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BIRS Workshop Lecture Videos

Existence and stability of spike solution in SIRS model with diffusion Gai, Chunyi


We investigate an SIRS epidemic PDE system with nonlinear incident rates. In the limit of small diffusion rate of infected class $D_I$, and on a finite interval, an equilibrium spike solution (or epidemic hotspot) to the epidemic model is constructed asymptotically and the motion of the spike is studied. For sufficiently large diffusion rate of recovered class $D_R$, the interior spike is shown to be stable, however it becomes unstable and moves to the boundary when $D_R$ is sufficiently small. We also studied two types of bifurcation behavior of multi-spike solutions: self-replication and spike competition, and their stability thresholds are precisely computed by asymptotic analysis and verified by numerical experiments. Finally, we show that the spike-type solution can transition into an interface-type solutions when the diffusion rates of recovered and susceptible class are sufficiently small, and the transition regime is obtained precisely.

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