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Existence and asymptotic profiles of the steady state for a diffusive epidemic model with saturated incidence and spontaneous infection mechanism

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    * Corresponding author
Partially supported by National Natural Science Foundation of China (No. 12171125), Natural Science Foundation of Heilongjiang Province (No. LH2020A012), Heilongjiang Postdoctoral Science Foundation (No. LBH-Q17086) and the Fundamental Research Funds for Heilongjiang Provincial Universities (No. 2018-KYYWF-0996)
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  • In this paper, we are concerned with a reaction-diffusion SIS epidemic model with saturated incidence rate, linear source and spontaneous infection mechanism. We derive the uniform bounds of parabolic system and obtain the global asymptotic stability of the constant steady state in a homogeneous environment. Moreover, the existence of the positive steady state is established. We mainly analyze the effects of diffusion, saturation and spontaneous infection on the asymptotic profiles of the steady state. These results show that the linear source and spontaneous infection can enhance the persistence of an infectious disease. Our mathematical approach is based on topological degree theory, singular perturbation technique, the comparison principles for elliptic equations and various elliptic estimates.

    Mathematics Subject Classification: 35K57, 35B40, 35J57, 92D25.


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