Localized outbreaks in S-I-R model with diffusion

We investigate a SIRS epidemic model with spatial diffusion and nonlinear incidence rates. We show that for small diffusion rate of the infected class D_I, the infected population tends to be highly localized at certain points inside the domain, forming K spikes. We then study three distinct destabilization mechanisms, as well as a transition from localized spikes to plateau solutions. Two of the instabilities are due to coarsening (spike death) and self-replication (spike birth), and have well-known analogues in other reaction-diffusion systems such as the Schnakenberg model. The third transition is when a single spike becomes unstable and moves to the boundary. This happens when the diffusion of the recovered class, DR becomes sufficiently small. In all cases, the stability thresholds are computed asymptotically and are verified by numerical experiments. We also show that the spike solution can transit into a plateau-type solution when the diffusion rates of recovered and susceptible class are sufficiently small. Implications for disease spread and control through quarantine are discussed.