Compound matrices

For a stable matrix A with real entries, sufficient and necessary conditions for A y D to be stable for all non-negative diagonal matrices D are obtained. Implications of these conditions for the stability and instability of constant steadystate solutions to reaction᎐diffusion systems are discussed and an example is given to show applications.

A necessary and sufficient condition for the stability of n = n matrices with real entries is proved. Applications to asymptotic stability of equilibria for vector fields are considered. The results offer an alternative to the well-known Routh᎐Hurwitz conditions.

The dynamics of many epidemic models for infectious diseases that spread in a single host population demonstrate a threshold phenomenon. If the basic reproduction numberR 0 is below unity, the disease-free equilibrium P 0 is globally stable in the feasible region and the disease always dies out. If R 0 > 1 a unique endemic equilibrium P is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present. In this paper, this threshold phenomenon is established for two epidemic models of SEIR type using two recent approaches to the global-stability problem.

We provide the natural extension, from the dynamical point of view, of the Poincark-Hopf theorem to noncompact manifolds. On the other hand, given a compact set K being an attractor for a flow generated by a 'St tangent vector field X on an n-manifold, we prove that the Euler characteristic of its region of attraction .rQ, x(d), is defined and satisfies Ind.&X) = (-1)"x(&). Finally we prove that x(d) = x(K) when K is an euclidean neighbourhood retract being asymptotically stable and invariant.

A mathematical model for the Hepatitis A Virus (HAV) epidemiology with dual transmission mechanisms is developed and presented. The model considers vaccination and sanitation as mitigation strategies. The effective reproductive number was derived and employed to study the stability of the model. Using Routh's stability criteria, the local stability of a disease-free equilibrium was determined, whereas the global stability of the endemic equilibrium was attained through a suitable Lyapunov function. Furthermore, bifurcation analysis is carried out using the centre manifold theory to ascertain its nature and implication for disease control. It was revealed that the model exhibits a forward bifurcation indicating the possibility of disease eradication when the effective reproduction number is kept below unity. Numerical results indicate that infection rates decrease quantitatively when at least one control measure is effectively implemented. It was deduced that combining vaccination and sanitation yields even fewer cases, making it the best alternative for eliminating Hepatitis A (HA) infection from the community. A sensitivity analysis was conducted to ascertain the parameters of the strong influence that could significantly affect the system. It was revealed that constant recruitment and vaccination coverage were the most critical parameters affecting the system. In addition, the study found that direct transmission plays an essential role in the occurrence of HA infection. In contrast, indirect transmission contributes marginally but significantly to the prevalence of HA infection.

A mathematical model for the Hepatitis A Virus (HAV) epidemiology with dual transmission mechanisms is developed and presented. The model considers vaccination and sanitation as mitigation strategies. The effective reproductive number was derived and employed to study the stability of the model. Using Routh's stability criteria, the local stability of a disease-free equilibrium was determined, whereas the global stability of the endemic equilibrium was attained through a suitable Lyapunov function. Furthermore, bifurcation analysis is carried out using the centre manifold theory to ascertain its nature and implication for disease control. It was revealed that the model exhibits a forward bifurcation indicating the possibility of disease eradication when the effective reproduction number is kept below unity. Numerical results indicate that infection rates decrease quantitatively when at least one control measure is effectively implemented. It was deduced that combining vaccination and sanitation yields even fewer cases, making it the best alternative for eliminating Hepatitis A (HA) infection from the community. A sensitivity analysis was conducted to ascertain the parameters of the strong influence that could significantly affect the system. It was revealed that constant recruitment and vaccination coverage were the most critical parameters affecting the system. In addition, the study found that direct transmission plays an essential role in the occurrence of HA infection. In contrast, indirect transmission contributes marginally but significantly to the prevalence of HA infection.

A mathematical model for the Hepatitis A Virus (HAV) epidemiology with dual transmission mechanisms is developed and presented. The model considers vaccination and sanitation as mitigation strategies. The effective reproductive number was derived and employed to study the stability of the model. Using Routh's stability criteria, the local stability of a disease-free equilibrium was determined, whereas the global stability of the endemic equilibrium was attained through a suitable Lyapunov function. Furthermore, bifurcation analysis is carried out using the centre manifold theory to ascertain its nature and implication for disease control. It was revealed that the model exhibits a forward bifurcation indicating the possibility of disease eradication when the effective reproduction number is kept below unity. Numerical results indicate that infection rates decrease quantitatively when at least one control measure is effectively implemented. It was deduced that combining vaccination and sanitation yields even fewer cases, making it the best alternative for eliminating Hepatitis A (HA) infection from the community. A sensitivity analysis was conducted to ascertain the parameters of the strong influence that could significantly affect the system. It was revealed that constant recruitment and vaccination coverage were the most critical parameters affecting the system. In addition, the study found that direct transmission plays an essential role in the occurrence of HA infection. In contrast, indirect transmission contributes marginally but significantly to the prevalence of HA infection.

