Second Method of Lyapunov

Resumen: In this work we deal with global stability properties of two host-vector disease models using the Poicare-Bendixson Theorem and Second Method of Lyapunov. We construct a Lyapunov function for each Vector-Host model. We proved that the local and global stability are completely determined by the threshold parameter, R_0. If R_0 minor o equal 1, the disease-free equilibrium point is globally asymptotically stable. If R_0> 1, a unique endemic equilibrium point exists and is globally asymptotically stable in the interior ...

In this paper, we introduce and analyze two structured models for the transmission of a vector-borne infectious disease. The first of these models assumes that the level of contagiousness and the rate of removal (recovery) of infected individuals depends on the infection age. In the second model the hosts population is structured with respect to the physical age of the hosts, and the susceptibility of the hosts is assumed to be age-dependent. For these models, the threshold parameter for the existence of a positive (endemic) equilibrium state is determined, and the global asymptotic stability of the equilibrium states are established by the Lyapunov's direct method.

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra-type functions, composite quadratic functions and Volterra-type functionals, we provide the global stability for this model. If R 0 , the basic reproductive number, satisfies R 0 Ä 1, then the infection-free equilibrium state is globally asymptotically stable. Our system is persistent if R 0 > 1. On the other hand, if R 0 > 1, then infection-free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R 0 > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations.

The purpose of this paper is to study the global stability properties of equilibria for age-dependent epidemiological models in presence of recurrence phenomenon. In these systems, the recurrence rate depends on asymptomatic-infectionage. The models are appropriate for human herpes virus (HSV-1 and HSV-2) and varicella-zoster virus. We derived explicit formulas for the basic reproductive number, which completely characterizes the global behaviour of solutions to the models: if the basic reproductive number is less than or equal to unity, the disease will die out; if the basic reproductive number is greater than unity, the disease will be persistent. Volterra-type Lyapunov functions are constructed to establish the global asymptotic stability of the infection-free and endemic steady states.

We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccineage, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

In epidemic modeling, the term infection strength indicates the ratio of infection rate and cure rate. If the infection strength is higher than a certain threshold-which we define as the epidemic threshold-then the epidemic spreads through the population and persists in the long run. For a single generic graph representing the contact network of the population under consideration, the epidemic threshold turns out to be equal to the inverse of the spectral radius of the contact graph. However, in a real world scenario it is not possible to isolate a population completely: there is always some interconnection with another network, which partially overlaps with the contact network. Results for epidemic threshold in interconnected networks are limited to homogeneous mixing populations and degree distribution arguments. In this paper, we adopt a spectral approach. We show how the epidemic threshold in a given network changes as a result of being coupled with another network with fixed infection strength. In our model, the contact network and the interconnections are generic. Using bifurcation theory and algebraic graph theory, we rigorously derive the epidemic threshold in interconnected networks. These results have implications for the broad field of epidemic modeling and control. Our analytical results are supported by numerical simulations.

In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.

In epidemic modeling, the term infection strength indicates the ratio of infection rate and cure rate. If the infection strength is higher than a certain threshold-which we define as the epidemic threshold-then the epidemic spreads through the population and persists in the long run. For a single generic graph representing the contact network of the population under consideration, the epidemic threshold turns out to be equal to the inverse of the spectral radius of the contact graph. However, in a real world scenario it is not possible to isolate a population completely: there is always some interconnection with another network, which partially overlaps with the contact network. Results for epidemic threshold in interconnected networks are limited to homogeneous mixing populations and degree distribution arguments. In this paper, we adopt a spectral approach. We show how the epidemic threshold in a given network changes as a result of being coupled with another network with fixed infection strength. In our model, the contact network and the interconnections are generic. Using bifurcation theory and algebraic graph theory, we rigorously derive the epidemic threshold in interconnected networks. These results have implications for the broad field of epidemic modeling and control. Our analytical results are supported by numerical simulations.

In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.

We propose an SEIR epidemic model with latent period and a modified saturated incidence rate. This work investigates the fundamental role of the vaccination strategies to reduce the number of susceptible, exposed, and infected individuals and increase the number of recovered individuals. The existence of the optimal control of the nonlinear model is also proved. The optimality system is derived and then solved numerically using a competitive Gauss-Seidel-like implicit difference method.

This study presents a family of stochastic models for the dynamics of influenza in a closed human population. We consider treatment for the disease in the form of vaccination and incorporate the periods of effectiveness of the vaccine and infectiousness for the individuals in the population. Our model is a SVIR model, with trinomial transition probabilities, where all individuals who recover from the disease acquire permanent natural immunity against the strain of the disease. A special SVIR model in the stochastic family based on correlated vaccination and infection probabilities at any instant is presented. The methods of maximum likelihood and expectation–maximization algorithm are applied to find estimates for the parameters of the chain. Moreover, estimators for some special epidemiological control parameters, such as the basic reproduction number, are computed. A numerical simulation example is presented to find the MLE of the parameters of the model.

This study presents a family of stochastic models for the dynamics of influenza in a closed human population. We consider treatment for the disease in the form of vaccination, and incorporate the periods of effectiveness of the vaccine and infectiousness for the individuals in the population. Our model is a SVIR model, with trinomial transition probabilities, where all individuals who recover from the disease acquire permanent natural immunity against the strain of the disease. Special SVIR models in the family are presented, based on the structure of the probability of getting infection and vaccination at any instant. The methods of maximum likelihood, and expectation maximization are derived for the parameters of the chain. Moreover, estimators for some epidemiological assessment parameters, such as the basic reproduction number are computed. Numerical simulation examples are presented for the model.

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.

We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccine-age, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

Introduction: The risk of contact infection among susceptible individuals in a randomly mixed population can be reduced by the presence of immune individuals and this concept is referred as herd immunity. Although herd immunity is observed in vaccinated population for some infectious diseases, it has never been truly attained in compartmental models such as the susceptible-infectious-recovered (SIR) model. This paper introduces a new SIR framework to overcome the limitation of the conventional SIR model in attaining herd immunity.Methods: Two SIR models were newly developed based on the reduced risk of contact infection. The first model A assumes that the risk of contact infection reduces as soon as susceptible individuals are infected and move from class S(t) to I(t), therefore incorporating prevalence of both infectious and susceptible individuals into its force of infection. The second model B assumes the risk of contact infection would reduce after infected individuals have reco...

The present study considers a deterministic compartmental model for obesity dynamics. The model exhibits forward bifurcation at basic reproduction number,R0=1, that is; forR0<1, obesity is not sustained. However forR0>1the model approaches a locally asymptotically stable endemic equilibrium. To control this epidemic and reduce the obesity at the endemic equilibrium, we considered intervention strategies for the spread of overweight and obesity, where Pontryagin’s Maximum Principle is applied. The numerical technique was used to show that there are effective control strategies that include minimizing the social contact rate with the overweight and obese population and campaigning. Numerical results indicated the effects of the two controls (prevention and education/campaigning) to be different. In societies with lower obesity, the social contact rate with the overweight and obese population plays a more prominent role in spreading obesity than lack of educational programs/campa...

Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as an effective strategy for the elimination of infectious diseases. An SIR epidemic model with pulse vaccination and distributed time delay is proposed in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than some critical value. The permanence of the model is investigated analytically. Our results indicate that a large pulse vaccination rate is sufficient for the eradication of the disease.

In this paper, we consider a mathematical model of leptospirosis disease which is an infectious disease. The model we are considering is a system of nonlinear ordinary differential equations and it is difficult to find exact solution. In order to compute the approximate solution, He's homotopy perturbation method is used. The findings obtained by HPM and Runge-Kutta fourth order (RK4) methods are compared. To show the simplicity and reliability of the method, sample plots are given at the end of the paper.

The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals. We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method.

In this paper, we consider a mathematical model of leptospirosis disease which is an infectious disease. The model we are considering is a system of nonlinear ordinary differential equations and it is difficult to find exact solution. In order to compute the approximate solution, He's homotopy perturbation method is used. The findings obtained by HPM and Runge-Kutta fourth order (RK4) methods are compared. To show the simplicity and reliability of the method, sample plots are given at the end of the paper.

We derive and analyze a mathematical model of smoking in which the population is divided into four classes: potential smokers, smokers, temporary quitters, and permanent quitters. In this model we study the effect of smokers on temporary quitters. Two equilibria of the model are found: one of them is the smoking-free equilibrium and the other corresponds to the presence of smoking. We examine the local and global stability of both equilibria and we support our results by using numerical simulations.

