Global asymptotic stability in a pseudorabies virus model with age structure

Pseudorabies is a highly contagious disease caused by pseudorabies virus (PRV) or suid herpesvirus 1 (SuHV1), causing significant economic losses to the swine industry in countries where the disease exists. In this paper, we formulate an age structure model of pseudorabies virus that takes into account disease-related mortality and vertical transmission. We find a threshold to determine the stability and existence of the disease. We show that there is always a globally asymptotically stable boundary equilibrium if and only if $R_{02}<1+\theta$, which means that the disease always exists in piglets and will die out in adult pigs. When $R_{02}>1+\theta$, the boundary equilibrium is unstable and there exists a unique disease-endemic equilibrium, which is globally asymptotically stable. We give detailed proofs of our theoretical results and numerical examples. Brief concluding remarks are also provided.