Qualitative Analysis Of An Age-structured Influenza-epidemiological Model

In this paper, we study an age-structured in?uenza-epidemiological model, basedon the model in [23], adding the individuals in the latent period, we obtain a nonlinearsystem with three equations, in which have two thresholdsÏ„₁,Ï„₂ on the susceptibleclass and the individuals in the latent period respectively. For this model, we ?rst makesome necessary simpli?cations, transform it into a non-densely de?ned Cauchy prob-lem, using operator semigroup theory to justify the global existence and uniqueness ofthe positive solutions, and then discuss the conditions for the existence of equilibriumsolutions, so we know that whenδ>ν, the system has an unique positive equilibriumsolution, we calculate the linearized equation of the system around the equilibriumsolution and some relevant linear operators, deducing the characteristic equation. Bystudying the characteristic equation, we analysis the stability of equilibrium solution,?nd that when thresholdsÏ„₁ =Ï„₂ = 0 orÏ„₁ = 0Ï„₂ > 0, equilibrium solution ofthe system is locally asymptotically stable; whenÏ„₁ > 0,Ï„₂ = 0, we get the resultsin [23]; whenÏ„₁ =Ï„₂ > 0, we apply recent established by Pierre Megal–center mani-fold theory for semilinear equations with non-dense domain[12] and Hopf bifurcationtheory[13], we show that Hopf bifurcation occurs in the model, producing non-trivialperiodic solution. This demonstrates that the age-structured in?uenza-epidemiologicalmodel which considers the latent class has an intrinsic tendency to oscillate, also showsthe sensitivity of the model dynamics on the thresholdsÏ„₁ andÏ„₂, generalizing the re-sults in.