Mathematical analysis and optimal control for Chikungunya virus with two routes of infection with nonlinear incidence rate

2021 ◽  
pp. 2150088 ◽  
Author(s):  
Miled El Hajji ◽  
Abdelhamid Zaghdani ◽  
Sayed Sayari
Keyword(s):  
Optimal Control ◽  
Steady State ◽  
Chikungunya Virus ◽  
Optimal Solution ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
State Formula ◽  
Number Formula ◽  
Global Threat

Chikungunya fever, caused by Chikungunya virus (CHIKV) and transmitted to humans by infected Aedes mosquitoes, has posed a global threat in several countries. In this paper, we investigated a modified within-host Chikungunya virus (CHIKV) infection model with antibodies where two routes of infection are considered. In a first step, the basic reproduction number [Formula: see text] was calculated and the local and global stability analysis of the steady states is carried out using the local linearization and the Lyapunov method. It is proven that the CHIKV-free steady-state [Formula: see text] is globally asymptotically stable when [Formula: see text], and the infected steady-state [Formula: see text] is globally asymptotically stable when [Formula: see text]. In a second step, we applied an optimal strategy in order to optimize the infected compartment and to maximize the uninfected one. For this, we formulated a nonlinear optimal control problem. Existence of the optimal solution was discussed and characterized using some adjoint variables. Thus, an algorithm based on competitive Gauss–Seidel-like implicit difference method was applied in order to resolve the optimality system. The theoretical results are confirmed by some numerical simulations.

Dynamics of a diffusion-driven HBV infection model with capsids and time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750062 ◽  
Author(s):  
Kalyan Manna
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Dynamical Behavior ◽  
Hbv Infection ◽  
Infection Model ◽  
Direct Lyapunov Method ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Discrete Time Delay ◽  
Number Formula

In this paper, a diffusive hepatitis B virus (HBV) infection model with a discrete time delay is presented and analyzed, where the spatial mobility of both intracellular capsid covered HBV DNA and HBV and the intracellular delay in the reproduction of infected hepatocytes are taken into account. We define the basic reproduction number [Formula: see text] that determines the dynamical behavior of the model. The local and global stability of the spatially homogeneous steady states are analyzed by using the linearization technique and the direct Lyapunov method, respectively. It is shown that the susceptible uninfected steady state is globally asymptotically stable whenever [Formula: see text] and is unstable whenever [Formula: see text]. Also, the infected steady state is globally asymptotically stable when [Formula: see text]. Finally, numerical simulations are carried out to illustrate the results obtained.

Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response

2017 ◽  
Vol 10 (05) ◽  
pp. 1750070 ◽  
Author(s):  
A. M. Ełaiw ◽  
A. A. Raezah ◽  
Khalid Hattaf
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Dynamical Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
State Formula ◽  
Number Formula ◽  
Saturated Incidence ◽  
Ctl Immune Response ◽  
Hiv 1

This paper studies the dynamical behavior of an HIV-1 infection model with saturated virus-target and infected-target incidences with Cytotoxic T Lymphocyte (CTL) immune response. The model is incorporated by two types of intracellular distributed time delays. The model generalizes all the existing HIV-1 infection models with cell-to-cell transmission presented in the literature by considering saturated incidence rate and the effect of CTL immune response. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number ([Formula: see text]) and the CTL immune response activation number ([Formula: see text]). By using suitable Lyapunov functionals, we show that if [Formula: see text], then the infection-free steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text] [Formula: see text], then the CTL-inactivated infection steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text], then the CTL-activated infection steady state [Formula: see text] is globally asymptotically stable. Using MATLAB we conduct some numerical simulations to confirm our results. The effect of the saturated incidence of the HIV-1 dynamics is shown.

scholarly journals Global Dynamics of an HIV Infection Model with Two Classes of Target Cells and Distributed Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. Elaiw
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hiv Infection ◽  
Infection Model ◽  
Target Cells ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Infection Model ◽  
Nonlinear Delay

We investigate the global dynamics of an HIV infection model with two classes of target cells and multiple distributed intracellular delays. The model is a 5-dimensional nonlinear delay ODEs that describes the interaction of the HIV with two classes of target cells, CD4+T cells and macrophages. The incidence rate of infection is given by saturation functional response. The model has two types of distributed time delays describing time needed for infection of target cell and virus replication. This model can be seen as a generalization of several models given in the literature describing the interaction of the HIV with one class of target cells, CD4+T cells. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states of the model. We have proven that if the basic reproduction numberR0is less than unity then the uninfected steady state is globally asymptotically stable, and ifR0>1then the infected steady state exists and it is globally asymptotically stable.

Global dynamics for a live-virus-blood model with distributed time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750067 ◽  
Author(s):  
Ding-Yu Zou ◽  
Shi-Fei Wang ◽  
Xue-Zhi Li
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Live Virus ◽  
Number Formula ◽  
Distributed Time Delay ◽  
The Basic Reproduction Number

In this paper, the global properties of a mathematical modeling of hepatitis C virus (HCV) with distributed time delays is studied. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states. It is shown that if the basic reproduction number [Formula: see text] is less than unity, then the uninfected steady state is globally asymptotically stable. If the basic reproduction number [Formula: see text] is larger than unity, then the infected steady state is globally asymptotically stable.

