Infected Equilibrium Research Articles

Modeling and analysis of a two-strain immuno-epidemiological model with reinfection

In this paper, we formulate a two-strain model with reinfection that combines immunological and epidemiological dynamics across scales, using the COVID-19 pandemic as a case study. Firstly, we conduct a qualitative analysis of both within-host and between-host models. For the within-host model, we prove the existence and stability of equilibria, and Hopf bifurcation occurs from the infection equilibrium with immune response. This implies that, under specific immune states, the virus within an infected individual may persist, and its concentration may also oscillate periodically. For the between-host model, the disease-free equilibrium always exists and is locally asymptotically stable when the epidemiological basic reproduction number ℜ0<1. In addition, the model can have boundary equilibria of strain 1 or strain 2, which are locally asymptotically stable under specific conditions. However, the co-existence equilibrium does not exist. Secondly, to explore the infection and transmission mechanisms of two strain models and obtain reliable parameter values, we utilize statistical data to fit the immuno-epidemiological model. Simultaneously, we conduct an identifiability analysis of the immuno-epidemiological model to ensure the robustness of the fitted parameters. The results demonstrate the reliable estimation of parameter ranges for structurally unidentifiable parameters with minor measurement errors using the affine invariant ensemble Markov Chain Monte Carlo algorithm (GWMCMC). Moreover, simulations illustrate that enhancing treatment of patients infected with BA.2 strains to inhibit the number of viruses released by infected cells can significantly reduce disease spread.

Birth of Catastrophe and Strange Attractors through Generalized Hopf Bifurcations in Covid-19 Transmission Mathematical Model

Coronavirus can be transmitted through the things that people carry or the things where it sticks to after being spread by the sufferer. Instead, various preventive measures have been carried out. We create a new mathematical model that represents Coronavirus that exists in non-living objects, susceptible, and infected subpopulations interaction by considering the Coronavirus transmission through non-living objects caused by susceptible and infected subpopulations along with its prevention to characterize the dynamics of Coronavirus transmission in the population under those conditions. One disease-free and two infection equilibrium points along with their local stability and coexistence are identified. Global stability of the disease-free equilibria and basic reproduction number are also investigated. Changes in susceptible-Coronavirus interaction rate generate Fold and Hopf bifurcations which represent the emergence of a cycle and the collision of two infection equilibrium points respectively. Catastrophe generated by the collision between an attractor and a repeller is found around a Generalized Hopf bifurcation point by changing susceptible-Coronavirus interaction rate and increasing rate of Coronavirus originating from infected subpopulation. It represents a momentary unpredictable dynamics as the effect of Coronavirus addition and infection. Non-chaotic strange attractors that represent complex but still predictable dynamics are also triggered by Generalized Hopf bifurcation when the susceptible-Coronavirus interaction rate and one of the following parameters, i.e. increasing rate of Coronavirus originating from infected subpopulation or infected subpopulation recovery rate vary.

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.

Lyapunov functionals for a general time-delayed virus dynamic model with different CTL responses.

A time-delayed virus dynamic model is proposed with general monotonic incidence, different nonlinear CTL (cytotoxic T lymphocyte) responses [CTL elimination function pyg1(z) and CTL stimulation function cyg2(z)], and immune impairment. Indeed, the different CTL responses pose challenges in obtaining the dissipativeness of the model. By constructing appropriate Lyapunov functionals with some detailed analysis techniques, the global stability results of all equilibria of the model are obtained. By the way, we point out that the partial derivative fv(x,0) is increasing (but not necessarily strictly) in x>0 for the general monotonic incidence f(x,v). However, some papers defaulted that the partial derivative was strictly increasing. Our main results show that if the basic reproduction number R0≤1, the infection-free equilibrium E0 is globally asymptotically stable (GAS); if CTL stimulation function cyg2(z)=0 for z=0 and the CTL threshold parameter R1≤1<R0, then the immunity-inactivated infection equilibrium E1 is GAS; if the immunity-activated infection equilibrium E+ exists, then it is GAS. Two specific examples are provided to illustrate the applicability of the main results. The main results acquired in this paper improve or extend some of the existing results.

