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Global behavior of a multi-group SIS epidemic model with age structure. (English) Zbl 1083.35020
The authors consider the following system of integro-partial differential equations with non-local boundary conditions, describings the dynamics of the transmission of an SIS age-structured epidemic model for a disease that is vertically as well as horizontally transmitted: \begin{aligned} \left ( \frac {\partial}{\partial t}+ \frac {\partial}{\partial a}\right)s_i(t,a)&= - \mu_i (a)s_i(t,a)-\Lambda_i (a, u(t,.))s_i (t,a) + \gamma_i (a) u_i (t,a),\;a>0, t>0,\\ \left ( \frac {\partial}{\partial t}+ \frac {\partial}{\partial a}\right)u_i(t,a)&= - \mu_i (a)u_i(t,a)+\Lambda_i (a, u(t,.))s_i (t,a) - \gamma_i (a) u_i (t,a),\;a>0, t>0,\\ s_i (t,0)&= \int^\omega_0 b_i (a)\left[ s_i (t,a)+(1-q)u_i (t,a)\right]da, \quad t>0,\\ u_i (t,0)&=q_i\int^\omega_0 b_i (a)u_i (t,a)da,\quad 0<q<1,\;t>0, \\ s_i (0,a)&= \psi_i (a),\quad u_i (0,a) = \varphi_i (a),\quad i=1,2,\dots,n, \;a\geq 0, \end{aligned} where $$s_i (t,a), u_i (t,a)$$ are, respectively, the age-density for susceptible and infective individuals at time t. $$\Lambda_i (t,u(t,.)):= K_i (a) u_i (t,a) + \sum_{j=1}^n \int^\omega_0 K_{ij}(a,s)u_j (t,s)\,ds$$ is the force of infection, $$\gamma_i (a)$$ is the cure rate, $$b_i (a)$$ is the birth rate, $$\mu_i (a)$$ is the death rate, $$q_i \in (0,1)$$ is the vertical transmission parameter, and $$\psi_i (a), \varphi_i (a)$$ are the initial age-distributions.
The main purpose of the paper is to improve the existing stability results for the SIS age-structured epidemic model and generalize the single group results to multiple groups. The authors use semigroup theory and for, appropriate initial age-distributions, obtain convergence results for either the disease-free equilibrium or the endemic equilibrium, in terms of the dominant eigenvalue of an operator.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 92D25 Population dynamics (general) 45K05 Integro-partial differential equations 47D06 One-parameter semigroups and linear evolution equations 45M05 Asymptotics of solutions to integral equations 47N60 Applications of operator theory in chemistry and life sciences 47N20 Applications of operator theory to differential and integral equations 35L40 First-order hyperbolic systems 35R10 Partial functional-differential equations
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