Infection-free Equilibrium Point Research Articles

ANALYSIS AND OPTIMAL CONTROL OF HIV GROWTH MODEL IN THE BODY WITH ANTIRETROVIRAL THERAPY

Human Immunodeficiency Virus (HIV) is a virus that affects the human immune system. HIV infection causes a decrease in the body's immunity because the virus attacks immune-building cells, especially T-CD4 cells. Currently, there is no treatment that can cure or eliminate HIV, but antiretroviral therapy can be done. This study discusses the growth model of HIV in the body that is given control in an effort to maximize healthy T-CD4 cells. In this model, the infection-free and infected equilibrium points are also discussed and their stability is analyzed. Then the optimal control is solved using the Pontryagin Maximum Principle method and solved numerically using the fourth-order Runge Kutta method. Based on the analysis and simulation results, the system is asymptotically stable around the infection-free equilibrium point and unstable around the infected equilibrium point. Simulation results show that with the control of antiretroviral therapy, the T-CD4 cell population grows significantly which can improve the quality of life of patients. And the growth of HIV in the body can be inhibited until it cannot reproduce itself.

Analysis of a non-integer order mathematical model for double strains of dengue and COVID-19 co-circulation using an efficient finite-difference method

An efficient finite difference approach is adopted to analyze the solution of a novel fractional-order mathematical model to control the co-circulation of double strains of dengue and COVID-19. The model is primarily built on a non-integer Caputo fractional derivative. The famous fixed-point theorem developed by Banach is employed to ensure that the solution of the formulated model exists and is ultimately unique. The model is examined for stability around the infection-free equilibrium point analysis, and it was observed that it is stable (asymptotically) when the maximum reproduction number is strictly below unity. Furthermore, global stability analysis of the disease-present equilibrium is conducted via the direct Lyapunov method. The non-standard finite difference (NSFD) approach is adopted to solve the formulated model. Furthermore, numerical experiments on the model reveal that the trajectories of the infected compartments converge to the disease-present equilibrium when the basic reproduction number ({mathbb {R}}_0) is greater than one and disease-free equilibrium when the basic reproduction number is less than one respectively. This convergence is independent of the fractional orders and assumed initial conditions. The paper equally emphasized the outcome of altering the fractional orders, infection and recovery rates on the disease patterns. Similarly, we also remarked the importance of some key control measures to curtail the co-spread of double strains of dengue and COVID-19.

Dynamics of a time fractional order spatio-temporal [formula omitted] with vaccination and temporary immunity

The principal aim of this work it to study and analyze a spatio-temporal fractional Susceptible–Infected–Recovered SIR infection model. We will assume vaccination of susceptible population as well as the temporary immunity of the recovered. The dynamics of the infection is described by three partial differential equations with a fractional derivative order and diffusion for each equation. First, we will prove the existence, uniqueness and boundedness of the solutions. The infection-free equilibrium point and the endemic equilibrium point have been presented as function of the basic reproduction number R0. Next, we will concluded that the local stability of the each equilibrium depends mainly on the basic reproduction number. Finally, the theoretical results are verified by numerical simulations.

Global stability of a general HTLV-I infection model with Cytotoxic T-Lymphocyte immune response and mitotic transmission

Mathematical models have been considered as a robust tool to support biological and medical studies of human viral infections. The global stability of viral infection models remains an important and largely open research problem. Such results are necessary to evaluate treatment strategies for infections and to establish thresholds for treatment rates. Human T-lymphotropic virus class I (HTLV-I) is a retrovirus which infects the CD4+T cells and causes chronic and deadly diseases. In this article, we developed a general nonlinear system of ODEs which describes the within-host dynamics of HTLV-I under the effect Cytotoxic T-Lymphocytes (CTLs) immunity. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearancerates for all types of cells. The incidence rate of infection is also modeled by a general nonlinear function. These general functions are assumed to satisfy a set of suitable conditions and include several forms presented in the literature. We determine a bounded domain for the system’s solutions. We prove the existence of the system’s equilibrium points and determine two threshold numbers, the basic reproductive number R0 and the CTL immunity stimulation number R1. We establish the global stability of all equilibrium points by constructing Lyapunov function and applying Lyapunov-LaSalle asymptotic stability theorem. Under certain conditions it is shown that ifR0≤1, then the infection-free equilibrium point is globally asymptotically stable (GAS) and the HTLV-I infection is cleared. If R1≤1<R0, then the infected equilibrium point without CTL immunity is GAS and the HTLV-I infection becomes chronic with no sustained CTL immune response. If R1>1, then the infected equilibrium point with CTL immunity is GAS and the infection becomes chronic with persistent CTL immune response. We present numerical simulations for the system by choosing special shapes of the general functions. The effect of Crowley-Martin functional response and mitotic division of actively infected cells on the HTLV-I progression are studied. Our results cover and improve several ones presented in the literature.

