .........I I In n n n n    1 1 .(12)Then the next generation approach [7] shows that thebasic reproduction number is01 Nqi iiin .The reproduction number for the model (11) withoutbehavioural response is  N jjjn1.Integration of the equations for I1, I2, ..., In in (12)gives  1 1 2 2000I s ds I s ds I s dsn n( ) ( ) ( ) ,     (13)while integration of the equation for S givesln( )( ).SSp Nq I sT sdsj jjjnj010     (14)Also, integration of the sum of the equations for S andI1 in (12) gives1 10I s ds N S( ) .   (15)Since T(s) ≤ N, combination of (13), (14), (15) givesln ,SSSN01 1 with11p N qj jjjn  .To obtain an upper bound, we use the inequalityT ≥ min(p, q1, q2, ... , qn)N.If min(p, q1, q2, ... , qn) = p, then p/T ≤1/N, and wehaveln .SSSN00 1 If min(p, q1, q2, ... , qn) = qk, we obtainln ,SSSN02 1 with21  N qq j jk jjn.We may summarize these calculations by saying thatthe bounds obtained for the simple SIR model extend toBrauer BMC Public Health 2011, 11(Suppl 1):S3http://www.biomedcentral.com/1471-2458-11-S1-S3Page 4 of 5the staged progression model (12). There is no difficultyin extending to staged progression models with arbitra-rily distributed length of stay in each stage. The meantime in stage j replaces 1/aj in each estimate. We havenot carried out the calculations here because of thetechnical complications in writing the model equations,but these may be found in [3].ConclusionsBehavioural changes are an essential aspect of thecourse of an epidemic. The changes in behaviour byinfectious members of a population have differentcauses than the changes in behaviour by uninfectedmembers, and a model incorporating behaviouralchanges should reflect this. One implication is that amodel incorporating behavioural changes must includeheterogeneous mixing. One consequence of this is thatthe final size of an epidemic can not be determinedexactly from a final size relation but can only beapproximated. There is an effective reproduction num-ber which is less than the basic reproduction number inmany cases but not necessarily always.Epidemic models with age structure or other heteroge-neities in mixing can also be extended to incorporatebehavioural changes. This would result in models withcomplicated mixing behaviour that would be difficult toanalyze. There would be a system of final size equationswhich could not be solved exactly, but would still yieldfinal size estimates. An important question that has notyet been attacked is the formulation of models thatinclude behavioural responses, especially by uninfectedmembers of the population, that depend on the stateand history of the epidemic.AcknowledgementsThis article has been published as part of BMC Public Health Volume 11Supplement 1, 2011: Mathematical Modelling of Influenza. The full contentsof the supplement are available online at http://www.biomedcentral.com/1471-2458/11?issue=S1.Competing interestsThe author declares that he has no competing interests.Published: 25 February 2011References1. Bootsma MCJ, Ferguson NM: The effect of public health measures on the1918 influenza pandemic in U.S. cities. Proc Natl Acad Sci U S A. 2007,104(18):7588-7593.2. Brauer F: Age-of-infection and the final size relation. Math Biosci Eng.2008, 5(4):681-690.3. Brauer F, Castillo-Chavez C, Feng Z: Discrete epidemic models. Math BiosciEng. 2010, 7(1):1-15.4. del Valle S, Hethcote HW, Hyman JM, Castillo-Chavez C: Effects ofbehavioral changes in a smallpox attack model. Math Biosci. 2005,195(2):228-251.5. Kermack WO, McKendrick AG: A contribution to the mathematical theoryof epidemics. Proc. Royal Soc. London 1927, 115:700-721.6. Ma J, Earn DJD: Generality of the final size formula for an epidemic of anewly invading infectious disease. Bull. Math. Biol. 2006, 68:679-702.7. Van den Driessche P, Watmough J: Reproduction numbers andsubthreshold endemic equilibria for compartmental models of diseasetransmission. Math Biosci. 2000, 180:29-48.doi:10.1186/1471-2458-11-S1-S3Cite this article as: Brauer: A simple model for behaviour change inepidemics. BMC Public Health 2011 11(Suppl 1):S3.Submit your next manuscript to BioMed Centraland take full advantage of: • Convenient online submission• Thorough peer review• No space constraints or color figure charges• Immediate publication on acceptance• Inclusion in PubMed, CAS, Scopus and Google Scholar• Research which is freely available for redistributionSubmit your manuscript at www.biomedcentral.com/submitBrauer BMC Public Health 2011, 11(Suppl 1):S3http://www.biomedcentral.com/1471-2458-11-S1-S3Page 5 of 5


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