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In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R0 which is based on the disease-free equilibrium, and a new net reproduction number R0 (E*) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 ≤ 1 and unstable if R0 > 1. Moreover, the endemic equilibrium is locally asymptotically stable if R0(E*) < 1 < R0. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a population suppression model and a population replacement model with periodic impulsive releases of Nilaparvata lugens infected with wStri. The conditions for the stability of wild- $ N.\,lugens $ N. l u g e n s -eradication periodic solution of two systems are obtained by applying the Floquet theorem and comparison theorem. And the sufficient conditions for the persistence in the mean of wild $ N.\,lugens $ N. l u g e n s are also given. In addition, the sufficient conditions for the extinction and persistence of the wild $ N.\,lugens $ N. l u g e n s in the subsystem without wLug are also obtained. Finally, we give numerical analysis which shows that increasing the release amount or decreasing the release period are beneficial for controlling the wild $ N.\,lugens $ N. l u g e n s , and the efficiency of population replacement strategy in controlling wild populations is higher than that of population suppression strategy under the same release conditions. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion SIR epidemic model via environmental driven infection in heterogeneous space is proposed. To reflect the prevention and control measures of disease in allusion to the susceptible in the model, the nonlinear incidence function E f (S) is applied to describe the protective measures of susceptible. In the general spatially heterogeneous case of the model, the well-posedness of solutions is obtained. The basic reproduction number R 0 is calculated. When R 0 ≤ 1 the global asymptotical stability of the disease-free equilibrium is obtained, while when R 0 > 1 the model is uniformly persistent. Furthermore, in the spatially homogeneous case of the model, when R 0 > 1 the global asymptotic stability of the endemic equilibrium is obtained. Lastly, the numerical examples are enrolled to verify the open problems. [ABSTRACT FROM AUTHOR]

In this paper, a delayed human immunodeficiency virus (HIV) model with apoptosis of cells has been studied. Both immunological and intracellular delay have been incorporated to make the model more relevant. Firstly, the model has been investigated using local stability analysis. Next, the global stability analysis of steady states has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. The transition of the system from order to chaos has been explored, and the analytical results have been verified by numerical simulations. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been studied in the model numerically. [ABSTRACT FROM AUTHOR]

This paper studies a delayed viral infection model with diffusion and a general incidence rate. A discrete-time model was derived by applying nonstandard finite difference scheme. The positivity and boundedness of solutions are presented. We established the global stability of equilibria in terms of R 0 by applying Lyapunov method. The results showed that if R 0 is less than 1, then the infection-free equilibrium E 0 is globally asymptotically stable. If R 0 is greater than 1, then the infection equilibrium E ∗ is globally asymptotically stable. Numerical experiments are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, with eclipse stage in consideration, we propose an HIV-1 infection model with a general incidence rate and CTL immune response. We first study the existence and local stability of equilibria, which is characterized by the basic infection reproduction number R 0 and the basic immunity reproduction number R 1 . The local stability analysis indicates the occurrence of transcritical bifurcations of equilibria. We confirm the bifurcations at the disease-free equilibrium and the infected immune-free equilibrium with transmission rate and the decay rate of CTLs as bifurcation parameters, respectively. Then we apply the approach of Lyapunov functions to establish the global stability of the equilibria, which is determined by the two basic reproduction numbers. These theoretical results are supported with numerical simulations. Moreover, we also identify the high sensitivity parameters by carrying out the sensitivity analysis of the two basic reproduction numbers to the model parameters. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number R 01 and immune reproduction number R 02 are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when R 01 < 1 , the virus will be cleared out eventually and the CTL immune response will also disappear; when R 02 < 1 < R 01 , the virus persists within the host, but the CTL immune response disappears eventually; when R 02 > 1 , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given. [ABSTRACT FROM AUTHOR]

