Mathematical analysis of a two-patch model of tuberculosis disease with staged progression

The spread of tuberculosis is studied through a two-patch epidemiological system SE₁ ? E_nI which incorporates migrations from one patch to another just by susceptible individuals. Our model is consider with bilinear incidence and migration between two patches, where infected and infectious individuals cannot migrate from one patch to another, due to medical reasons. The existence and uniqueness of the associated endemic equilibria are discussed. Quadratic forms and Lyapunov functions are used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium (DFE) is globally asymptotically stable, and when it is greater than one there exists in each case a unique endemic equilibrium (boundary equilibria and endemic equilibrium) which is globally asymptotically stable. Numerical simulation results are provided to illustrate the theoretical results.     

Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate

In this paper, an SVEIS epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated. The role that temporary immunity (natural, disease induced, vaccination induced) plays in the spread of disease, is incorporated in the model. The total host population is bounded and the incidence term is of the Holling-type II form. It is shown that the model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. The global dynamics are completely determined by the basic reproduction number R₀. If R₀<1, the disease-free equilibrium is globally stable which leads to the eradication of disease from population. If R₀>1, a unique endemic equilibrium exists and is globally stable in the feasible region under certain conditions. Further, the transcritical bifurcation at R₀=1 is explored by projecting the flow onto the extended center manifold. We use the geometric approach for ordinary differential equations which is based on the use of higher-order generalization of Bendixson’s criterion. Further, we obtain the threshold vaccination coverage required to eradicate the disease. Finally, taking biologically relevant parametric values, numerical simulations are performed to illustrate and verify the analytical results.     

The effect of population dispersal on the spread of a disease

The effect of population dispersal among n patches on the spread of a disease is investigated. Population dispersal does not destroy the uniqueness of a disease free equilibrium and its attractivity when the basic reproduction number of a disease R₀<1. When R₀>1, the uniqueness and global attractivity of the endemic equilibrium can be obtained if dispersal rates of susceptible individuals and infective individuals are the same or very close in each patch. However, numerical calculations show that population dispersal may result in multiple endemic equilibria and even multi-stable equilibria among patches, and also may result in the extinction of a disease, even though it cannot be eradicated in each isolated patch, provided the basic reproduction numbers of isolated patches are not very large.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

Threshold of disease transmission in a patch environment

An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that reproduction number is a threshold of the uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence the persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when the population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when the population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen. It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.     

On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

Birds movement impact on the transmission of West Nile virus between patches

Spatial heterogeneity plays an important role in the distribution and persistence of many infectious disease. In the paper, a multi-patch model for the spread of West Nile virus among $n$ discrete geographic regions is presented that incorporates a mobility process. In the mobility process, we assume that the birds can move among regions, but not the mosquitoes based on scale-space. We show that the movement of birds between patches is sufficient to maintain disease persistence in patches. We compute the basic reproduction number $R_{0}$. We prove that if $R_{0}<1$, then the disease-free equilibrium of the model is globally asymptotically stable. When $R_{0}>1$, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between two regions.     

Global stability for an HIV/AIDS epidemic model with different latent stages and treatment

An HIV/AIDS epidemic model with different latent stages and treatment is constructed. The model allows for the latent individuals to have the slow and fast latent compartments. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are determined by the basic reproduction number under some conditions. If R₀ < 1, the disease free equilibrium is globally asymptotically stable, and if R₀ > 1, the endemic equilibrium is globally asymptotically stable for a special case. Some numerical simulations are also carried out to confirm the analytical results.     

Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates

In this paper, we present a new delay multigroup SEIR model with group mixing and nonlinear incidence rates and investigate its global stability. We establish that the global dynamics of the models are completely determined by the basic reproduction number R₀. It is shown that, if R₀?1, then the disease free equilibrium is globally asymptotically stable and the disease dies out; if R₀>1, there exists a unique endemic equilibrium that is globally asymptotically stable and thus the disease persists in the population. Finally, a numerical example is also discussed to illustrate the effectiveness of the results.     

Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity

In this paper, some SEIRS epidemiological models with vaccination and temporary immunity are considered. First of all, previously published work is reviewed. In the next section, a general model with a constant contact rate and a density-dependent death rate is examined. The model is reformulated in terms of the proportions of susceptible, incubating, infectious, and immune individuals. Next the equilibrium and stability properties of this model are examined, assuming that the average duration of immunity exceeds the infectious period. There is a threshold parameter R_o and the disease can persist if and only if R_o exceeds one. The disease-free equilibrium always exists and is locally stable if R_o < 1 and unstable if R_o > 1. Conditions are derived for the global stability of the disease-free equilibrium. For R_o > 1, the endemic equilibrium is unique and locally asymptotically stable.For the full model dealing with numbers of individuals, there are two critical contact rates. These give conditions for the disease, respectively, to drive a population which would otherwise persist at a finite level or explode to extinction and to cause a population that would otherwise explode to be regulated at a finite level. If the contact rate β(N) is a monotone increasing function of the population size, then we find that there are now three threshold parameters which determine whether or not the disease can persist proportionally. Moreover, the endemic equilibrium need no longer be locally asymptotically stable. Instead stable limit cycles can arise by supercritical Hopf bifurcation from the endemic equilibrium as this equilibrium loses its stability. This is confirmed numerically.     

Modeling diseases with latency and nonlinear incidence rates: global dynamics of a multi‐group model

In this paper, we perform global stability analysis of a multi‐group SEIR epidemic model in which we can consider the heterogeneity of host population and the effects of latency and nonlinear incidence rates. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the basic reproduction number is defined by the theory of the next generation operator and proved to be a sharp threshold determining whether or not disease spread. Under certain assumptions, the disease‐free equilibrium is globally asymptotically stable if R₀≤1 and there exists a unique endemic equilibrium which is globally asymptotically stable if R₀>1. The proofs of global stability of equilibria exploit a matrix‐theoretic method using Perron eigenvetor, a graph‐theoretic method based on Kirchhoff's matrix tree theorem and Lyapunov functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

Spread of disease in a patchy environment

For a two patches SIR model, it is shown that its dynamic behavior is determined by several quantities. We have shown that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable, otherwise it is unstable. Some sufficient conditions for the local stability of boundary equilibria are obtained. Numerical simulations indicate that travel between patches can reduces oscillations in both patches; may enhances oscillations in both patches; or travel switches oscillations from one patch to another.     

The analysis and application of an HBV model

A mathematical model is formulated to describe the spread of hepatitis B. The stability of equilibria and persistence of disease are analyzed. The results shows that the dynamics of the model is completely determined by the basic reproductive number ρ₀. If ρ₀ < 1, the disease-free equilibrium is globally stable. When ρ₀ > 1, the disease-free equilibrium is unstable and the disease is uniformly persistent. Furthermore, under certain conditions, it is proved that the endemic equilibrium is globally attractive. Numerical simulations are conducted to demonstrate our theoretical results. The model is applied to HBV transmission in China. The parameter values of the model are estimated based on available HBV epidemic data in China. The simulation results matches the HBV epidemic data in China approximately.     

Analysis of a delayed HIV/AIDS epidemic model with saturation incidence

In this paper, a delayed HIV/AIDS epidemic model with saturation incidence is proposed and analyzed. The equilibria and their stability are investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is found that if the threshold R ₀<1, then the disease-free equilibrium is globally asymptotically stable, and if the threshold R ₀>1, the system is permanent and the endemic equilibrium is asymptotically stable under certain conditions.     

Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R₀. It shows that, the basic reproduction number R₀ is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.     

Global stability of a stage-structured epidemic model with a nonlinear incidence

In this paper, a stage-structured epidemic model with a nonlinear incidence with a factor S^p is investigated. By using limit theory of differential equations and Theorem of Busenberg and van den Driessche, global dynamics of the model is rigorously established. We prove that if the basic reproduction number R₀ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model.     

Global analysis of an environmental disease transmission model linking within-host and between-host dynamics

In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R₀ for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R₀ < 1, and the endemic equilibrium is globally asymptotically stable when R₀ > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.     

Mathematical Analysis of West Nile Virus Model with Discrete Delays

The paper presents the basic model for the transmission dynamics of West Nile virus (WNV). The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease-free equilibrium whenever the associated reproduction number (?₀) is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity). It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever ?₀ > 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by Δ_b and Δ_v are negative. On the other hand, increasing the length of the incubation period increases disease burden if Δ_b > 0 and Δ_v > 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence (considered in [2]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease).