A discrete state-structured model on networks with two transmission modes: Global dynamics analysis

Abstract

In this paper, we formulate and analyze a class of discrete state-structured epidemic models that spread through both horizontal and vertical transmissions on networks, where infected individuals can move from one infected state to any other state so that our models include all possible state-transfers (disease deterioration and amelioration) among different states. Many epidemic transmissions with or without vertical transmission in nature can be analyzed by referring to our models, such as HIV-1, viral hepatitis, and Covid-19. We derive the basic reproduction number $ \mathcal{R}_0 $ = $ \mathcal{R}_h $+ $ \mathcal{R}_v $, and prove that the global dynamics are completely determined by the basic reproduction number: if $ \mathcal{R}_0\leq 1 $, the disease-free equilibrium is globally asymptotically stable and the disease always dies out; if $ \mathcal{R}_0>1 $, the disease-free equilibrium is unstable, and there exists a unique endemic equilibrium that is globally asymptotically stable, and the disease persists at a positive level in the population. It also implies that vertical transmission has an impact on maintaining infectious diseases when horizontal transmission cannot sustain the disease on its own. The proof of global stability is based on the graph-theoretic approach and answer the open problem left in [1]. Finally, numerical simulations are performed to illustrate the theoretical results.