In this paper, we study a more general diffusive spatially dependent vaccination model for infectious disease. In our diffusive vaccination model, we consider both therapeutic impact and nonlinear incidence rate. Also, in this model, the number of compartments of susceptible, vaccinated and infectious individuals are considered to be functions of both time and location, where the set of locations (equivalently, spatial habitats) is a subset of R n with a smooth boundary. Both local and global stability of the model are studied. Our study shows that if the threshold level R 0 ≤ 1 , the disease-free equilibrium E 0 is globally asymptotically stable. On the other hand, if R 0 > 1 then there exists a unique stable disease equilibrium E ∗ . The existence of solutions of the model and uniform persistence results are studied. Finally, using finite difference scheme, we present a number of numerical examples to verify our analytical results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R 0 . [ABSTRACT FROM AUTHOR]

Scarlet fever is an acute respiratory infectious disease and the incidence rate is increasing from 2011 throughout the world. In this paper, the mathematical models are proposed, which incorporate both direct transmissions and indirect transmissions of scarlet fever. The threshold conditions for disease invasion are obtained in terms of the basic reproduction number. The peak value, final size and epidemic time in a seasonal prevalence are investigated numerically. Furthermore, the effects of seasonal fluctuations on disease outbreak are also studied on the basis of real data in China. [ABSTRACT FROM AUTHOR]

This paper proposes a malaria transmission model to describe the dynamics of malaria transmission in the human and mosquito populations. This model emphasizes the impact of limited resource on malaria transmission. We derive a formula for the basic reproductive number of infection and investigate the existence of endemic equilibria. It is shown that this model may undergo backward bifurcation, where the locally stable disease-free equilibrium co-exists with an endemic equilibrium. Furthermore, we determine conditions under which the disease-free equilibrium of the model is globally asymptotically stable. The global stability of the endemic equilibrium is also studied when the basic reproductive number is greater than one. Finally, numerical simulations to illustrate our findings and brief discussions are provided. [ABSTRACT FROM AUTHOR]

Vector-borne infectious diseases may involve both horizontal transmission between hosts and transmission from infected vectors to susceptible hosts. In this paper, we incorporate these two transmission modes into a vector-borne disease model that includes general nonlinear incidence rates and the age of infection for both hosts and vectors. We show the existence, uniqueness, nonnegativity, and boundedness of solutions for the model. We study the existence and local stability of steady states, which is shown to be determined by the basic reproduction number. By showing the existence of a global compact attractor and uniform persistence of the system, we establish the threshold dynamics using the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable and otherwise the disease will be established when there is initial infection force for the hosts. We also study a model with the standard incidence rate and discuss the effect of different incidence rates on the disease dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a stage-structured predator–prey model with mutual interference, in which includes two discrete delays. By theoretical analysis, we establish the stability of the unique positive equilibrium and the existence of Hopf bifurcation when the maturation delay for predators is used as the bifurcation parameter. Our results exhibit that the maturation delay for preys does not affect the stability of the positive equilibrium. However, the maturation delay for predator is able to destabilize the positive equilibrium and causes periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and display the differential impacts of two type delays and mutual interference. [ABSTRACT FROM AUTHOR]

This paper studies the global stability of discrete-time viral infection models with humoural immunity. We consider both latently and actively infected cells. We study also a model with general production and clearance rates of all compartments as well as general incidence rate of infection. We use nonstandard finite difference method to discretize the continuous-time models. The positivity and boundedness of solutions of the discrete models are established. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number R 0 and the humoural immune response activation number R 1 . The results signify that the infection dies out if R 0 ≤ 1. Moreover, the infection persists with inactive immune response if R 1 ≤ 1 < R 0 and with active immune response if R 1 > 1. We illustrate our theoretical results by using numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number R 0 is obtained such that R 0 ≤ 1 indicates the possibility of dysentery eradication in the community while R 0 > 1 represents uniform persistence of the disease. The Lyapunov–LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for R 0 > 1. Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation. [ABSTRACT FROM AUTHOR]

