ALBERT

Description: 〈p〉Publication date: 15 November 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 265, Issue 10〈/p〉 〈p〉Author(s): Zhipeng Qiu, Michael Y. Li, Zhongwei Shen〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we derive and analyze a state-structured epidemic model for infectious diseases in which the state structure is nonlocal. The state is a measure of infectivity of infected individuals or the intensity of viral replications in infected cells. The model gives rise to a system of nonlinear integro-differential equations with a nonlocal term. We establish the well-posedness and dissipativity of the associated nonlinear semigroup. We establish an equivalent principal spectral condition between the linearized operator and the next-generation operator and show that the basic reproduction number 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is a sharp threshold: if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉, the disease-free equilibrium is globally asymptotically stable, and if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉, the disease-free equilibrium is unstable and a unique endemic equilibrium is globally asymptotically stable. The proof of global stability of the endemic equilibrium utilizes a global Lyapunov function whose construction was motivated by the graph-theoretic method for coupled systems on networks developed in [24].〈/p〉〈/div〉