Cruz Vargas-De-León

Global stability for an HIV-1 infection model including an eclipse stage of infected cells

Abstract 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 Bendixsonʼs 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.

Stability and bifurcation analysis of a vector-bias model of malaria transmission

The vector-bias model of malaria transmission, recently proposed by Chamchod and Britton, is considered. Nonlinear stability analysis is performed by means of the Lyapunov theory and the LaSalle Invariance Principle. The classical threshold for the basic reproductive number, R(0), is obtained: if R(0)>1, then the disease will spread and persist within its host population. If R(0)<1, then the disease will die out. Then, the model has been extended to incorporate both immigration and disease-induced death of humans. This modification has been shown to strongly affect the system dynamics. In particular, by using the theory of center manifold, the occurrence of a backward bifurcation at R(0)=1 is shown possible. This implies that a stable endemic equilibrium may also exists for R(0)<1. When R(0)>1, the endemic persistence of the disease has been proved to hold also for the extended model. This last result is obtained by means of the geometric approach to global stability.

Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes.

A delayed vector-bias model for malaria transmission with incubation period in mosquitoes is studied. The delay t corresponds to the time necessary for a latently infected vector to become an infectious vector. We prove that the global stability is completely determined by the threshold parameter, R₀(τ). If R₀(τ) ≥ 1, the disease-free equilibrium is globally asymptotically stable. If R₀(τ) > 1 a unique endemic equilibrium exists and is globally asymptotically stable. We apply our results to Ross-MacDonald malaria models with an incubation period (extrinsic or intrinsic).

Qualitative analysis and optimal control of an epidemic model with vaccination and treatment

We focus on an epidemic model which incorporates a non-linear force of infection and two controls: an imperfect preventive vaccine given to susceptible individuals and therapeutic treatment given to infectious. We study both the cases of constant and non constant controls. In the case of constant controls we perform a qualitative analysis based on Lyapunov stability which allows to integrate the bifurcation analysis performed in a previous paper. The occurrence of a backward bifurcation is discussed in the perspective of disease control. The case of time-dependent controls is studied by means of the optimal control theory. The strategy is to minimize both the disease burden and the intervention costs. We derive the optimality system and solve it numerically. The characterization of the optimal time profile of the controls, together with the qualitative analysis provides a rather complete picture of the possible outcomes of the model.

Age-dependency in host-vector models: The global analysis

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 hosts 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 Lyapunovs direct method.

Stability and Hopf bifurcation in a delayed viral infection model with mitosis transmission

In this paper we study a model of HCV with saturation and delay, we stablish the local and global stability of system also we stablish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. We present a sensitivity analysis for the basic reproductive number.

Global Stability Results in a SVIR Epidemic Model with Immunity Loss Rate Depending on the Vaccine-Age

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.

Global stability of infectious disease models with contact rate as a function of prevalence index

In this paper, we consider a SEIR epidemiological model with information--related changes in contact patterns. One of the main features of the model is that it includes an information variable, a negative feedback on the behavior of susceptible subjects, and a function that describes the role played by the infectious size in the information dynamics. Here we focus in the case of delayed information. By using suitable assumptions, we analyze the global stability of the endemic equilibrium point and disease--free equilibrium point. Our approach is applicable to global stability of the endemic equilibrium of the previously defined SIR and SIS models with feedback on behavior of susceptible subjects.

GLOBAL PROPERTIES FOR A VIRUS DYNAMICS MODEL WITH LYTIC AND NON-LYTIC IMMUNE RESPONSES, AND NONLINEAR IMMUNE ATTACK RATES

We consider a mathematical model that describes a viral infection with lytic and non-lytic immune responses. One of the main features of the model is that it includes a rate of linear activation of cytotoxic T lymphocytes (CTLs) immune response, a constant production rate of CTLs export from thymus, and a nonlinear attack rate for each immune effector mechanism. Stability of the infection-free equilibrium, and existence, uniqueness and stability of an immune-controlled equilibrium, are investigated. The stability results are given in terms of the basic reproductive number. We use the method of Lyapunov functions to study the global stability of the infection-free equilibrium and the immune-controlled equilibrium. We give a sufficient condition on the non-lytic-immune attack rate for the global asymptotic stability of the immune-controlled equilibrium. By theoretical analysis and numerical simulations, we show that the lytic and non-lytic activities are required to combat a viral infection.

Dynamics of High-Risk Nonvaccine Human Papillomavirus Types after Actual Vaccination Scheme

Human papillomavirus (HPV) has been identified as the main etiological factor in the developing of cervical cancer (CC). This finding has propitiated the development of vaccines that help to prevent the HPVs 16 and 18 infection. Both genotypes are associated with 70% of CC worldwide. In the present study, we aimed to determine the emergence of high-risk nonvaccine HPV after actual vaccination scheme to estimate the impact of the current HPV vaccines. A SIR-type model was used to study the HPV dynamics after vaccination. According to the results, our model indicates that the application of the vaccine reduces infection by target or vaccine genotypes as expected. However, numerical simulations of the model suggest the presence of the phenomenon called vaccine—induced pathogen strain replacement. Here, we report the following replacement mechanism: if the effectiveness of cross-protective immunity is not larger than the effectiveness of the vaccine, then the high-risk nonvaccine genotypes emerge. In this scenario, further studies of infection dispersion by HPV are necessary to ascertain the real impact of the current vaccines, primarily because of the different high-risk HPV types that are found in CC.