Stability Of Disease-free Equilibrium Research Articles

A model for the control of transmission dynamics of human monkeypox disease in Sub-Saharan Africa

The Human Monkeypox virus has received significant research interest in recent times. While few researchers have studied the effects of vaccination on human-to-animal or animal-to-human Monkeypox transmission, others just studied the effects of treatment on human Monkeypox. In this article, we made the proposition of a deterministic vaccine model that deals with the dynamics of the effects of vaccination and treatment on human Monkeypox in sub-Saharan Africa. We investigated the effects of vaccination on the various epidemiological classes qualitatively. The findings from the analysis of the model are that the model possesses two equilibria, locally asymptotically stable disease-free equilibrium (DFE) when an epidemiological threshold - the effective reproductive number is less than unity, and locally asymptotically stable endemic equilibrium when the number is greater than unity. We then corroborated the theoretical findings with numerical simulations, which reveal that when the rate of vaccination is increased resulting in many newly born persons in the populace being vaccinated, the prevalence of the deadly scourge is significantly reduced, while newly born individuals that miss vaccination experience a drastic torment of the deadly disease that often occasion death. Further revelation from the simulations is that when greater efforts

Mathematical Modeling of the Transmission Dynamics of Gumboro Disease

Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number Rog is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever Rog<1 and an asymptotic stable EE whenever Rog>1. The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical findings.

Estimation of Parameters for the Mathematical Model of the Spread of Hepatitis B in Burkina Faso Using Grey Wolf Optimizer

In this paper, we developed a mathematical model of differential susceptibility, taking into account vaccination and treatment, to simulate the transmission of the hepatitis B virus in the population of Burkina Faso. The existence and uniqueness of non-negative solutions are proved. The model has a globally asymptotically stable disease free equilibrium when the basic reproduction number R0 < 1 and an endemic equilibrium when R0 > 1. We estimated the parameters of the model based on hepatitis B cases from 1997 to 2020 by using a Grey Wolf Optimizer Algorithm (GWO). The results demonstrated the efficacy of the GWO algorithm in estimating the model parameters. A sensitivity analysis was carried out to determine the determining factors in the spread of hepatitis B in Burkina Faso. The estimated parameters were used to simulate the spread of hepatitis B in Burkina Faso from 1997 to 2020.

ANALYSIS AND SIMULATION OF THE SIR MODEL ON THE SPREAD OF COVID-19 BY CONSIDERING THE VACCINATION FACTOR

Covid-19 is a serious respiratory disease that can be fatal for those affected. Governments have tried various strategies to conquer the Covid-19 pandemic. One of them is to vaccinate people with 6 years old and over. The vaccination program aims to form herd immunity so that the number of confirmed positive cases can be reduced. The purpose of this research is to form a mathematical model of the SIR (Susceptible-Infected-Recovery) spread of Covid-19 by considering vaccination factors. The SIR model is combined with a vaccination factor to forestall the unfold of Covid-19. The research method includes deriving models of nonlinear differential equation systems, solving qualitative models, deriving the basic reproduction ratio ( ), analysis of equilibrium points, and building simulation models. This model has an asymptotically stable disease-free equilibrium point. At the same time, the endemic equilibrium point is unstable. Model simulation is obtained by using different parameter values. This is proven through the outcomes of the model analysis vaccination coverage is a key parameter that can be controlled to reduce so that the pandemic ends soon.

Mathematical Modelling and Numerical Simulation of Hepatitis B Viral Infection: The Case of Burkina Faso