In this work we considered nonlinear ordinary differential equations to study the dynamics of hepatitis B virus (HBV) epidemics within the host. We proved that the invariant and bounded ness of the solution of the dynamical system. We used a nonlinear stability analysis method for proving the local and global stability of the existing equilibrium points. We found that the diseases free equilibrium point and endemic equilibrium point exist for some conditions.We proved that the disease free equilibrium point is locally asymptotically stable and also globally asymptotically stable. We found that the basic reproduction number for the system is R0 = pθ δ+ω μ+π+θ + qπ λ+η μ+π+θ which depends on nine parameters. Using standard parameter estimation we found that the numerical value of the basic reproduction number is R0 = 2.944234214. From this numerical value we conclude that the disease spreads in the host. Out of these nine parameters we identified four effective parameters which contri...

We consider a plant-pathogen interaction model and perform a bifurcation analysis at the threshold where the pathogen-free equilibrium loses its hyperbolicity. We show that a stimulatory-inhibitory host response to infection load may be responsible for the occurrence of multiple steady states via backward bifurcations. We also find sufficient conditions for the global stability of the pathogen-present equilibrium in case of null or linear inhibitory host response. The results are discussed in the framework of the recent literature on the subject.

Korobeinikov and Wake [9] introduced a family of Lyapunov functions for three-compartmental epidemiological models which appear to be useful for more sophisticated models. In this paper we have reinvestigated the models of Korobeinikov and Wake [9] with different incidences. The basic reproduction number 0 Â is identified and local stability of the equilibrium states is discussed. The Global stability of the equilibrium states is proved by constructing a Lyapunov function. Some numerical simulations are given to illustrate the analytical results.

In this paper a mathematical model has been proposed and analyzed to study the role of reserved zone on the dynamical behavior of prey-predator system in two different cases. In the first case it is assumed that there is a wholly dependent predator, while in the second case it is assumed that the predator is a partially dependent. The dynamical behaviors of the proposed systems have been investigated locally as well as globally. The conditions for the systems to be uniformly persistence have been derived. The existence of a Hopf bifurcation is discussed. Finally numerical simulation is carried out to confirm our obtained results and understand the role of each parameter.

A necessary and sufficient condition for the stability of n = n matrices with real entries is proved. Applications to asymptotic stability of equilibria for vector fields are considered. The results offer an alternative to the well-known Routh᎐Hurwitz conditions.

For a stable matrix A with real entries, sufficient and necessary conditions for A y D to be stable for all non-negative diagonal matrices D are obtained. Implications of these conditions for the stability and instability of constant steadystate solutions to reaction᎐diffusion systems are discussed and an example is given to show applications.

We consider the mathematical model for the viral dynamics of HIV-1 introduced in Rong et al. (2007) [37]. One main feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class. The viral dynamics is described by four nonlinear ordinary differential equations. In Rong et al. (2007) [37], the stability of the infected equilibrium has been analyzed locally. Here, we perform the global stability analysis using two techniques, the Lyapunov direct method and the geometric approach to stability, based on the higher-order generalization of Bendixson's criterion. We obtain sufficient conditions written in terms of the system parameters. Numerical simulations are also provided to give a more complete representation of the system dynamics.

. where ) 0 and f : 0,ϱ ª ‫ޒ‬ is monotonically increasing and concave with Ž . Ž . Ž f 0 -0 semipositone . We establish that f should be appropriately concave by . establishing conditions on f to allow multiple positive solutions. For any ) 0, Ž . we obtain the exact number of positive solutions as a function of f t rt. We follow Ž . Ž . several families of nonlinearities f for which f Ј ϱ [ lim f Ј t ) 0 and study t ª ϱ how the positive solution curves to the above problem evolve. Also, we give 1 The results in this paper were made public at the Conference on Applied Analysis of

We consider the basic model of virus dynamics with noncytolytic loss of infected cells for the infection with viral hepatitis B. Stability of the infection-free steady state and existence, uniqueness and stability of the infected steady state is investigated. The stability results are given in terms of the basic reproductive number R0. Here, we perform the global stability analysis using two techniques, the method of Lyapunov functions and the theory of competitive three dimensional systems. We shall use suitable linear combinations of known functions, common quadratic, composite quadratic and Volterra-type functions, for obtain a suitable Lyapunov function for each steady state. Numerical simulations are presented to illustrate the results.

Chronic HBV affects 350 million people and can lead to death through cirrhosisinduced liver failure or hepatocellular carcinoma. We analyze the dynamics of a model considering logistic hepatocyte growth and a standard incidence function governing viral infection. This model also considers an explicit time delay in virus production. With this model formulation all model parameters can be estimated from biological data; we also simulate a course of lamivudine therapy and find that the model gives good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or a chronic steady state. Our model admits a third possibility of sustained oscillations. We show that when the basic reproductive number is greater than 1 there exists a biologically meaningful chronic steady state, and the stability of this steady state is dependent upon both the rate of hepatocyte regeneration and the virulence of the disease. When the chronic steady state is unstable, simulations show the existence of an attracting periodic orbit. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.

Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV). In this paper, the transmission dynamics of hepatitis B is formulated with a mathematical model with considerations of different classes of individuals, namely immunized, susceptible, latent,infected and recovered class. The role of vaccination of new born babies against hepatitis B and the treatment of both latently and actively infected individuals in controlling the spread are factored into the model. The model in this study is based on the standard SEIR model. The disease-free equilibrium state of the model was established and its stability analyzed using the Routh-Hurwitz theorem. The result of the analysis of the stability of the disease-free equilibrium state shows that hepatitis B can totally be eradicated if effort is made to ensure that the sum of the rate of recovery of the latent class, the rate at which latently infected individuals become actively infected and the rate ...

International audience In this work, we propose a mathematical model to describe the dynamics of the hepatitis B virus (HBV) infection by taking into account the cure of infected cells, the export of precursor cytotoxic T lympho-cytes (CTL) cells from the thymus and both modes of transmission that are the virus-to-cell infection and the cell-to-cell transmission. The local stability of the disease-free equilibrium and the chronic infection equilibrium is obtained via characteristic equations. Furthermore, the global stability of both equilibria is established by using two techniques, the direct Lyapunov method for the disease-free equilibrium and the geometrical approach for the chronic infection equilibrium. Dans ce travail, nous proposons un modèle mathématique pour décrire la dynamique du virus d'hépatite B (HBV) en prenant en compte le taux de guérison de cellules infectées, l'exportation de précurseur cytotoxic des lymphocytes T (CTL) des cellules du thymus et les deux ...

A mathematical model is formulated that enables the understanding of the dynamics of the transmission of serogroup A meningococcal (MenA) infection. We provide the theoretical analysis of the model. We compute the basic reproduction number R 0 and shows that the outcome of the disease depends on a threshold parameter ξ. More precisely, we show that when R 0 ≤ ξ , the disease-free equilibrium is globally asymptotically stable, while when ξ ≤ R 0 ≤ 1, the model exhibits the phenomenon of backward bifurcation, when a stable diseasefree equilibrium co-exists with one or more stable endemic equilibria. Sensitivity analysis of the model has been performed in order to determine the impact of related parameters on meningitis outbreak. Random perturbation of the vaccine efficacy was performed to gain insight into the role of the vaccine efficacy on the stability of the disease-free equilibrium. Theoretical results are supported by numerical simulations, which further suggest that the control of the epidemic of MenA pass through a combination of a large coverage vaccination of young susceptible individuals and the production of a vaccine with a high level of efficacy.

This paper discusses a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growing and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the lost of immunity of newborns. The presence of outsider infectious is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-byimmunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and its suitable reference one.

We formulate and systematically study the global dynamics of a simple model of HBV 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.

Chronic hepatitis B (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined the within-host dynamics of the disease. Most previous models assumed that infected hepatocytes do not proliferate; however, the effect of HBV infection on hepatocyte proliferation is controversial, with conflicting data showing both induction and inhibition of proliferation. With a family of ordinary differential equation (ODE) models, we explored the dynamical impact of proliferation among HBV-infected hepatocytes. Here, we show that infected hepatocyte proliferation in this class of models generates a threshold that divides the dynamics into two categories. Sufficiently compromised proliferation in infected cells produces complex dynamics characterized by oscillating viral loads, whereas higher proliferation generates straightforward dynamics that always results in chronic infection, sometimes with liver failure. A global stability result of the liver failure...