In this paper, an Susceptible-Vaccines-Exposed-Infectious-Recovered model with continuous age-structure in the exposed and infectious classes is investigated. These two ages are assumed to have arbitrary distributions that are represented by age-specific rates leaving the exposed and the infectious classes. We investigate the global dynamics of this model in the sense of basic reproduction number via constructing Lyapunov functions. The asymptotic smoothness of solutions and uniform persistence of the system is shown from reformulating the system as a system of Volterra integral equations. [ABSTRACT FROM PUBLISHER]

In this paper, we study the stability analysis of latent Chikungunya virus (CHIKV) dynamics models. The incidence rate between the CHIKV and the uninfected monocytes is modelled by a general nonlinear function which satisfies a set of conditions. The model is incorporated by intracellular discrete or distributed time delays. Using the method of Lyapunov function, we established the global stability of the steady states of the models. The theoretical results are confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven 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, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global stability of the models which incorporate humoural immunity or cell-mediated immunity. We consider the effect of loss of a pathogen, which is called the absorption effect when it infects an uninfected cells. We construct Lyapunov functions for these models under some conditions of parameters, and prove the global stability of the interior equilibria. It is impossible to remove the condition of parameters for the model incorporating humoural immunity. [ABSTRACT FROM AUTHOR]

In this paper, we study the effects of Beddington-DeAngelis interference and squabbling, respectively, on the minimal rate of predator release required to drive a pest population to zero. A two-dimensional system of coupled ordinary differential equations is considered, augmented by an impulsive component depicting the periodic release of predators into the system. This periodic release takes place independently of the detection of the pests in the field. We establish the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. In particular, we show that with the interference effects, the minimal rate will only guarantee eradication if the releases are carried out frequently enough. When Beddington-DeAngelis behaviour is considered, an additional constraint for the existence itself of a successful release rate is that the pest growth rate should be less than the predation pressure, the latter explicitly formulated in terms of the predation function and the interference parameters. [ABSTRACT FROM AUTHOR]

Many models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models offacultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable. [ABSTRACT FROM AUTHOR]

This paper deals with the global property of a drinking model with public health educational campaigns. With the help of Lyapunov function, global stability of equilibria of the model is derived. The alcohol-free equilibrium is globally asymptotically stable and the alcohol problems are eliminated from population if R0 ≤ 1. A unique alcohol present equilibrium is globally asymptotically stable if R0 > 1. Furthermore, the basic reproductive R0 for the model is compared with the basic reproductive number R1 for the absence of public health educational campaigns. We conclude that the public health educational campaigns of drinking individuals can slow down the drinking dynamics. Some numerical simulations are also given to explain our conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyse a virus dynamics model with humoral immune response including latently infected cells. The incidence rate is given by Beddington–DeAngelis functional response. We have derived two threshold parameters, the basic infection reproduction numberand the humoral immune response activation numberwhich completely determined the basic and global properties of the virus dynamics model. By constructing suitable Lyapunov functions and applying LaSalle's invariance principle we have proven that if, then the infection-free equilibrium is globally asymptotically stable (GAS), if, then the chronic-infection equilibrium without humoral immune response is GAS, and if, then the chronic-infection equilibrium with humoral immune response is globally asymptotically stable. These results are further illustrated by numerical simulations. [ABSTRACT FROM PUBLISHER]

We provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles that a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of, what we consider, the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to obtain two models, a Beverton–Holt adult model and a Ricker adult model and discuss the global dynamics of both models. [ABSTRACT FROM AUTHOR]

