Dynamics and Bifurcations in a Nondegenerate Homogeneous Diffusive SIR Rabies Model

SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 632-660, April 2024.
Abstract. In this paper, we are interested in the spatiotemporal pattern formations and bifurcations for a nondegenerate reaction-diffusion rabies SIR model which was used to explain the epidemiological patterns observed in Europe. First, by using the iteration methods, we are able to show the global existence and boundedness of in-time solutions of the parabolic system. Second, for the ODEs, we analytically prove the phenomena observed by Anderson et al. [Nature, 289 (1981), pp. 765–771]: if the carrying capacity [math] is smaller than some positive [math], then rabies eventually dies out; if [math] is larger than [math], then the rabies prevails. Moreover, if [math] for some positive [math], then the endemic equilibrium solution is (locally asymptotically) stable, while it is unstable if [math]. In particular, at [math], the loss of the stability of the endemic equilibrium leads to a Hopf bifurcation. Finally, for the PDEs, we derive sufficient conditions on the diffusion rates so that under these conditions, Turing instability of both the endemic equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions can occur. Once Turing instability of the solution (equilibrium or periodic solution) occurs, it is observed numerically that the system might have new spatiotemporal patterns.