Threshold and stability results for an age-structured epidemic model (Q1174280)

The author treats the following age-structured SIR-type epidemic model: \[ S_ t(a,t)+S_ a(a,t)=-[\lambda(a,t)+\mu(a)]S(a,t),\quad I_ t(a,t)+I_ a(a,t)=\lambda(a,t)S(a,t)-[\mu(a)+\gamma]I(a,t), \] \[ S(0,t)=N,\quad I(0,t)=0=S(a,0)=S_ 0(a),\quad I(a,0)=I_ 0(a). \] Here \(\int^ \omega_ 0 S(a,t)da\) is the total population of susceptibles at time \(t\), \(\int^ \omega_ 0 I(a,t)da\) is the total population of infectives at time \(t\), \(\lambda(a,t)=\int^ \omega_ 0 \beta(a,\sigma)I(\sigma,t)d\sigma\) is the force of infection, and \(N\) is the constant birth rate of susceptibles. It is assumed that a susceptible who contacts the disease will eventually recover with permanent immunity. The model is formulated as an abstract Cauchy problem in a Banach space and existence and uniqueness of solutions is established. Conditions are derived for the existence and uniqueness of nontrivial steady states and their stability is analyzed using a linearization method. A threshold condition is established in terms of the spectral radius of a certain operator for the existence and stability of an endemic equilibrium. The author uses operator semigroup methods in this analysis.