We propose a hand, foot and mouth disease (HFMD) transmission model for children with behaviour change and imperfect quarantine. The symptomatic and quarantined states obey constant behaviour change while others follow variable behaviour change depending on the numbers of new and recent infections. The basic reproduction number $ \mathcal {R}_0 $ R 0 of the model is defined and shown to be a threshold for disease persistence and eradication. Namely, the disease-free equilibrium is globally asymptotically stable if $ \mathcal {R}_0\le 1 $ R 0 ≤ 1 whereas the disease persists and there is a unique endemic equilibrium otherwise. By fitting the model to weekly HFMD data of Shanghai in 2019, the reproduction number is estimated at 2.41. Sensitivity analysis for $ \mathcal {R}_0 $ R 0 shows that avoiding contagious contacts and implementing strict quarantine are essential to lower HFMD persistence. Numerical simulations suggest that strong behaviour change not only reduces the peak size and endemic level dramatically but also impairs the role of asymptomatic transmission. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the long-term behaviour of solutions of the periodic Sigmoid Beverton–Holt equation where the a n and δ n are p-periodic positive sequences. Under certain conditions, there are shown to exist an asymptotically stable p-periodic state and a p-periodic Allee state with the property that populations smaller than the Allee state are driven to extinction while populations greater than the Allee state approach the stable state, thus accounting for the long-term behaviour of all initial states. This appears to be the first study of the equation with variable δ. The results are discussed with possible interpretations in Population Dynamics with emphasis on fish populations and smooth cordgrass. [ABSTRACT FROM PUBLISHER]

In this paper, we formulate a discrete-time model with the reproductive and overwintering periods to assess the impact of avian influenza transmission in poultry. It is shown that the disease is extinct if the basic reproduction number is less than one and is persistent if the basic reproductive number is greater than one. Furthermore, the model admits a closed invariant cycle, which means that avian influenza fluctuates in poultry. [ABSTRACT FROM AUTHOR]

A susceptible-vaccinated-infectious-recovered epidemic model with infection age and relapse age has been formulated. We first address the asymptotic smoothness of the solution semiflow, existence of a global attractor, and uniform persistence of the model. Then by constructing suitable Volterra-type Lyapunov functionals, we establish a global threshold dynamics of the model, which is determined by the basic reproduction number. Biologically, it is confirmed that neglecting the possibility of vaccinees getting infected will over-estimate the effect of vaccination strategies. The obtained results generalize some existing ones. [ABSTRACT FROM PUBLISHER]

Based on evolutionary game theory and Darwinian evolution, we propose and study discrete-time competition models of two species where at least one species has an evolving trait that affects their intra-specific, but not their inter-specific competition coefficients. By using perturbation theory, and the theory of the limiting equations of non-autonomous discrete dynamical systems, we obtain global stability results. Our theoretical results indicate that evolution may promote and/or suppress the stability of the coexistence equilibrium depending on the environment. This relies crucially on the speed of evolution and on how the intra-specific competition coefficient depends on the evolving trait. In general, equilibrium destabilization occurs when α > 2 , when the speed of evolution is sufficiently slow. In this case, we conclude that evolution selects against complex dynamics. However, when evolution proceeds at a faster pace, destabilization can occur when α < 2. In this case, if the competition coefficient is highly sensitive to changes in the trait v, destabilization and complex dynamics occur. Moreover, destabilization may lead to either a period-doubling bifurcation, as in the non-evolutionary Ricker equation, or to a Neimark-Sacker bifurcation. [ABSTRACT FROM AUTHOR]

In this study, a model for the spread of cyst echinococcosis with interventions is formulated. The disease-free and endemic equilibrium points of the model are calculated. The control reproduction number R c for the model is derived, and the global dynamics are established by the values of R c . The disease-free equilibrium is globally asymptotically stable if and only if R c < 1. For R c > 1 , using Volterra–Lyapunov stable matrices, it is proven that the endemic equilibrium is globally asymptotically stable. Sensitivity analysis to identify the most influential parameters in the dynamics of CE is carried out. To establish the long-term behaviour of the disease, numerical simulations are performed. The impact of control strategies is investigated. It is shown that, whenever vaccination of sheep is carried out solely or in combination with cleaning or disinfecting of the environment, cyst echinococcosis can be wiped out. [ABSTRACT FROM AUTHOR]

