Abstract
A strongly coupled cooperative parabolic system, which describes fecally-orally epidemic model with cross-diffusion in a heterogeneous environment, was formulated and analyzed. The basic reproduction number R_{0} ^{D}, which serves as a threshold parameter that predicts whether the coexistence will exist or not, is introduced by the next infection operator and the related eigenvalue problems. By applying upper and lower solutions method, we present the sufficient conditions for the existence of the coexistence solution. The true positive solutions can also be obtained by monotone iterative method. Our results imply that the fecally-orally epidemic model with cross-diffusion admits at least one coexistence solution when the basic reproduction number exceeds one and the cross-diffusion coefficient is sufficiently small, while no coexistence exists when the basic reproduction number is smaller than one or the cross-diffusion coefficient is large enough. Finally, some numerical simulations are exhibited to confirm our analytical findings.
Highlights
Since the most influential and theoretical model, the SIR model, was formulated by Kermack and McKendrick in, the geographic transmission of infectious diseases has been becoming an important issue in mathematical epidemiology
D, d, a, a and a are the positive constants, u(x, t) and v(x, t), respectively, represent the spatial densities of the bacterial population and of the human population infected by the bacteria at location x in the habitat and at time t is the mean lifetime of the agent in the environment
In present paper, based on the model in [ ], we will focus on the following cross-diffusion epidemic model in a spatially heterogeneous environment with Dirichlet boundary condition:
Summary
Since the most influential and theoretical model, the SIR model, was formulated by Kermack and McKendrick in , the geographic transmission of infectious diseases has been becoming an important issue in mathematical epidemiology. In the past few years, a great deal of mathematical models have been developed to investigate the impact of diffusion and spatial heterogeneity on the dynamics of diseases [ ,. In ecology, different concentration levels of species can affect the diffusive direction of another interacting species, which is called cross-diffusion [ ]. This is another hot issue and attracts much attention in recent years; see [ – ] and the references therein. In present paper, based on the model in [ ], we will focus on the following cross-diffusion epidemic model in a spatially heterogeneous environment with Dirichlet boundary condition:.
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