Stability and Hopf Bifurcation Analysis of a Fractional-order Leslie-Gower Prey-predator-parasite System with Delay

A fractional-order Leslie-Gower prey-predator-parasite system with delay is proposed in this article. The existence and uniqueness of the solutions, as well as their non-negativity and boundedness, are studied. Based on the characteristic equations and the conditions of stability and Hopf bifurcation, the local asymptotic stability of each equilibrium point and Hopf bifurcation of interior equilibrium point are investigated. Moreover, a Lyapunov function is constructed to prove the global asymptotic stability of the infection-free equilibrium point. Lastly, numerical examples are studied to verify the validity of the obtained newly results

Global Dynamics of a Diffusive Two-Strain Epidemic Model with Non-Monotone Incidence Rate.

In this article, we investigate a diffusive two-strain epidemic model with non-monotone incidence rate and virus mutation. The positivity, existence and uniform boundedness of the solutions of the model system are studied. It is found that the system has three equilibrium points, namely the infection-free equilibrium point, the strain-2 endemic equilibrium point and both the strain-1 and strain-2 endemic equilibrium points. The global asymptotic stability analysis of the diffusive model system near all the equilibrium points is carried out by constructing appropriate Lyapunov functional. It is found that the system has no strain-1 endemic equilibrium point possibly due to the virus mutation. So, in this type of diseases, the infection due to strain-1 cannot be persistent in the community.

Optimal control of combined antiretroviral therapies in an HIV infection model with cure rate and fusion effect

This paper mainly targets to deduce an optimal treatment strategy for combined antiretroviral drugs, which can maximize healthy CD4+ T cells level with minimum side effects and cost. For this purpose, we consider a within-host treatment model for the HIV infection with two controls incorporating full logistic proliferation of healthy CD4+ T cells, cure rate and fusion effect. These two controls represent the effects of reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs), respectively. The model analysis begins with proving different basic properties like non-negativity, boundedness of the model solutions and calculation of the basic reproduction number of the model under consideration. Then, stability results are obtained for HIV infection-free equilibrium point and also, a critical efficacy for the combined therapies is calculated. After that, the optimal control problem is proposed and solved numerically using a forward–backward iterative method. Finally, we obtain an optimal treatment strategy that can maximize healthy CD4+ T cells count and control the viral load, and HIV-infected CD4+ T cells count to an undetectable level resulting in improved health conditions of infected individuals.

Optimal Control Studies on Age Structured Modeling of COVID-19 in Presence of Saturated Medical Treatment of Holling Type III.

COVID-19 pandemic has caused the most severe health problems to adults over 60 years of age, with particularly fatal consequences for those over 80. In this case, age-structured mathematical modeling could be useful to determine the spread of the disease and to develop a better control strategy for different age groups. In this study, we first propose an age-structured model considering two different age groups, the first group with population age below 30 years and the second with population age above 30 years, and discuss the stability of the equilibrium points and the sensitivity of the model parameters. In the second part of the study, we propose an optimal control problem to understand the age-specific role of treatment in controlling the spread of COVID -19 infection. From the stability analysis of the equilibrium points, it was found that the infection-free equilibrium point remains locally asymptotically stable when R_0 < 1, and when R_0 is greater than one, the infected equilibrium point remains locally asymptotically stable. The results of the optimal control study show that infection decreases with the implementation of an optimal treatment strategy, and that a combined treatment strategy considering treatment for both age groups is effective in keeping cumulative infection low in severe epidemics. Cumulative infection was found to increase with increasing saturation in medical treatment.

Stability analysis of general delayed HTLV-I dynamics model with mitosis and CTL immunity.