Co-infection dynamics between HIV-HTLV-I disease with the effects of Cytotoxic T-lymphocytes, saturated incidence rate and study of optimal control

The spreading of HIV or HTLV-I among the cells has received the great attention in recent modelling study to explore the virus infection dynamics. The co-infection of HIV and HTLV-I with the effect of Cytotoxic T-lymphocytes (CTLs) immune response is also important from epidemiological point of view. To identify the co-infection scenario of HIV and HTLV-I with the CTLs effect we proposed in this paper a six compartmental ODE-model with uninfected, HIV-infected, HTLV-I infected CD4+T cells and free HIV virus particles with HIV specific CTLs and HTLV-I specific CTLs. The rates of infection of the cases are considered here saturated type and proliferation rate of uninfected and HIV infected CD4+ T-cells are of logistic terms. To establish the well-posedness of the model we have shown that the solution of the proposed model is non-negative and bounded. We obtain the basic reproduction number which is the maximum of the HIV-related reproduction and the HTLV-I related reproduction number. Along with the disease free equilibrium point the system contains other seven endemic equilibrium points containing infection by single disease or both. Analytically, we establish the local and global stability conditions of the equilibrium points and also we establish that the system experiences transcritical bifurcation by the generation of only HIV or HTLV-I infected endemic equilibrium point. Using numerical simulations, we validate the theoretical results and found two infection paths, one initiating with HIV and other with HTLV-I, both cases ultimately become co-infected. Finally, using the optimal control analysis we found the optimal policy for treatment using AVR, RTI & PI for HIV or AZT for HTLV-I control and lastly concluded by some recommendations.

Numerical analysis of age-structured HIV model with general transmission mechanism

In this paper, we discuss the numerical representation of the linearly implicit Euler method for an age-structured HIV infection model with a general transmission mechanism. We first define the basic reproduction number of the continuous model, and present the stability results of the equilibriums. For the numerical process, we establish the solvability of the system and the non-negativity and convergence of numerical solutions. In the analysis of the long-term dynamical behavior, this paper mainly focus on the existence of the infection equilibrium determined by the numerical reproduction number R0Δt. To overcome the difficulty caused by the complexity of epidemic transmission mechanisms, the 1-order convergence analysis of numerical basic reproduction numbers R0Δt is implemented by using the properties of the fundamental solution matrix. By a comparison principle, we show that the disease-free equilibrium is globally asymptotically stable if R0Δt<1. Moreover, for R0Δt>1, a unique numerical endemic equilibrium exists, which converges to the exact one, is locally asymptotically stable. Hence, numerical processes visually represent the dynamic properties of nonlinear age-structured HIV models. Finally, some numerical experiments demonstrate the verification and the efficiency of our results.

Global stability of secondary DENV infection models with non-specific and strain-specific CTLs

Dengue virus (DENV) is a highly perilous virus that is transmitted to humans through mosquito bites and causes dengue fever. Consequently, extensive efforts are being made to develop effective treatments and vaccines. Mathematical modeling plays a significant role in comprehending the dynamics of DENV within a host in the presence of cytotoxic T lymphocytes (CTL) immune response. This study examines two models for secondary DENV infections that elucidate the dynamics of DENV under the influence of two types of CTL responses, namely non-specific and strain-specific responses. The first model encompasses five compartments, which consist of uninfected monocytes, infected monocytes, free DENV particles, non-specific CTLs, and strain-specific CTLs. In the second model, latently infected cells are introduced into the model. We posit that the CTL responsiveness is determined by a combination of self-regulating CTL response and a predator-prey-like CTL response. The model's solutions are verified to be nonnegativity and bounded and the model possesses two equilibrium states: the uninfected equilibrium EQ0 and the infected equilibrium EQ⁎. Furthermore, we calculate the basic reproduction number R0, which determines the existence and stability of the model's equilibria. We examine the global stability by constructing suitable Lyapunov functions. Our analysis reveals that if R0≤1, then EQ0 is globally asymptotically stable (G.A.S), and if R0>1, then EQ0 is unstable while EQ⁎ is G.A.S. To illustrate our findings analytically, we conduct numerical simulations for each model. Additionally, we perform sensitivity analysis to demonstrate how the parameter values of the proposed model impact R0 given a set of data. Finally, we discuss the implications of including the CTL immune response and latently infected cells in the secondary DENV infection model. Our study demonstrates that incorporating the CTL immune response and latently infected cells diminishes R0 and enhances the system's stability around EQ0.

A novel within-host model of HIV and nutrition.

