Open AccessThe risk index for an SIR epidemic model and spatial spreading of the infectious disease
Author(s) -
Min Zhu,
Xiaofei Guo,
Zhigui Lin
Publication year - 2017
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2017081
Subject(s) - basic reproduction number , advection , epidemic model , dirichlet boundary condition , index (typography) , boundary (topology) , disease , infectious disease (medical specialty) , boundary value problem , epidemic disease , diffusion , disease transmission , statistical physics , mathematics , econometrics , computer science , biology , mathematical analysis , physics , medicine , virology , environmental health , population , pathology , world wide web , thermodynamics
In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number RDA0 for an associated model with Dirichlet boundary condition, we introduce the risk index RF0(t) for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if RF0(t0) ≤ 1 for some t0 and the disease is vanishing if RF0(∞) < 1, while if RF0 (0) < 1, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.