Found in the genital mucosa are antigen presenting Langerhans cells that have characteristics and features that attract R5 HIV towards them. Some characteristics include their ability to capture and degrade R5 HIV and the loss of their antigen presenting properties when they are overwhelmed. Hence, they play both inhibitory and mediatory roles for R5 HIV infection. The project investigates the benefits of modelling R5 HIV dynamics in the Langerhans cells during early HIV infection within the host incorporating treatment with maraviroc as an entry inhibitor. Critical thresholds, the reproduction number (R 0), important in determining the invasion of R5 HIV in the Langerhans cells were derived. The results in this study showed that there exist two equilibrium points. These are the disease free equilibrium (DFE) point and the endemic equilibrium point. Stability analysis of the equilibrium points using the Routh-Hurwitz and the center manifold theory techniques was carried out. Our res...

For a stable matrix A with real entries, sufficient and necessary conditions for A y D to be stable for all non-negative diagonal matrices D are obtained. Implications of these conditions for the stability and instability of constant steadystate solutions to reaction᎐diffusion systems are discussed and an example is given to show applications.

In this paper, we construct a new Lyapunov function for a variety of SIR and SEIR model in epidemiology. Lyapunov functions are used to show that when the basic reproduction ratio is less than or equal to one, the disease-free equilibrium is globally asymptotically stable and the basic reproduction ratio is greater than one, and the endemic equilibrium is also globally asymptotically stable for both models.

The existence of certain m-dimensional structures in a dynamical system implies that the Hamdorff dimension of its attractor is at least m + 1. A Bendixson criterion for the nonexistence of periodic orbits for systems in Hilbert spaces is found.

dedicated to professor jack k. hale on the occasion of his 70th birthday The simplification resulting from reduction of dimension involved in the study of invariant manifolds of differential equations is often difficult to achieve in practice. Appropriate coordinate systems are difficult to find or are essentially local in nature thus complicating analysis of global dynamics. This paper develops an approach which avoids the selection of coordinate systems on the manifold. Conditions are given in terms compound linear differential equations for the stability of equilibria and periodic orbits. Global results include criteria for the nonexistence of periodic orbits and a discussion of the nature of limit sets. As an application, a global stability criterion is established for the endemic equilibrium in an epidemiological model.

A concept of phase asymptotic semiflow is defined. It is shown that any Lagrange stable orbit at which the semiflow is phase asymptotic limits to a stable periodic orbit. A Lagrange stable solution of a C 1 differential equation is considered. When the second compound of the variational equation with respect to this solution is uniformly asymptotically stable and the omega limit set contains no equilibrium, then the semiflow is phase asymptotic at the orbit of the solution and the omega limit set is a stable periodic orbit. Analogous results are obtained for discrete semiflows and periodic differential equations.

Chronic hepatitis B (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined the within-host dynamics of the disease. Most previous models assumed that infected hepatocytes do not proliferate; however, the effect of HBV infection on hepatocyte proliferation is controversial, with conflicting data showing both induction and inhibition of proliferation. With a family of ordinary differential equation (ODE) models, we explored the dynamical impact of proliferation among HBV-infected hepatocytes. Here, we show that infected hepatocyte proliferation in this class of models generates a threshold that divides the dynamics into two categories. Sufficiently compromised proliferation in infected cells produces complex dynamics characterized by oscillating viral loads, whereas higher proliferation generates straightforward dynamics that always results in chronic infection, sometimes with liver failure. A global stability result of the liver failure...

Chronic HBV affects 350 million people and can lead to death through cirrhosisinduced liver failure or hepatocellular carcinoma. We analyze the dynamics of a model considering logistic hepatocyte growth and a standard incidence function governing viral infection. This model also considers an explicit time delay in virus production. With this model formulation all model parameters can be estimated from biological data; we also simulate a course of lamivudine therapy and find that the model gives good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or a chronic steady state. Our model admits a third possibility of sustained oscillations. We show that when the basic reproductive number is greater than 1 there exists a biologically meaningful chronic steady state, and the stability of this steady state is dependent upon both the rate of hepatocyte regeneration and the virulence of the disease. When the chronic steady state is unstable, simulations show the existence of an attracting periodic orbit. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.