This paper formulates and analyzes a general delayed mathematical model which describe the within-host dynamics of Human T-cell lymphotropic virus class I (HTLV-I) under the effect Cytotoxic T Lymphocyte (CTL) immunity. The models consist of four components: uninfected CD$ 4^{+} $T cells, latently infected cells, actively infected cells and CTLs. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearance rates for all types of cells. The incidence rate of infection is also modeled by a general nonlinear function. These general functions are assumed to be satisfy some suitable conditions. To account for series of events in the infection process and activation of latently infected cells, we introduce two intracellular distributed-time delays into the models: (ⅰ) delay in the formation of latently infected cells, (ⅱ) delay in the activation of latently infected cells. We determine a bounded domain for the system's solutions. We calculate two threshold numbers, the basic reproductive number $ R_{0} $ and the CTL immunity stimulation number $ R_{1} $. We determine the conditions for the existence and global stability of the equilibrium points. We study the global stability of all equilibrium points using Lyapunov method. We prove the following: (a) if $ R_{0}\leq 1 $, then the infection-free equilibrium point is globally asymptotically stable (GAS), (b) if $ R_{1}\leq 1 < R_{0} $, then the infected equilibrium point without CTL immunity is GAS, (c) if $ R_{1} > 1 $, then the infected equilibrium point with CTL immunity is GAS. We present numerical simulations for the system by choosing special shapes of the general functions. The effects of proliferation of CTLs and time delay on the HTLV-I progression is investigated. We noted that the CTL immunity does not play the role in clearing the HTLV-I from the body, but it has an important role in controlling and suppressing the viral infection. On the other hand, we observed that, increasing the time delay intervals can have similar influences as drug therapies in removing viruses from the body. This gives some impression to develop two types of treatments, the first type aims to extend the intracellular delay periods, while the second type aims to activate and stimulate the CTL immune response.

STABILITY ANALYSIS AND OPTIMAL CONTROL OF AN INTRACELLULAR HIV INFECTION MODEL WITH ANTIRETROVIRAL TREATMENT

In this paper, a deterministic model describing the dynamics of the in-host HIV infection of CD4[Formula: see text] T-cells is proposed. The model incorporates the presence of the CD[Formula: see text] T-cells and two types of antiretroviral drugs, for disrupting new infection and for inhibiting virus production, respectively. First, the existence, boundedness and positivity of the model solutions are shown, the basic reproduction number [Formula: see text] being then derived and shown to be a threshold value as far as the stability of the equilibria is concerned. When [Formula: see text] the infection-free equilibrium point is globally stable, whereas when [Formula: see text] the system is uniformly persistent and the infected equilibrium point is globally asymptotically stable. Further, we develop an optimal control model by taking the effect of the antiretroviral drugs to be control variables in order to minimize the HIV infection in different scenarios. By using Pontryagin’s Minimum Principle and solving the model numerically, the results show that each antiretroviral drug in isolation can play a key role in reducing the count of both infected CD4[Formula: see text] T-cells and HIV viruses. However, a combination of both drugs could reduce the in-host HIV infection more significantly.

Dynamics of an SEIR model with infectivity in incubation period and homestead-isolation on the susceptible

In this paper, we present an SEIR epidemic model with infectivity in incubation period and homestead-isolation on the susceptible. We prove that the infection-free equilibrium point is locally and globally asymptotically stable with condition R0<1. We also prove that the positive equilibrium point is locally and globally asymptotically stable with condition R0>1. Numerical simulations are employed to illustrate our results. In the absence of vaccines or antiviral drugs for the virus, our results suggest that the governments should strictly implement the isolation system to make every effort to curb propagation of disease during the epidemic.

Control strategy design for the anti-HBV mathematical model.

Currently, the anti-viral therapy has been extensively utilised to reduce the viral burden and switch off certain infectious sources for hepatitis B virus (HBV) infected patients in clinical treatment. Several pieces of existing evidence have demonstrated that large-scale coverage with anti-viral therapy has obtained a certain great contribution in hygiene and disease control. In this study, an anti-HBV mathematical model is considered and its control strategy of the drug treatment is designed. Based on the Lyapunov theory, this study derives three main theorems to propose three different control strategies, respectively, for drug treatments [inline-formula removed] and [inline-formula removed], such that all states of the anti-HBV model can finally converge to the infection-free equilibrium point [inline-formula removed] asymptotically. Especially, the designed drug treatment [inline-formula removed] or [inline-formula removed] is not a fixed value, but it is time-varying and dependent on states. In Theorem 1, the single drug treatment [inline-formula removed] without [inline-formula removed] is synthesised. Theorem 2 considers the single drug treatment [inline-formula removed] without [inline-formula removed]. In Theorem 3, the combination therapy of [inline-formula removed] and [inline-formula removed] is designed. Finally, there are several simulations to show that the proposed combination therapy is much more effective to cure HBV infected patients than the drug treatment with solely single [inline-formula removed] or single [inline-formula removed].