Theoretical description of the metastable phase decay kinetics in the presence of specific connections between the embryos of small sizes has been given. The theory of the decay kinetics in the presence of relaxation processes is constructed in analytical manner. Them-mers nucleation is investigated and the global kinetics of decay is also constructed in this case analytically.

We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccine-age, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

We develop and analyze an age-structured SVIRS epidemic model with ages of vaccination [1] and recovery which allows for variations in the exit rate of the removed as a function of age (time since recovery) and variations in the vaccine loss rate as a function of age (time since vaccination). We show that the age-structured model has one disease-free steady state and an endemic steady state. Local and global stability analysis show that the disease-free steady state is locally and globally stable if the reproduction number is below one. This means that if the reproduction number is reduced below one, say through vaccination or treatment, the disease will be eliminated. In addition, to provide a better understanding of the interaction between treatment and vaccination, we formulate an optimal control problem [2] for the age-structured model and then derive the existence of solutions to the optimal control problem from optimal control theory. The optimal treatment and vaccination stra...

We discuss a stochastic SIR epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction numberR0. We deduce the globally asymptotic stability of the disease-free equilibrium whenR0≤ 1and the perturbation is small, which means that the disease will die out. WhenR0>1, we derive that the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average. The key to our analysis is choosing appropriate Lyapunov functions.

We investigate a stochastic SIS model with nonlinear incidence rate. We show that there exists a unique nonnegative solution to the system, and condition for the infectious individualsI(t)to be extinct is given. Moreover, we prove that the system has ergodic property. Finally, computer simulations are carried out to verify our results.

In this work we deal with global stability properties of two host - vector disease models using the Poicare-Bendixson Theorem and Second Method of Lyapunov. We construct a Lyapunov function for each Vector- Host model. We proved that the local and global stability are completely determined by the threshold parameter, R_0. If R_0 minor o equal 1, the disease-free equilibrium point is globally asymptotically stable. If R_0 > 1, a unique endemic equilibrium point exists and is globally asymptotically stable in the interior of the feasible region

The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals. We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method.

A classical epidemiological framework is used to qualitatively assess the impact of early detection and treatment on the dynamics of HIV/AIDS. Within this theoretical framework, two classes of infected populations: those infected but unaware of their serological status and those who are aware of their disease status, are considered. In this context, we formulate and analyze a deterministic model for the transmission dynamics of HIV/AIDS and assess the potential population-level impact of early detection in curtailing the epidemic. A critical threshold parameter for which case detection will have a positive impact is derived. Model parameters sensitivity analysis indicates that the number of partners is the most sensitive in increasing the average number of secondary transmission parameter. However, the case detection coverage is the main drivers in reducing the initial disease transmission. Numerical simulations of the model are provided to support the analytical results. Early detection and treatment alone are insufficient to eliminate the disease, and other control strategies are to be explored.

We derive and analyze a mathematical model of smoking in which the population is divided into four classes: potential smokers, smokers, temporary quitters, and permanent quitters. In this model we study the effect of smokers on temporary quitters. Two equilibria of the model are found: one of them is the smoking-free equilibrium and the other corresponds to the presence of smoking. We examine the local and global stability of both equilibria and we support our results by using numerical simulations.

We examine possible approximate solutions of both integer and noninteger systems of nonlinear differential equations describing tuberculosis disease population dynamics. The approximate solutions are obtained via the relatively new analytical technique, the homotopy decomposition method (HDM). The technique is described and illustrated with numerical example. The numerical simulations show that the approximate solutions are continuous functions of the noninteger-order derivative. The technique used for solving these problems is friendly, very easy, and less time consuming.

We derive and analyze a mathematical model of smoking in which the population is divided into four classes: potential smokers, smokers, temporary quitters, and permanent quitters. In this model we study the effect of smokers on temporary quitters. Two equilibria of the model are found: one of them is the smoking-free equilibrium and the other corresponds to the presence of smoking. We examine the local and global stability of both equilibria and we support our results by using numerical simulations.

The stability of the SIR epidemic model with information variable and limited medical resources was studied. When the basic reproduction ratio R 0 < 1, there exists the disease-free equilibrium and when the basic reproduction ratio R 0 > 1, we obtain the sufficient conditions of the existence of the endemic equilibrium. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the effectiveness of the main results.

