Spatial Segregation in Reaction-Diffusion Epidemic Models

Abstract

.In this paper, we formulate susceptible-infected-susceptible reaction-diffusion epidemic models with cognition and show the impact of movement strategies on disease outbreak and mitigation under a spatially heterogeneous environment. The cognitive diffusion either takes a Fokker–Planck type diffusion obtained by Chapman's diffusion law (called random diffusion) or follows Fick's diffusion law (called symmetric diffusion). We derive a variational expression of the basic reproduction number \(\mathcal{R}_0\) for both models and prove that the disease-free equilibrium is unique and globally asymptotically stable if \(\mathcal{R}_0\lt 1\) . Furthermore, if \(\mathcal{R}_0\gt 1\) , the model following Fick's diffusion law admits at least one endemic equilibrium and the model following Chapman's diffusion law has a unique endemic equilibrium. The theoretical results are illustrated by numerical simulations, which additionally show the segregation phenomenon between susceptible and infected populations regulated by different movement strategies. Spatial segregation here is natural, not caused by an isolation policy, and thus is the most important indicator for an infectious disease to spread or wane in the absence of intervention. The first example shows that a heterogeneous random diffusion segregates infected and susceptible populations further than an ODE model and thus reduces the infection size. However, symmetric diffusion never does that. The second example shows that a heterogeneous random diffusion detriments segregation but still reduces the infection severity by moving infected individuals to a disease-free region. In a certain situation, a heterogeneous random diffusion may increase the infection severity as shown in the last example.Keywordsspatial heterogeneitycognitive movementbasic reproduction numberdisease-free equilibriumendemic equilibriumsegregationMSC codes35K5792D3091D2537N25

Topics & Concepts