A mathematical model is formulated for the transmission and spread of cholera in a heterogeneous host population that consists of several patches of homogeneous host populations sharing a common water source. The basic reproduction number ℛ0is derived and shown to determine whether or not cholera dies out. Explicit formulas are derived for target/type reproduction numbers that measure the control strategies required to eradicate cholera from all patches. [ABSTRACT FROM AUTHOR]

A general three-stage discrete-time population model is studied. The inherent net reproductive number for this model is derived. Global stability of the origin is established provided that the inherent net reproductive number is less than one. If it is larger than one the existence of a unique positive fixed point is proved and the persistence of the system is established. Finally, for certain parameter ranges global stability of the positive fixed point is proved. [ABSTRACT FROM AUTHOR]

We analysed a epidemiological model with varying populations of susceptible, carriers, infectious and recovered (SCIR) and a general non-linear incidence rate of the form f (S) [ g (C) + h (I) ]. We show that this model exhibits two positive equilibriums: the disease-free and disease equilibrium. We proved using the Lyapunov direct method that these two equilibriums are globally asymptotically stable under some sufficient conditions over the functions f, g, h. [ABSTRACT FROM AUTHOR]

We develop a delay differential equation model for the interactive wild and sterile mosquitoes. Different from the existing modelling studies, we assume that only those sexually active sterile mosquitoes play a role for the interactive dynamics. We consider the cases where the release amount is either constant or described by a given function of time. For the constant releases, we establish a threshold of releases to determine whether the wild mosquito suppression succeeds or fails. We study the existence and stability of the model equilibria. When the releases are described by given functions, the trivial equilibrium is no longer globally but locally uniformly asymptotically stable if the amount of releases is below the threshold whereas it is still globally uniformly asymptotically stable if the release amount is above the threshold. Numerical examples demonstrating the model dynamical features and brief discussions of our findings are also provided. [ABSTRACT FROM AUTHOR]

The long and binding treatment of tuberculosis (TB) at least 6–8 months for the new cases, the partial immunity given by BCG vaccine, the loss of immunity after a few years doing that strategy of TB control via vaccination and treatment of infectious are not sufficient to eradicate TB. TB is an infectious disease caused by the bacillus Mycobacterium tuberculosis. Adults are principally attacked. In this work, we assess the impact of vaccination in the spread of TB via a deterministic epidemic model (SVELI) (Susceptible, Vaccinated, Early latent, Late latent, Infectious). Using the Lyapunov–Lasalle method, we analyse the stability of epidemic system (SVELI) around the equilibriums (disease-free and endemic). The global asymptotic stability of the unique endemic equilibrium whenever R 0 > 1 is proved, where R 0 is the reproduction number. We prove also that when R 0 is less than 1, TB can be eradicated. Numerical simulations, using some TB data found in the literature in relation with Cameroon, are conducted to approve analytic results, and to show that vaccination coverage is not sufficient to control TB, effective contact rate has a high impact in the spread of TB. [ABSTRACT FROM AUTHOR]

A drinking model with immigration is constructed. For the model with problem drinking immigration, the model admits only one problem drinking equilibrium. For the model without problem drinking immigration, the model has two equilibria, one is problem drinking-free equilibrium and the other is problem drinking equilibrium. By employing the method of Lyapunov function, stability of all kinds of equilibria is obtained. Numerical simulations are also provided to illustrate our analytical results. Our results show that alcohol immigrants increase the difficulty of the temperance work of the region. [ABSTRACT FROM AUTHOR]