Most of the current epidemic models assume that the infectious period follows an exponential distribution. However, due to individual heterogeneity and epidemic diversity, these models fail to describe the distribution of infectious periods precisely. We establish a SIS epidemic model with multistaged progression of infectious periods on complex networks, which can be used to characterize arbitrary distributions of infectious periods of the individuals. By using mathematical analysis, the basic reproduction number 0 for the model is derived. We verify that the 0 depends on the average distributions of infection periods for different types of infective individuals, which extend the general theory obtained from the single infectious period epidemic models. It is proved that if 0 < 1, then the disease-free equilibrium is globally asymptotically stable; otherwise the unique endemic equilibrium exists such that it is globally asymptotically attractive. Finally numerical simulations hold for the validity of our theoretical results is given.

The main idea of this work is to present and study the dynamical behavior of an age-dependent SIR endemic model. First, the age-dependent SIR endemic model is formulated from existing SIR epidemic models by adding age-dependent recruitment rate. The behavior of the model is analyzed by using the basic reproductive number R 0. The stability theory is used for the analysis of disease-free and endemic equilibrium. Finally, finite difference method of characteristics is used to present numerical results for the suggested age-structured endemic model.

The main idea of this work is to present and study the dynamical behavior of an age-dependent SIR endemic model. First, the age-dependent SIR endemic model is formulated from existing SIR epidemic models by adding age-dependent recruitment rate. The behavior of the model is analyzed by using the basic reproductive number R 0. The stability theory is used for the analysis of disease-free and endemic equilibrium. Finally, finite difference method of characteristics is used to present numerical results for the suggested age-structured endemic model.

The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals. We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method.

In this work we deal with global stability properties of classic SIS, SIR and SIRS epidemic models with constant recruitment rate, mass action incidence and variable population size. The usual approach to determine global stability of equilibria is the direct Lyapunov method which requires the construction of a function with specific properties. In this work we construct different Lyapunov functions for the systems mentioned above using combinations of suitable composite quadratic, simple quadratic and logarithmic functions. ...

This paper is aimed at designing a robust vaccination strategy capable of eradicating an infectious disease from a population regardless of the potential uncertainty in the parameters defining the disease. For this purpose, a control theoretic approach based on a sliding-mode control law is used. Initially, the controller is designed assuming certain knowledge of an upper-bound of the uncertainty signal. Afterwards, this condition is removed while an adaptive sliding control system is designed. The closed-loop properties are proved mathematically in the nonadaptive and adaptive cases. Furthermore, the usual sign function appearing in the sliding-mode control is substituted by the saturation function in order to prevent chattering. In addition, the properties achieved by the closed-loop system under this variation are also stated and proved analytically. The closed-loop system is able to attain the control objective regardless of the parametric uncertainties of the model and the lack of a priori knowledge on the system.

This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a diseasefree equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is shown, using linear and nonlinear Lyapunov functions, coupled with the LaSalle Invariance Principle, that this equilibrium is globally-asymptotically stable for a special case. The model has a resistant strain-only equilibrium, which is globally-asymptotically stable whenever its reproduction number is greater than unity and exceeds that of the wild strain. In this case, the two strains undergo competitive exclusion, where the strain with the higher reproduction number displaces the other. Further, for the case when treatment does not lead to resistance development, the model can have no coexistence equilibrium or a continuum of coexistence equilibria. When treatment leads to resistance development, the model can have a unique coexistence equilibrium or a resistant-only equilibrium. This coexistence equilibrium is shown to be locally-asymptotically stable, using a technique based on Krasnoselskii sub-linearity argument. Numerical simulations of the model show that for high treatment rates, the resistant strain can dominate, and drive out, the wild strain. Finally, when the two strains co-exist, the proportion of individuals with the resistant strain at steady-state decreases with increasing rate of resistance development.

