## Abstract

We investigate an 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, \(D_{R}\) 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 an 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.

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## Notes

- 1.
Let \(t=\frac{{\hat{t}}}{\nu },S=\sqrt{\frac{\nu }{\chi }}{\hat{S}},I=\sqrt{\frac{\nu }{\chi }}{\hat{I}},R=\sqrt{\frac{\nu }{\chi }} {\hat{R}}\) and define new parameters by \(\gamma =\nu {\hat{\gamma }},D_{S}=\nu \hat{D_{S}},D_{I}=\nu \varepsilon ^{2},D_{R}=\nu \hat{D_{R}}.\) Upon dropping the hats, this yields (3).

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## Acknowledgements

We would like to thank an anonymous referees for many useful suggestions which improved the exposition significantly. T.K. and D.I are funded by NSERC discovery grant. C.G. is funded in part by Nova Scotia Graduate Scholarship.

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Gai, C., Iron, D. & Kolokolnikov, T. Localized outbreaks in an S-I-R model with diffusion.
*J. Math. Biol.* (2020). https://doi.org/10.1007/s00285-020-01466-1

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### Mathematics Subject Classification

- 35B36
- 35Q92
- 92D25