Human African Trypanosomiasis (HAT) and Nagana in cattle, commonly called sleeping sickness, is caused by trypanosome protozoa transmitted by bites of infected tsetse flies. We present a deterministic model for the transmission of HAT caused by Trypanosoma brucei gambiense between human hosts, cattle hosts and tsetse flies. The model takes into account the growth of the tsetse fly, from its larval stage to the adult stage. Disease in the tsetse fly population is modeled by three compartments, and both the human and cattle populations are modeled by four compartments incorporating the two stages of HAT. We provide a rigorous derivation of the basic reproduction number R0. For R0 < 1, the disease free equilibrium is globally asymptotically stable, thus HAT dies out; whereas (assuming no return to susceptibility) for R0 > 1, HAT persists. Elasticity indices for R0 with respect to different parameters are calculated with baseline parameter values appropriate for HAT in West Africa; indicating parameters that are important for control strategies to bring R0 below 1. Numerical simulations with R0 > 1 show values for the infected populations at the endemic equilibrium, and indicate that with certain parameter values, HAT could not persist in the human population in the absence of cattle. [ABSTRACT FROM AUTHOR]

We study a susceptible–infected–susceptible model with distributed delays. By constructing suitable Lyapunov functionals, we demonstrate that the global dynamics of this model is fully determined by the basic reproductive ratioR0. To be specific, we prove that ifR0≤ 1, then the disease-free equilibrium is globally asymptotically stable. On the other hand, ifR0>1, then the endemic equilibrium is globally asymptotically stable. It is remarkable that the model dynamics is independent of the probability of immunity lost. [ABSTRACT FROM AUTHOR]

A mathematical model is formulated for the transmission and spread of cholera in a heterogeneous host population that consists of several patches of homogeneous host populations sharing a common water source. The basic reproduction number R0 is derived and shown to determine whether or not cholera dies out. Explicit formulas are derived for target/type reproduction numbers that measure the control strategies required to eradicate cholera from all patches. [ABSTRACT FROM AUTHOR]

A multi-strain model of a vector-borne disease with distributed delay in the vector and the host is investigated. It is shown that if the reproduction number of the model ℛ0<1, the unique disease-free equilibrium is globally asymptotically stable. Without loss of generality, strain one is assumed to have the largest reproduction number. In this case, the dominance equilibrium of strain one is shown to be locally stable. The basic reproduction number for a straini() is written as a product of the reproduction number of the vectorand the reproduction number of the host, i.e.. The competitive exclusion principle is derived under the somewhat stronger condition that if strain one maximizes both the reproduction number of the hostand the reproduction number of the vector, strain one dominance equilibrium is globally asymptotically stable. [ABSTRACT FROM PUBLISHER]

Influenza A virus evolves through two types of evolutionary mechanisms – drift and shift. These two evolutionary mechanisms allow the pathogen to infect us repeatedly, as well as occasionally create pandemics with large morbidity and mortality. Here we introduce a novel model that incorporates both evolutionary mechanisms. This necessitates the modelling of three types of strains – seasonal human strains, bird-to-human transmittable H5N1 strains and evolved pandemic H5N1 strain. We define reproduction and invasion reproduction numbers and use them to establish the presence of dominant and coexistence equilibria. We find that the amino acid substitution structure of human influenza can destabilize the human influenza equilibrium and sustained oscillations are possible. We find that for low levels of infection in domestic birds, these oscillations persist, inducing oscillations in the number of humans infected with the avian flu strain. The oscillations have a period of 365 days, similar to the one that can be observed in the cumulative number of human H5N1 cases reported by the World Health Organization (WHO). Furthermore, we establish some partial global results on the competition of the strains. [ABSTRACT FROM PUBLISHER]

We consider a population model consisting of d species interacting in a p-periodic environment and modelled by a d-dimensional system of Leslie-Gower-type difference equations (coupled Beverton-Holt equations). It is shown that if the interspecific competition (coupling) is sufficiently small and the inherent growth rate of each species is such that in the absence of competition each species will grow to its (positive) individual carrying capacity, then there is a positive asymptotically stable p-periodic state that globally attracts all positive initial states. Three examples are studied numerically in which the competition is large and the principle of competitive exclusion is observed. The rate of decay to extinction is observed to be sensitive to the inherent growth rate of the dying species. The individual carrying capacities are seen to play a determining role in the case of equal and large competition and equal inherent growth rates. [ABSTRACT FROM AUTHOR]