The control of severe acute respiratory syndrome (SARS), a fatal contagious viral disease that spread to over 32 countries in 2003, was based on quarantine of latently infected individuals and isolation of individuals with clinical symptoms of SARS. Owing to the recent ongoing clinical trials of some candidate anti-SARS vaccines, this study aims to assess, via mathematical modelling, the potential impact of a SARS vaccine, assumed to be imperfect, in curtailing future outbreaks. A relatively simple deterministic model is designed for this purpose. It is shown, using Lyapunov function theory and the theory of compound matrices, that the dynamics of the model are determined by a certain threshold quantity known as the control reproduction number (Rv). If Rv ≤ 1, the disease will be eliminated from the community; whereas an epidemic occurs if Rv > 1. This study further shows that an imperfect SARS vaccine with infection-blocking efficacy is always beneficial in reducing disease spread within the community, although its overall impact increases with increasing efficacy and coverage. In particular, it is shown that the fraction of individuals vaccinated at steady-state and vaccine efficacy play equal roles in reducing disease burden, and the vaccine must have efficacy of at least 75% to lead to effective control of SARS (assuming R 0 = 4). Numerical simulations are used to explore the severity of outbreaks when Rv > 1. 2000 Mathematics Subject Classification. 92D30.

Analysis of previously published target-cell limited viral dynamic models for pathogens such as HIV, hepatitis, and influenza generally rely on standard techniques from dynamical systems theory or numerical simulation. We use a quasi-steady-state approximation to derive an analytic solution for the model with a non-cytopathic effect, that is, when the death rates of uninfected and infected cells are equal. The analytic solution provides time evolution values of all three compartments of uninfected cells, infected cells, and virus. Results are compared with numerical simulation using clinical data for equine infectious anemia virus, a retrovirus closely related to HIV, and the utility of the analytic solution is discussed.

We consider the dynamics of a general stage-structured predator-prey model which generalizes several known predator-prey, SEIR, and virus dynamics models, assuming that the intrinsic growth rate of the prey, the predation rate, and the removal functions are given in an unspecified form. Using the Lyapunov method, we derive sufficient conditions for the local stability of the equilibria together with estimations of their respective domains of attraction, while observing that in several particular but important situations these conditions yield global stability results. The biological significance of these conditions is discussed and the existence of the positive steady state is also investigated.

In this work, we investigate the hepatitis C virus infection under treatment. We first derive a nonlinear ordinary differential equation model for the studied biological phenomenon. The obtained initial value problem is completely analysed. To begin with the analysis of the model, we use the standard theory of ordinary differential equations to prove existence, uniqueness and boundedness of the solution. Morever, the basic reproduction number R 0 determining the extinction or the persistence of the HCV infection is computed and used to express the equilibrium points. Also the global asymptotic stability of the HCV-uninfected equilibrium point and the HCV-infected equilibrium point of the model are derived by means of appropriate Lyapunov functions. Finally numerical simulations are carried out to confirm theoretical results obtained at HCV-unfected equilibrium.

In this paper, the aim is to analyze the global dynamics of Hepatitis C Virus (HCV) cellular mathematical model under therapy with uninfected hepatocytes proliferation. We prove that the solution of the model with positive initial values are global, positive and bounded. In addition, firstly we show that the model is locally asymptotically stable at free virus equilibrium and also at infected equilibrium. Secondly we show that the model is globally asymptotically stable at the free virus equilibrium by using an appropriate lyapunov function.

The aim of this work is to analyse the global dynamics of an extended mathematical model of Hepatitis C virus (HCV) infection in vivo with cellular proliferation, spontaneous cure and hepatocyte homeostasis. We firstly prove the existence of local and global solutions of the model and establish some properties of this solution as positivity and asymptotic behaviour. Secondly we show, by the construction of appropriate Lyapunov functions, that the uninfected equilibrium and the unique infected equilibrium of the mathematical model of HCV are globally asymptotically stable respectively when the threshold number 0 1 I q d q < − +  and when 0 1 >  .

Mathematical models are used to provide insights into the mechanisms and dynamics of the progression of viral infection in vivo. Untangling the dynamics between HIV and CD4+ cellular populations and molecular interactions can be used to investigate the effective points of interventions in the HIV life cycle. With that in mind, we develop and analyze a stochastic model for In-Host HIV dynamics that includes combined therapeutic treatment and intracellular delay between the infection of a cell and the emission of viral particles. The unique feature is that both therapy and the intracellular delay are incorporated into the model.

