An infection age-space structured SIR epidemic model with Neumann boundary condition

ABSTRACT

In this paper, we are concerned with an SIR epidemic model with infection age and spatial diffusion in the case of Neumann boundary condition. The original model is constructed as a nonlinear age structured system of reaction–diffusion equations. By using the method of characteristics, we reformulate the model into a system of a reaction–diffusion equation and a Volterra integral equation. For the reformulated system, we define the basic reproduction number ${\mathcal{R}}_0$ by the spectral radius of the next generation operator, and show that if ${\mathcal{R}}_0<1$, then the trivial disease-free steady state is globally attractive, whereas if ${\mathcal{R}}_0>1$, then the disease in the system is persistent. Moreover, under an additional assumption that there exists a finite maximum age of infectiousness, we show the global attractivity of a constant endemic steady state for ${\mathcal{R}}_0>1$.

KEYWORDS:

• SIR epidemic model
• infection age
• diffusion
• basic reproduction number
• global attractivity

AMS CLASSIFICATIONS:

• 35Q92
• 37N25
• 92D30

Acknowledgement

The authors are deeply grateful to the editor and the anonymous reviewers for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Japan Society for the Promotion of Science under Grant-in-Aid for Young Scientists (B) [No. 15K17585] and the Japan Agency for Medical Research and Development, AMED under the program of the Japan Initiative for Global Research Network on Infectious Diseases (J-GRID).