In this paper we develop a four compartment within-host model of nutrition and HIV. We show that the model has two equilibria: an infection-free equilibrium and infection equilibrium. The infection free equilibrium is locally asymptotically stable when the basic reproduction number $ \mathcal{R}_0 < 1 $, and unstable when $ \mathcal{R}_0 > 1 $. The infection equilibrium is locally asymptotically stable if $ \mathcal{R}_0 > 1 $ and an additional condition holds. We show that the within-host model of HIV and nutrition is structured to reveal its parameters from the observations of viral load, CD4 cell count and total protein data. We then estimate the model parameters for these 3 data sets. We have also studied the practical identifiability of the model parameters by performing Monte Carlo simulations, and found that the rate of clearance of the virus by immunoglobulins is practically unidentifiable, and that the rest of the model parameters are only weakly identifiable given the experimental data. Furthermore, we have studied how the data frequency impacts the practical identifiability of model parameters.

Stability analysis for a HIV model with cell-to-cell transmission, two immune responses and induced apoptosis

&lt;abstract&gt;&lt;p&gt;In this paper, a dynamic HIV model with cell-to-cell transmission, two immune responses, and induced apoptosis is proposed and studied. First, the non-negativity and boundedness of the solutions of the model are given, and then the exact expression of the basic reproduction number $ R_{0} $ is obtained by using the next generation matrix method. Second, criteria are obtained for the local stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses. Furthermore, the threshold conditions are also derived for the global asymptotic stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses by constructing the suitable Lyapunov function. Finally, some numerical simulations are conducted to verify the theoretical results; the numerical simulation results show that the increase of apoptosis rate had a positive role in the control of viral infection.&lt;/p&gt;&lt;/abstract&gt;

Mathematical modeling and analysis of monkeypox 2022 outbreak with the environment effects using a Cpauto fractional derivative

Monkeypox is a serious global challenge to human health after the COVID-19 pandemic. Although this infection is not new, still many variations have been noticed in its epidemiology. Numerous approaches have been applied to analyze the dynamics of this infection. In this study, we present a mathematical model to study various epidemiological aspects of monkeypox. Transmission from human to animal, human to human, and through the environment (surface) are considered while formulating the proposed model. The model is constructed based on a classical system of seven nonlinear differential equations. Further, the classical epidemic model is reconstructed using the standard Caputo derivative to examine the dynamical aspects of monkeypox disease in the presence of memory effects. Initially, the necessary mathematical properties of the fractional model are carried out. The model exhibits three equilibrium points: monkeypox-free equilibrium, infected animal-free endemic equilibrium, and coexistence equilibrium. Additionally, we give a thorough theoretical analysis that considers solution positivity and stability results of equilibriums of the Caputo monkeypox model. Furthermore, the parameters of the proposed model are estimated using the nonlinear least square method from the reported cases of monkeypox in the United States in a recent outbreak in 2022. Finally, the numerical solution of the model is carried out using the well-known Adams-Bashforth-Moulton scheme and simulation is performed to explore the role of memory index and various preventing measures on the disease incidence.

Stability Switches, Hopf Bifurcation and Chaotic Dynamics in Simple Epidemic Model with State-Dependent Delay

Dynamic behavior investigations of infectious disease models are central to improve our understanding of emerging characteristics of model states interaction. Here, we consider a Susceptible-Infected (SI) model with a general state-dependent delay, which covers an immuno-epidemiological model of pathogen transmission, developed in our early study, using a threshold delay to examine the effects of multiple exposures to a pathogen. The analysis in the previous work showed the appearance of forward as well as backward bifurcations of endemic equilibria when the basic reproductive ratio [Formula: see text] is less than unity. The analysis, in the present work, of the endemically infected equilibrium behavior, through the study of a second order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation on the upper branch of the backward bifurcation diagram and gives the criteria for stability switches. Furthermore, the inclusion of state-dependent delays is shown to entirely change the dynamics of the SI model and give rise to rich behaviors including periodic, torus and chaotic dynamics.

Dynamical behavior of a stochastic HIV model with logistic growth and Ornstein-Uhlenbeck process

In this paper, we investigate a stochastic human immunodeficiency virus (HIV) model with logistic growth and Ornstein-Uhlenbeck process, which is used to describe the pathogenesis and transmission dynamics of HIV in the population. We first validate that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient conditions for the existence of a stationary distribution of the system, which shows the coexistence of all CD4+ T cells and free viruses. Especially, under some mild conditions which are used to ensure the local asymptotic stability of the quasi-chronic infection equilibrium of the stochastic system, we obtain the specific expression of covariance matrix in the probability density around the quasi-chronic infection equilibrium of the stochastic system. In addition, for completeness, we also obtain sufficient criteria for elimination of all infected CD4+ T cells and free virus particles. Finally, several examples together with comprehensive numerical simulations are conducted to support our analytic results.