Global stability and optimal control of a two-patch tuberculosis epidemic model using fractional-order derivatives

2020 ◽  
Vol 13 (03) ◽  
pp. 2050008
Author(s):  
Hossein Kheiri ◽  
Mohsen Jafari
Keyword(s):  
Optimal Control ◽  
Effective Treatment ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Maximum Level ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Fractional Differential ◽  
The Impact

In this paper, we propose a fractional-order and two-patch model of tuberculosis (TB) epidemic, in which susceptible, slow latent, fast latent and infectious individuals can travel freely between the patches, but not under treatment infected individuals, due to medical reasons. We obtain the basic reproduction number [Formula: see text] for the model and extend the classical LaSalle’s invariance principle for fractional differential equations. We show that if [Formula: see text], the disease-free equilibrium (DFE) is locally and globally asymptotically stable. If [Formula: see text] we obtain sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. We extend the model by inclusion the time-dependent controls (effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches), and formulate a fractional optimal control problem to reduce the spread of the disease. The numerical results show that the use of all controls has the most impact on disease control, and decreases the size of all infected compartments, but increases the size of susceptible compartment in both patches. We, also, investigate the impact of the fractional derivative order [Formula: see text] on the values of the controls ([Formula: see text]). The results show that the maximum levels of effective treatment controls in both patches increase when [Formula: see text] is reduced from 1, while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when [Formula: see text] limits to 1.

scholarly journals Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Sun ◽  
Lequan Min
Keyword(s):  
Drug Resistance ◽  
Antiretroviral Therapy ◽  
Hiv Infection ◽  
Equilibrium Point ◽  
Infection Rate ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Rna

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive numberR0of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; ifR0of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive numberR0<1then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’sR0<1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.

scholarly journals Threshold dynamics and threshold analysis of HIV infection model with treatment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zhimin Chen ◽  
Xiuxiang Liu ◽  
Liling Zeng
Keyword(s):  
Hiv Infection ◽  
Time Delays ◽  
Dynamical Behavior ◽  
Infection Model ◽  
Threshold Dynamics ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Immunodeficiency Virus ◽  
Hiv Infection Model

Abstract In this paper, a human immunodeficiency virus (HIV) infection model that includes a protease inhibitor (PI), two intracellular delays, and a general incidence function is derived from biologically natural assumptions. The global dynamical behavior of the model in terms of the basic reproduction number $\mathcal{R}_{0}$ R 0 is investigated by the methods of Lyapunov functional and limiting system. The infection-free equilibrium is globally asymptotically stable if $\mathcal{R}_{0}\leq 1$ R 0 ≤ 1 . If $\mathcal{R}_{0}>1$ R 0 > 1 , then the positive equilibrium is globally asymptotically stable. Finally, numerical simulations are performed to illustrate the main results and to analyze thre effects of time delays and the efficacy of the PI on $\mathcal{R}_{0}$ R 0 .

Dynamical analysis of a reaction–diffusion SEI epidemic model with nonlinear incidence rate

2021 ◽  
pp. 2150041
Author(s):  
Jianpeng Wang ◽  
Binxiang Dai
Keyword(s):  
Steady State ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Positive Steady State

In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] if [Formula: see text], the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if [Formula: see text]. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

scholarly journals Global Stability and Optimal Control of Dengue with Two Coexisting Virus Serotypes

MATEMATIKA ◽  
2019 ◽  
Vol 35 (4) ◽  
pp. 149-170
Author(s):  
Afeez Abidemi ◽  
Rohanin Ahmad ◽  
Nur Arina Bazilah Aziz
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Boundary Equilibrium

This study presents a two-strain deterministic model which incorporates Dengvaxia vaccine and insecticide (adulticide) control strategies to forecast the dynamics of transmission and control of dengue in Madeira Island if there is a new outbreak with a different virus serotypes after the first outbreak in 2012. We construct suitable Lyapunov functions to investigate the global stability of the disease-free and boundary equilibrium points. Qualitative analysis of the model which incorporates time-varying controls with the specific goal of minimizing dengue disease transmission and the costs related to the control implementation by employing the optimal control theory is carried out. Three strategies, namely the use of Dengvaxia vaccine only, application of adulticide only, and the combination of Dengvaxia vaccine and adulticide are considered for the controls implementation. The necessary conditions are derived for the optimal control of dengue. We examine the impacts of the control strategies on the dynamics of infected humans and mosquito population by simulating the optimality system. The disease-freeequilibrium is found to be globally asymptotically stable whenever the basic reproduction numbers associated with virus serotypes 1 and j (j 2 {2, 3, 4}), respectively, satisfy R01,R0j 1, and the boundary equilibrium is globally asymptotically stable when the related R0i (i = 1, j) is above one. It is shown that the strategy based on the combination of Dengvaxia vaccine and adulticide helps in an effective control of dengue spread in the Island.

Stability of a CD4+ T cell viral infection model with diffusion

2018 ◽  
Vol 11 (05) ◽  
pp. 1850071 ◽  
Author(s):  
Zhiting Xu ◽  
Youqing Xu
Keyword(s):  
T Cell ◽  
Viral Infection ◽  
Lyapunov Functions ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Viral Infection Model

This paper is devoted to the study of the stability of a CD[Formula: see text] T cell viral infection model with diffusion. First, we discuss the well-posedness of the model and the existence of endemic equilibrium. Second, by analyzing the roots of the characteristic equation, we establish the local stability of the virus-free equilibrium. Furthermore, by constructing suitable Lyapunov functions, we show that the virus-free equilibrium is globally asymptotically stable if the threshold value [Formula: see text]; the endemic equilibrium is globally asymptotically stable if [Formula: see text] and [Formula: see text]. Finally, we give an application and numerical simulations to illustrate the main results.

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