A reaction-diffusion system consisting of one, two or three chemical species and taking place in an arbitrary number of spatial dimensions cannot exhibit Turing instability if none of the reaction steps express cross-inhibition. A corollary of this result -obtained by elementary calculations -underlines the importance of nonlinearity in the formation of stationary structures, a kind of self-organization on a chemical basis. Relations to global stability of reaction-diffusion systems, and results on multispecies systems are also mentioned. The statements are not restricted to mass action type models. As a by-product, the solution of a basic inverse problem of formal kinetics is also presented which extends a previous result by to models with arbitrary -including rational -functions as reaction rates so often occurring, e.g., in enzyme kinetics.

In this work we considered nonlinear ordinary differential equations to study the dynamics of hepatitis B virus (HBV) epidemics within the host. We proved that the invariant and bounded ness of the solution of the dynamical system. We used a nonlinear stability analysis method for proving the local and global stability of the existing equilibrium points. We found that the diseases free equilibrium point and endemic equilibrium point exist for some conditions.We proved that the disease free equilibrium point is locally asymptotically stable and also globally asymptotically stable. We found that the basic reproduction number for the system is 𝑅0 = 𝑝𝜃 𝛿+𝜔 𝜇+𝜋+𝜃 + 𝑞𝜋 𝜆+𝜂 𝜇+𝜋+𝜃 which depends on nine parameters. Using standard parameter estimation we found that the numerical value of the basic reproduction number is 𝑅0 = 2.944234214. From this numerical value we conclude that the disease spreads in the host. Out of these nine parameters we identified four effective parameters which contribute significant role in the spread of the disease; and these are the death rate of free virus𝜇, rate of infection of healthy blood cell 𝜋, the rate of cure of infected blood cell 𝜆and the death rate of infected blood cell𝜂. Out of these four effective parameters we identified thatthe most influential parameter is the death rate of free virus 𝜇. We also conduct numerical simulations which support the finding in the sensitivity analysis.

In this study, a system of first order ordinary differential equations is used to analyse the dynamics of cholera disease via a mathematical model extended from Fung (2014) cholera model. The global stability analysis is conducted for the extended model by suitable Lyapunov function and LaSalle's invariance principle. It is shown that the disease free equilibrium (DFE) for the extended model is globally asymptotically stable if Ro < 1 and the disease eventually disappears in the population with time while there exists a unique endemic equilibrium that is globally asymptotically stable whenever Ro> 1 for the extended model or Ro > 1 for the original model and the disease persists at a positive level though with mild waves (i.e few cases of cholera) in the case of Ro > 1. Numerical simulations for strong, weak, and no prevention and control measures are carried out to verify the analytical results and Maple 18 is used to carry out the computations.

The existence of certain m-dimensional structures in a dynamical system implies that the Hamdorff dimension of its attractor is at least m + 1. A Bendixson criterion for the nonexistence of periodic orbits for systems in Hilbert spaces is found.

The global stability for a delayed HIV-1 infection model is investigated. It is shown that the global dynamics of the system can be completely determined by the reproduction number, and the chronic infected equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [

Using patient data from a unique single source outbreak of hepatitis B virus (HBV) infection, we have characterized the kinetics of acute HBV infection by monitoring viral turnover in the serum during the late incubation and clinical phases of the disease in humans. HBV replicates rapidly with minimally estimated doubling times ranging between 2.2 and 5.8 d (mean 3.7 Ϯ 1.5 d). After a peak viral load in serum of nearly 10 10 HBV DNA copies/ml is attained, clearance of HBV DNA follows a two or three phase decay pattern with an initial rapid decline characterized by mean half-life ( t 1/2 ) of 3.7 Ϯ 1.2 d, similar to the t 1/2 observed in the noncytolytic clearance of covalently closed circular DNA for other hepadnaviruses. The final phase of virion clearance occurs at a variable rate ( t 1/2 of 4.8 to 284 d) and may relate to the rate of loss of infected hepatocytes. Free virus has a mean t 1/2 of at most 1.2 Ϯ 0.6 d. We estimate a peak HBV production rate of at least 10 13 virions/day and a maximum production rate of an infected hepatocyte of 200-1,000 virions/day, on average. At this peak rate of virion production we estimate that every possible single and most double mutations would be created each day.