Global Dynamics of Viral Infection with Two Distinct Populations of Antibodies

This paper presents two viral infection models that describe dynamics of the virus under the effect of two distinct types of antibodies. The first model considers the population of five compartments, target cells, infected cells, free virus particles, antibodies type-1 and antibodies type-2. The presence of two types of antibodies can be a result of secondary viral infection. In the second model, we incorporate the latently infected cells. We assume that the antibody responsiveness is given by a combination of the self-regulating antibody response and the predator–prey-like antibody response. For both models, we verify the nonnegativity and boundedness of their solutions, then we outline all possible equilibria and prove the global stability by constructing proper Lyapunov functions. The stability of the uninfected equilibrium EQ0 and infected equilibrium EQ* is determined by the basic reproduction number R0. The theoretical findings are verified through numerical simulations. According to the outcomes, the trajectories of the solutions approach EQ0 and EQ* when R0≤1 and R0&gt;1, respectively. We study the sensitivity analysis to show how the values of all the parameters of the suggested model affect R0 under the given data. The impact of including the self-regulating antibody response and latently infected cells in the viral infection model is discussed. We showed that the presence of the self-regulating antibody response reduces R0 and makes the system more stabilizable around EQ0. Moreover, we established that neglecting the latently infected cells in the viral infection modeling leads to the design of an overflow of antiviral drug therapy.

Threshold dynamics of an age-structured infectious disease model with limited medical resources

In this paper, an age-structured infectious disease dynamical model that considers two diseases simultaneously but with limited medical resources is proposed and analyzed. The asymptotic smoothness and persistence of the solution semi-flow are investigated. Then conditions for the existence of a global attractor are derived, which means that disease persists when ℜ0>1. By using a Lyapunov function, it is shown that the infection-free equilibrium is globally asymptotically stable if ℜ0<1 and the infection equilibrium is globally asymptotically stable if ℜ0>1. In the presence of limited medical resources, the results suggest that equitable distribution for the limited medical resources is significant when treating low-risk and high-risk diseases and that keeping a resource sharing coefficient at a moderate level helps to eliminate the disease.

Analysis of metapopulation models of the transmission of SARS-CoV-2 in the United States.

During the COVID-19 pandemic, renewal equation estimates of time-varying effective reproduction numbers were useful to policymakers in evaluating the need for and impact of mitigation measures. Our objective here is to illustrate the utility of mechanistic expressions for the basic and effective (or intrinsic and realized) reproduction numbers, [Formula: see text] and related quantities derived from a Susceptible-Exposed-Infectious-Removed (SEIR) model including features of COVID-19 that might affect transmission of SARS-CoV-2, including asymptomatic, pre-symptomatic, and symptomatic infections, with which people may be hospitalized. Expressions from homogeneous host population models can be analyzed to determine the effort needed to reduce [Formula: see text] from [Formula: see text] to 1 and contributions of modeled mitigation measures. Our model is stratified by age, 0-4, 5-9, …, 75+ years, and location, the 50 states plus District of Columbia. Expressions from such heterogeneous host population models include subpopulation reproduction numbers, contributions from the above-mentioned infectious states, metapopulation numbers, subpopulation contributions, and equilibrium prevalence. While the population-immunity at which [Formula: see text] has captured the popular imagination, the metapopulation [Formula: see text] could be attained in an infinite number of ways even if only one intervention (e.g., vaccination) were capable of reducing [Formula: see text] However, gradients of expressions derived from heterogeneous host population models,[Formula: see text] can be evaluated to identify optimal allocations of limited resources among subpopulations. We illustrate the utility of such analytical results by simulating two hypothetical vaccination strategies, one uniform and other indicated by [Formula: see text] as well as the actual program estimated from one of the CDC's nationwide seroprevalence surveys conducted from mid-summer 2020 through the end of 2021.

Traveling waves for a diffusive virus infection model with humoral immunity, cell‐to‐cell transmission, and nonlinear incidence

This paper is concerned with the existence and nonexistence of traveling waves of a virus infection model with humoral immunity, cell‐to‐cell transmission, and general nonlinear incidence. Our results show that the existence of traveling wave solutions is determined not only by the basic reproduction number of virus infection and the antibody response reproduction number but also by the critical wave speed . More precisely, we obtain the existence of traveling wave solution connecting infection‐free equilibrium and antibody‐free infection equilibrium for , and and connecting infection‐free equilibrium and antibody‐present infection equilibrium for , and . The existence of traveling wave solution connecting infection‐free equilibrium and antibody‐present infection equilibrium is discussed. Some numerical simulations are carried out to illustrate our analytical results.

