Global behavior of a multi-group SIS epidemic model with age structure.

*(English)*Zbl 1083.35020The 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.

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.

Reviewer: Mohamed O. El-Doma (Beirut)

##### 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 |

##### Keywords:

quasi-irreducibility; multiple group model; dominant eigenvalue; horizontal transmission; vertical transmission; non-local boundary conditions
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\textit{Z. Feng} et al., J. Differ. Equations 218, No. 2, 292--324 (2005; Zbl 1083.35020)

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