A REACTION-DIFFUSION MODEL FOR NESTED WITHIN-HOST AND BETWEEN-HOST DYNAMICS IN AN ENVIRONMENTALLY-DRIVEN INFECTIOUS DISEASE

Abstract

<abstract> A reaction-diffusion model for nested within-host and between-host dynamics in an environmentally-driven infectious disease is proposed. The model is composed of the within-host virus infectious fast time model of ordinary differential equations and the between-host disease transmission slow time model of reaction-diffusion equations. The isolated fast model has been investigated in previous literature, and the main results are summarized. For the isolated slow model, the well-posedness of solutions, and the basic reproduction number $ R_{b} $ are obtained. When $ R_{b}\leq1 $, the model only has the disease-free equilibrium which is globally asymptotically stable, and when $ R_{b}&gt;1 $ the model has a unique endemic equilibrium which is globally asymptotically stable. For the nested slow model, the positivity and boundedness of solutions, the basic reproduction number $ R_{c} $ and the existence of equilibrium are firstly obtained. Particularly, the nested slow model can exist two positive equilibrium when $ R_{c}&lt;1 $ and a unique endemic equilibrium when $ R_{c}&gt;1 $. When $ R_{c}&lt;1 $ the disease-free equilibrium is locally asymptotically stable, and when $ R_{c}&gt;1 $ and an additional condition is satisfied the unique endemic equilibrium is locally asymptotically stable. When there are two positive equilibria, then a positive equilibria is locally asymptotically stable under an additional condition and the other one is unstable, which implies that the nested slow model occurs the backward bifurcation at $ R_c = 1 $. Lastly, numerical examples are given to verify the main conclusions. The research shows that the nested slow model has more complex dynamical behavior than the corresponding isolated slow model. </abstract>