Partial Differential Equations of an Epidemic Model with Spatial Diffusion

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

[1]  V. Capasso.  Mathematical Structures of Epidemic Systems , 1993, Lecture Notes in Biomathematics.

[2]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[3]  Chris Cosner,et al.  On the Dynamics of Predator–Prey Models with the Beddington–DeAngelis Functional Response☆ , 2001 .

[4]  Shouming Zhong,et al.  Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response , 2013 .

[5]  David Abend,et al.  Maximum Principles In Differential Equations , 2016 .

[6]  K. Hattaf,et al.  A delayed SIR epidemic model with a general incidence rate , 2013 .

[7]  R. Ruth,et al.  Stability of dynamical systems , 1988 .

[8]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[9]  Nicholas D. Alikakos,et al.  An application of the invariance principle to reaction-diffusion equations , 1979 .

[10]  J. Beddington,et al.  Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency , 1975 .

[11]  Xueyong Zhou,et al.  GLOBAL STABILITY OF THE VIRAL DYNAMICS WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE , 2011 .

[12]  Donald L. DeAngelis,et al.  A Model for Tropic Interaction , 1975 .

[13]  Donald L. DeAngelis,et al.  A MODEL FOR TROPHIC INTERACTION , 1975 .

[14]  Philip H. Crowley,et al.  Functional Responses and Interference within and between Year Classes of a Dragonfly Population , 1989, Journal of the North American Benthological Society.

[15]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[16]  Khalid Hattaf,et al.  Global stability for reaction-diffusion equations in biology , 2013, Comput. Math. Appl..