Global asymptotic stability of a delay differential equation model for SARS-CoV-2 virus infection mediated by ACE2 receptor protein

The COVID-19 pandemic has brought a serious threat to human life safety worldwide. SARS-CoV-2 virus mainly binds to the target cell surface receptor ACE2 (Angiotensin-converting enzyme 2 ) through the S protein expressed on the surface of the virus, resulting in infection of target cells. During this infection process, the target cell ACE2 receptor plays a very important mediating role. In this paper, a delay differential equation model containing the mediated effect of target cell receptor is established according to the mechanism of SARS-CoV-2 virus invasion of target cells, and the global stability of the infection-free equilibrium and the infected equilibrium of the model is obtained by using the basic reproduction number ℛ0 and constructing the appropriate Lyapunov functional. The expression of the basic reproduction number ℛ0 intuitively gives the dependence on the expression ratio of the target cell surface ACE2 receptor, which is helpful for the understanding of the mechanism of SARS-CoV-2 virus infection.

Modeling the interplay between albumin-globulin metabolism and HIV infection.

Human immunodeficiency virus (HIV) infection is a major public health concern with 1.2 million people living with HIV in the United States. The role of nutrition in general, and albumin/globulin in particular in HIV progression has long been recognized. However, no mathematical models exist to describe the interplay between HIV and albumin/globulin. In this paper, we present a family of models of HIV and the two protein components albumin and globulin. We use albumin, globulin, viral load and target cell data from simian immunodeficiency virus (SIV)-infected monkeys to perform model selection on the family of models. We discover that the simplest model accurately and uniquely describes the data. The selection of the simplest model leads to the observation that albumin and globulin do not impact the infection rate of target cells by the virus and the clearance of the infected target cells by the immune system. Moreover, the recruitment of target cells and immune cells are modeled independently of globulin in the selected model. Mathematical analysis of the selected model reveals that the model has an infection-free equilibrium and a unique infected equilibrium when the immunological reproduction number is above one. The infection-free equilibrium is locally stable when the immunological reproduction number is below one, and unstable when the immunological reproduction number is greater than one. The infection equilibrium is locally stable whenever it exists. To determine the parameters of the best fitted model we perform structural and practical identifiability analysis. The structural identifiability analysis reveals that the model is identifiable when the immune cell infection rate is fixed at a value obtained from the literature. Practical identifiability reveals that only seven of the sixteen parameters are practically identifiable with the given data. Practical identifiability of parameters performed with synthetic data sampled a lot more frequently reveals that only two parameters are practically unidentifiable. We conclude that experiments that will improve the quality of the data can help improve the parameter estimates and lead to better understanding of the interplay of HIV and albumin-globulin metabolism.

Analysis of a stochastic HIV model with cell-to-cell transmission and Ornstein–Uhlenbeck process

In this paper, we establish and analyze a stochastic human immunodeficiency virus model with both virus-to-cell and cell-to-cell transmissions and Ornstein–Uhlenbeck process, in which we suppose that the virus-to-cell infection rate and the cell-to-cell infection rate satisfy the Ornstein–Uhlenbeck process. First, we validate that there exists a unique global solution to the stochastic model with any initial value. Then, we adopt a stochastic Lyapunov function technique to develop sufficient criteria for the existence of a stationary distribution of positive solutions to the stochastic system, which reflects the strong persistence of all CD4+ T cells and free viruses. In particular, under the same conditions as the existence of a stationary distribution, we obtain the specific form of the probability density around the quasi-chronic infection equilibrium of the stochastic system. Finally, numerical simulations are conducted to validate these analytical results. Our results suggest that the methods used in this paper can be applied to study other viral infection models in which the infected CD4+ T cells are divided into latently infected and actively infected subgroups.

Stability analysis of a fractional model for the transmission of the cochineal

Scale insects are parasitic insects that attack many indoor and outdoor plants, including cacti and succulents. These insects are among the frequent causes of diseases in cacti: for the reason that they are tough, multiply in record time and could be destructive to these plants, although they are considered resistant. Mealybugs feed on the sap of plants, drying them out and discoloring them. In this research, we propose and investigate a fractional model for the transmission of the Cochineal. In the first place, we prove the positivity and boundedness of solutions in order to ensure the well-posedness of the proposed model. The local stability of the disease-free equilibrium and the chronic infection equilibrium is established. Numerical simulations are presented in order to validate our theoretical results.