HIV drug resistance: Insights from mathematical modelling

In 2013 the World Health Organization recommended the initiation of antiretroviral therapy (ART) to any person who tests HIV positive irrespective of his/her CD4+ count. However, implementation of the new guidelines poses a lot of challenges especially in Sub-Sahara Africa such as: drug side effects, drug resistance-mutations and significant financial burdens. Most importantly, it has been established that HIV resistance and subsequent virologic failure occur in a substantial proportion of HIV-infected patients receiving HAART. This study therefore, seeks to investigate the emergence of drug resistant HIV virus during treatment with the aim of determining the proper use of HIV therapy that would lessen drug resistance. To carry out the analysis a ten dimensional in-vivo mathematical model is proposed for HIV dynamics. The model is formulated in such away that it takes into account two virus strain, that is, the wild type as well as the naive type HIV virus. The in-vivo model is shown to be both biologically meaningful and mathematically well posed. The existence of unique infection-free equilibrium point is determined and both its local and global stability investigated. In addition, the basic reproduction number for each viral strain is computed using the next generation matrix method. An optimal control model is proposed and analysed by applying Pontryagin maximum principle, to obtain the optimal drug combination for HIV treatment. Here two drugs, that is, Reverse Transcriptase inhibitor and Protease inhibitor are used as the controls in the model. We provide an objective function for the minimisation of the number of wild type HIV virus and the drug resistant virus as well as the costs associated with the use of Reverse Transcriptase inhibitor and protease inhibitor. The forward backward sweep method is applied to numerically solve the optimality system. From the numerical simulations, it is evident that protease inhibitor is the most effective drug in controlling HIV infection. The results suggest that prolonged use of HAART leads to development of drug resistant and that people with drug-resistant infection could play a core role in the epidemic of HIV.

Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate

The main objective of this manuscript is to study the dynamical behaviour and numerical solution of a HIV/AIDS dynamics model with fusion effect and cure rate. Local and global asymptotic stability of the model is established by Routh-Hurwitz criterion and Lyapunov functional method for infection-free equilibrium point. The numerical solutions of the model has also examined for support of analysis, through Mathematica software.

HIV Infection Control Design With the Form of Constant or State Function

Antiviral therapy widely used in HIV treatment has obtained some achievements in public health. In this study, an HIV infection dynamic mathematical model is considered and three types of the control strategy for this model are designed. On the basis of the Lyapunov theory, we propose three main theorems in which three types of the control strategy for drug treatments $u_{1}(t)$ or/and $u_{2} (t)$ are designed to push all states to the infection-free equilibrium point $E_{1}$ . Theorem 1 or Theorem 2 proposes the monotherapy with $u_{1} (t)$ only or with $u_{2} (t)$ only. The combination therapy with $u_{1}(t)$ and $u_{2} (t)$ together is constructed in Theorem 3 . In these three theorems, $u_{1}(t)$ or/and $u_{2} (t)$ can be a function of states or a fixed constant within a certain range. Finally, there are several simulation results to illustrate that the proposed controls are effective to treat the HIV infection. In the simulation section, there is a detailed discussion about the state responses with the proposed three types of control strategy for drug treatments.

Global stability and optimal control for a hepatitis B virus infection model with immune response and drug therapy

In this paper, a nonlinear mathematical model describing the relationship between hepatitis B virus (HBV), the immune response and the drug therapy is studied. Two main equilibrium points (infection-free and endemic) are obtained. The basic reproduction number is also determined and becomes the threshold for equilibrium point stabilities. We show that when the basic reproduction number is less than one, the infection-free equilibrium point is both locally and globally stable whereas when it is greater than one, the system is uniformly persistent i.e. the virus is endemic and the endemic equilibrium point is globally asymptotically stable. The sensitivity analysis is carried out to seek for potential parameters that could reduce overall HBV infection. Further, by using Pontryagin’s minimum principle, the optimal control problem is constructed with two drug therapy controls. Finally, the numerical simulations are established to show the role of these optimal therapies in controlling viral replication and HBV infection. Our results show that the treatment by inhibiting viral production gives more significant result than the treatment by blocking new infection, however the combination of both treatments is the best strategy to reduce overall HBV infection and the concentration of free virus.

