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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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an in-host model for the viral dynamics of HIV-1 infection and its interaction with the CTL immune response. The model is sufficiently general to allow nonlinear forms for both viral infection and CTL response. Threshold parameters are identified that completely determine the global dynamics and outcomes of the virus-target cell-CTL interactions. Impacts of key parameter values for CTL functions and viral budding rate on the HIV-1 viral load and CD4 count are investigated using numerical simulations. Results support clinical evidence for important differences between HIV-1 nonprogressors and progressors. [ABSTRACT FROM AUTHOR]

In this paper, we derive and analyze a state-structured epidemic model for infectious diseases in which the state structure is nonlocal. The state is a measure of infectivity of infected individuals or the intensity of viral replications in infected cells. The model gives rise to a system of nonlinear integro-differential equations with a nonlocal term. We establish the well-posedness and dissipativity of the associated nonlinear semigroup. We establish an equivalent principal spectral condition between the linearized operator and the next-generation operator and show that the basic reproduction number R 0 is a sharp threshold: if R 0 < 1 , the disease-free equilibrium is globally asymptotically stable, and if R 0 > 1 , the disease-free equilibrium is unstable and a unique endemic equilibrium is globally asymptotically stable. The proof of global stability of the endemic equilibrium utilizes a global Lyapunov function whose construction was motivated by the graph-theoretic method for coupled systems on networks developed in [24] . [ABSTRACT FROM AUTHOR]

Population migration and immigration have greatly increased the spread and transmission of many infectious diseases at a regional, national and global scale. To investigate quantitatively and qualitatively the impact of migration and immigration on the transmission dynamics of infectious diseases, especially in heterogeneous host populations, we incorporate immigration/migration terms into all sub-population compartments, susceptible and infected, of two types of well-known heterogeneous epidemic models: multi-stage models and multi-group models for HIV/AIDS and other STDs. We show that, when migration or immigration into infected sub-population is present, the disease always becomes endemic in the population and tends to a unique asymptotically stable endemic equilibrium P*. The global stability of P* is established under general and biological meaningful conditions, and the proof utilizes a global Lyapunov function and the graph-theoretic techniques developed in Guo et al. (2008). [ABSTRACT FROM AUTHOR]

We propose a general class of multistage epidemiological models that allow possible deterioration and amelioration between any two infected stages. The models can describe disease progression through multiple latent or infectious stages as in the case of HIV and tuberculosis. Amelioration is incorporated into the models to account for the effects of antiretroviral or antibiotic treatment. The models also incorporate general nonlinear incidences and general nonlinear forms of population transfer among stages. Under biologically motivated assumptions, we derive the basic reproduction number R0 and show that the global dynamics are completely determined by R0: if R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if R0 > 1, then the disease persists in all stages and a unique endemic equilibrium is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

Abstract: We consider a mathematical model for HIV/AIDS that incorporates staged progression and amelioration. Amelioration as a result of HAART treatment is allowed to occur across any number of stages. The global dynamics are completely determined by the basic reproduction number . If , then the disease-free equilibrium (DFE) is globally asymptotically stable and the disease always dies out. If , DFE is unstable and a unique endemic equilibrium (EE) is globally asymptotically stable, and the disease persists at the endemic equilibrium. The proof of global stability utilizes a global Lyapunov function. [Copyright &y& Elsevier]

The dynamics of an in-host model with general form of target-cell dynamics, nonlinear incidence, and distributed delay are investigated. The model can describe the in vivo infection dynamics of many viruses such as HIV-I, HCV, and HBV. We derive the basic reproduction number R0 for the viral infection and establish that the global dynamics are completely determined by the values of R0: if R0 ≤ 1, the infection-free equilibrium is globally asymptotically stable, and the virus is cleared; if R0 > 1, then the infection persists, and the chronic-infection equilibrium is globally asymptotically stable. An implication of our results is that intracellular delays will lead to periodic oscillations in in-host models only with the right kind of target-cell dynamics. [ABSTRACT FROM AUTHOR]

Abstract: The global-stability problem of equilibria is investigated for coupled systems of differential equations on networks. Using results from graph theory, we develop a systematic approach that allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. The approach is applied to several classes of coupled systems in engineering, ecology and epidemiology, and is shown to improve existing results. [Copyright &y& Elsevier]

Abstract: We investigate a class of multi-group epidemic models with distributed delays. We establish that the global dynamics are completely determined by the basic reproduction number . More specifically, we prove that, if , then the disease-free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium and it is globally asymptotically stable. Our proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach to the method of Lyapunov functionals. [Copyright &y& Elsevier]

The state of an infectious disease can represent the degree of infectivity of infected individuals, or susceptibility of susceptible individuals, or immunity of recovered individuals, or a combination of these measures. When the disease progression is long such as for HIV, individuals often experience switches among different states. We derive an epidemic model in which infected individuals have a discrete set of states of infectivity and can switch among different states. The model also incorporates a general incidence form in which new infections are distributed among different disease states. We discuss the importance of the transmission–transfer network for infectious diseases. Under the assumption that the transmission–transfer network is strongly connected, we establish that the basic reproduction number R 0 is a sharp threshold parameter: if R 0 ≤ 1 , the disease-free equilibrium is globally asymptotically stable and the disease always dies out; if R 0 > 1 , the disease-free equilibrium is unstable, the system is uniformly persistent and initial outbreaks lead to persistent disease infection. For a restricted class of incidence functions, we prove that there is a unique endemic equilibrium and it is globally asymptotically stable when R 0 > 1. Furthermore, we discuss the impact of different state structures on R 0 , on the distribution of the disease states at the unique endemic equilibrium, and on disease control and preventions. Implications to the COVID-19 pandemic are also discussed. • A discrete state-structured epidemic model was developed for HIV and Tuberculosis. • The model was rigorously analysed and investigated the global dynamics. • Graph-theoretic approach to Lyapunov functions was further improved and adapted. • Some interesting impacts of state structures were discussed by numerical simulations. • Main results include as special cases earlier results on multistage epidemic models. [ABSTRACT FROM AUTHOR]