We consider a four-compartment tuberculosis model including exogenous reinfection. We derive sufficient conditions, in terms of the parameters of the system, which guarantee the occurrence of backward bifurcation. We also discuss the global stability of the endemic state by using a generalization of the Poincare-Bendixson criterion. An application is given for the case of Internally Displaced People's Camps in North Uganda. The study suggests how important it is to provide qualitative indications on the threshold value of the population density in the area occupied by the camps, in order to possibly eradicate the disease. [ABSTRACT FROM AUTHOR]

We analyse, from a mathematical point of view, the global stability of equilibria for models describing the interaction between infectious agents and humoral immunity. We consider the models that contain the variables of pathogens explicitly. The first model considers the situation where only a single strain exists. For the single strain model, the disease steady state is globally asymptotically stable if the basic reproductive ratio is greater than one. The other models consider the situations where multiple strains exist. For the multi-strain models, the disease steady state is globally asymptotically stable. In the model that does not explicitly contain an immune variable, only one strain with the maximum basic reproductive ratio can survive at the steady state. However, in our models explicitly involving the immune system, multiple strains coexist at the steady state. [ABSTRACT FROM AUTHOR]

We establish the threshold dynamics of a delayed reaction diffusion equation subject to the homogeneous Dirichlet boundary condition when the delayed reaction term is non-monotone. We illustrate the main results by two examples, including the delayed Nicholson's blowflies diffusion equation. [ABSTRACT FROM AUTHOR]

We formulate and systematically study the global dynamics of a simple model of hepatitis B virus in terms of delay differential equations. This model has two important and novel features compared to the well-known basic virus model in the literature. Specifically, it makes use of the more realistic standard incidence function and explicitly incorporates a time delay in virus production. As a result, the infection reproduction number is no longer dependent on the patient liver size (number of initial healthy liver cells). For this model, the existence and the component values of the endemic steady state are explicitly dependent on the time delay. In certain biologically interesting limiting scenarios, a globally attractive endemic equilibrium can exist regardless of the time delay length. [ABSTRACT FROM AUTHOR]

We analyze the global dynamics of a mathematical model for infectious diseases that progress through distinct stages within infected hosts with possibility of amelioration. An example of such diseases is HIV/AIDS that progresses through several stages with varying degrees of infectivity; amelioration can result from a host's immune action or more commonly from antiretroviral therapies, such as highly active antiretroviral therapy. For a general n-stage model with constant recruitment and bilinear incidence that incorporates amelioration, we prove that the global dynamics are completely determined by the basic reproduction number R0. If R0≤1, then the disease-free equilibrium P0 is globally asymptotically stable, and the disease always dies out. If R0>1, P0 is unstable, a unique endemic equilibrium P* is globally asymptotically stable, and the disease persists at the endemic equilibrium. Impacts of amelioration on the basic reproduction number are also investigated. [ABSTRACT FROM AUTHOR]

We consider a three-stage discrete-time population model with density-dependent survivorship and time-dependent reproduction. We provide stability analysis for two types of birth mechanisms: continuous and seasonal. We show that when birth is continuous there exists a unique globally stable interior equilibrium provided that the inherent net reproductive number is greater than unity. If it is less than unity, then extinction is the population's fate. We then analyze the case when birth is a function of period two and show that the unique two-cycle is globally attracting when the inherent net reproductive number is greater than unity, while if it is less than unity the population goes to extinction. The two birth types are then compared. It is shown that for low birth rates the adult average number over a one-year period is always higher when reproduction is continuous. Numerical simulations suggest that this remains true for high birth rates. Thus periodic birth rates of period two are deleterious for the three-stage population model. This is different from the results obtained for a two-stage model discussed by Ackleh and Jang (J. Diff. Equ. Appl., 13, 261-274, 2007), where it was shown that for low birth rates seasonal breeding results in higher adult averages. [ABSTRACT FROM AUTHOR]