Analysis of SVEIR worm attack model with saturated incidence and partial immunization

Internet worms can propagate across networks at terrifying speeds, reduce network security to a remarkable extent, and cause heavy economic losses. Thus, the rapid elimination of Internet worms using partial immunization becomes a significant matter for sustaining Internet infrastructure. This paper addresses this issue by presenting a novel worm susceptible-vaccinated-exposed-infectious-recovered model, named the SVEIR model. The SVEIR model extends the classical susceptible-exposed-infectious-recovered model (refer to SEIR model) through incorporating a saturated incidence rate and a partial immunization rate. The basic reproduction number in the SVEIR model is obtained. By virtue of the basic reproduction number, we prove the global stabilities of an infection-free equilibrium point and a unique endemic equilibrium point. Numerical methods are used to verify the proposed SVEIR model. Simulation results show that partial immunization is highly effective for eliminating worms, and the SVEIR model is viable for controlling and forecasting Internet worms.

Estudio del Punto Libre de Infección de un Modelo Matemático de la Dinámica de Transmisión del VIH/SIDA en una Población Heterosexual Activa en el Perú con Aplicación a la Salud Pública

Se presenta el análisis de un modelo matemático aplicado a la Epidemiología que explica la dinámica de transmisión de la enfermedad del Síndrome de Inmunodeficiencia Adquirida (SIDA) entre parejas heterosexualmente activas donde se considera el uso de protección por medios naturales y no naturales. Se estudia la sensibilidad del modelo con respecto a su estabilidad local para el Punto Libre de Infección complementando mediante Simulaciones Computacionales. Esto brinda un entendimiento dinámico del modelo con lo cual permitirá tomar decisiones en el área de Salud Pública teniendo en cuenta el aspecto social y económico.

Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response.

Resistance of developed bacteria to antibiotic treatment is a very important issue, because introduction of any new antibiotic is after a little while followed by the formation of resistant bacterial isolates in the clinic. The significant increase in clinical resistance to antibiotics is a troubling situation especially in nosocomial infections, where already defenseless patients can be unsuccessful to respond to treatment, causing even greater health issue. Nosocomial infections can be identified as those happening within 2 days of hospital acceptance, 3 days of discharge or 1 month of an operation. They influence 1 out of 10 patients admitted to hospital. Annually, this outcomes in 5000 deaths only in UK with a cost to the National Health Service of a billion pounds. Despite these problems, antibiotic therapy is still the most common method used to treat bacterial infections. On the other hand, it is often mentioned that immune system plays a major role in the progress of infections. In this context, we proposed a mathematical model defining population dynamics of both the specific immune cells produced according to the properties of bacteria by host and the bacteria exposed to multiple antibiotics synchronically, presuming that resistance is gained through mutations due to exposure to antibiotic. Qualitative analysis found out infection-free equilibrium point and other equilibrium points where resistant bacteria and immune system cells exist, only resistant bacteria exists and sensitive bacteria, resistant bacteria and immune system cells exist. As a result of this analysis, our model highlights the fact that when an individual’s immune system weakens, he/she suffers more from the bacterial infections which are believed to have been confined or terminated. Also, these results was supported by numerical simulations.

Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model.

This study proposes a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. This model has an infection-free equilibrium point and an endemic infection equilibrium point. Using Lyapunov functions and LaSalle's invariance principle shows that if the model's basic reproductive number R0 < 1, the infection-free equilibrium point is globally asymptotically stable, otherwise the endemic infection equilibrium point is globally asymptotically stable. It is shown that a forward bifurcation will occur when R0 = 1. The basic reproductive number R0 of the modified model is independent of plasma total CD4⁺ T cell counts and thus the modified model is more reasonable than the original model proposed by Buonomo and Vargas-De-León. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of two group patients' anti-HIV infection treatments. The simulation results have shown that the first 4 weeks' treatments made the two group patients' R'0 < 1, respectively. After the period, drug resistance made the two group patients' R'0 > 1. The results explain why the two group patients' mean CD4⁺ T cell counts raised and mean HIV RNA levels declined in the first period, but contrary in the following weeks.