This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a diseasefree equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is shown, using linear and nonlinear Lyapunov functions, coupled with the LaSalle Invariance Principle, that this equilibrium is globally-asymptotically stable for a special case. The model has a resistant strain-only equilibrium, which is globally-asymptotically stable whenever its reproduction number is greater than unity and exceeds that of the wild strain. In this case, the two strains undergo competitive exclusion, where the strain with the higher reproduction number displaces the other. Further, for the case when treatment does not lead to resistance development, the model can have no coexistence equilibrium or a continuum of coexistence equilibria. When treatment leads to resistance development, the model can have a unique coexistence equilibrium or a resistant-only equilibrium. This coexistence equilibrium is shown to be locally-asymptotically stable, using a technique based on Krasnoselskii sub-linearity argument. Numerical simulations of the model show that for high treatment rates, the resistant strain can dominate, and drive out, the wild strain. Finally, when the two strains co-exist, the proportion of individuals with the resistant strain at steady-state decreases with increasing rate of resistance development.

We analytically study the dynamical behavior of a two-neuron network with a timedelayed self-connection. The effect of the time delay on the stability of the trivial solution and on the existence of self-sustained periodic solution are investigated. These results could be applied to understand the temporal activity appearing in the olfactory bulb.

This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a diseasefree equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is shown, using linear and nonlinear Lyapunov functions, coupled with the LaSalle Invariance Principle, that this equilibrium is globally-asymptotically stable for a special case. The model has a resistant strain-only equilibrium, which is globally-asymptotically stable whenever its reproduction number is greater than unity and exceeds that of the wild strain. In this case, the two strains undergo competitive exclusion, where the strain with the higher reproduction number displaces the other. Further, for the case when treatment does not lead to resistance development, the model can have no coexistence equilibrium or a continuum of coexistence equilibria. When treatment leads to resistance development, the model can have a unique coexistence equilibrium or a resistant-only equilibrium. This coexistence equilibrium is shown to be locally-asymptotically stable, using a technique based on Krasnoselskii sub-linearity argument. Numerical simulations of the model show that for high treatment rates, the resistant strain can dominate, and drive out, the wild strain. Finally, when the two strains co-exist, the proportion of individuals with the resistant strain at steady-state decreases with increasing rate of resistance development.

We present in this paper an SIRS epidemic model with saturated incidence rate and disease-inflicted mortality. The Global stability of the endemic equilibrium state is proved by constructing a Lyapunov function. For the stochastic version, the global existence and positivity of the solution is showed, and the global stability in probability and pth moment of the system is proved under suitable conditions on the intensity of the white noise perturbation.

This paper addresses the global stability of a two-species Lotka-Volterra cooperative system with four discrete delays via the method of Lyapunov functionals. We apply our Lyapunov functional techniques to prove stability of Lotka-Volterra models of mutualism with two delays or without delays. We note that, if all the delayed feedbacks are positive then the stability properties are independent of the values of the delays. We compare our results with known results.

Keywords: Lotka-Volterra cooperative system, discrete delays, delayed positive feedback, Lyapunov functional, global stability.

We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccineage, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

In this paper, we formulate a susceptible-vaccinated-infected-recovered (SVIR) model by  incorporating the vaccination of newborns, vaccine-age and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

Keyword: Age-dependent epidemic model; Vaccine-age; Waning vaccine-induced immunity; Volterra-type Lyapunov function; Global stability

"In this paper, we study the global stability conditions of  two epidemiological model with relapse, and bilinear and standard incidence rates, respectively, that includes recruitment rate of susceptible individuals into the community and that the disease produces non-negligible death in the infectious class. The models are appropriate for tuberculosis, including  tuberculosis in human and bovine, and for human herpes virus.  It is also included the dynamics of tobacco and alcohol use. We presents the construction of Lyapunov functions for the systems mentioned above using suitable combinations of known functions, common quadratic and Volterra--type, and of a novel function we call composite Volterra--type function.

Keywords: Epidemiological model, Relapse, Global stability,  Lyapunov's second method, LaSalle's invariance principle"

"""Lyapunov functions for some mathematical models of ecological commensalism between two species are introduced. Global stability of the unique positive equilibrium is thereby established.

Keyword: Models of ecological commensalism; Lyapunov's asymptotic stability theorem; Global stability"""

""By means of  Lyapunov functions, we have established the global stability of the steady state of two models that include latent infected cells, a linear immune response and a generalized immune attack term.

Keywords. Virus dynamics models, Immune response, Global stability, Method of Lyapunov functions, LaSalle's invariance principle.""