Hepatitis is a viral infection that can cause inflammation of the liver and lead to severe liver damage and even death. The study of hepatitis in Burkina Faso is crucial for several reasons. Indeed, understanding the epidemiology of hepatitis in Burkina Faso can help develop effective prevention and control strategies, and its study can contribute to a better understanding of the global burden of the disease and the development of effective interventions in other parts of the world. To this aim, a new differential susceptibility and infectivity mathematical model of Hepatitis B transmission was developed in order to simulate the potential spread of the Hepatitis B virus in the population of Burkina Faso. Once the mathematical model is presented, the existence and uniqueness of non-negative solutions are proved. The model has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R01 and a globally asymptotically stable endemic equilibrium when R0 > 1. The global asymptotic stability of the disease-free equilibrium has been studied using the Castillo Chavez method [5]. The Lyapunov function and the LaSalle invariance principle are used to prove the global asymptotic stability of the endemic equilibrium [26, 4, 18]. To simulate the proposed model, a Matlab numerical code has been developed. Numerical simulations are performed using data of Burkina Faso. The obtained numerical results confirm analytical results as well as the evolution of hepatitis B in Burkina Faso.

Dynamic Analysis of a Mathematical Model of the Anti-Tumor Immune Response

This study discusses the dynamic analysis, the Hopf bifurcation, and numerical simulations. The mathematical model of the anti-tumor immune response consists of three compartments namely Immature T Lymphocytes (L1), Mature T Lymphocytes (L2) and Tumor Cells (T). This research was conducted to represent the behavior between immune cells and tumor cells in the body with five γ conditions. Where γ is the intrinsic growth rate of mature T lymphocytes. This study produces R0 > 1 in conditions 1 to 4 while in condition 5 produces R0 < 1. The disease-free equilibrium point is stable only in condition 5, while the endemic equilibrium point is stable only in conditions 2 and 4. Hopf bifurcation occurs at the endemic equilibrium point. Numerical simulation graph in condition 1 shows that tumor cells will increase uncontrollably. In condition 2 the graph show that the endemic equilibrium point for large tumors is stable. In condition 3 the graph show that there will be a bifurcation from the endemic equilibrium point by the disturbance of the parameter value γ. In condition 4 the graph show the small tumor endemic equilibrium point is stable. Finally, in condition 5, the graph show a stable disease-free equilibrium point.

Mathematical dynamics for HIV infections with public awareness and viral load detectability

In this paper, we develop a nonlinear deterministic model that incorporates public awareness and treatment to describe the dynamics of HIV/AIDS in an infected population with detectable and undetectable viral load. The model undergoes backward bifurcation in which a stable disease-free equilibrium coexists with a stable endemic equilibrium. Numerical simulations carried out show the behavior of the state variables and the impact of public awareness in controlling the spread of HIV. The results show that public awareness will help in curtailing the spread of HIV infection, and when treatment is applied to infected individuals with detectable viral load can easily suppress their virus to become undetectable so that they cannot transmit HIV through sexual intercourse.

Treatment outcome in an SI model with evolutionary resistance: a Darwinian model for the evolution of resistance

We consider a Darwinian (evolutionary game theoretic) version of a standard susceptible-infectious SI model in which the resistance of the disease causing pathogen to a treatment that prevents death to infected individuals is subject to evolutionary adaptation. We determine the existence and stability of all equilibria, both disease-free and endemic, and use the results to determine conditions under which the treatment will succeed or fail. Of particular interest are conditions under which a successful treatment in the absence of resistance adaptation (i.e. one that leads to a stable disease-free equilibrium) will succeed or fail when pathogen resistance is adaptive. These conditions are determined by the relative breadths of treatment effectiveness and infection transmission rate distributions as functions of pathogen resistance.

A study on Zika–Dengue coinfection model with microcephaly newborn dynamics

A study of data on the Zika outbreak in Brazil in 2015–2016 provides knowledge that Zika infection can trigger brain disorders such as Guillain–Barré Syndrome in adults and microcephaly in newborns. Zika infection is a vector-borne disease most commonly transmitted to humans through an infected Aedes mosquito bite, which is also the primary vector for dengue. Thus, many cases of these two diseases were recorded in the same population. In this study, we formulated a Zika–dengue coinfection model that describes the combined dynamics of Zika and dengue, involving the birth of microcephaly-malformed infants from pregnant women infected with the Zika virus. The analysis of the model states that three equilibrium points can be calculated explicitly, namely a disease-free equilibrium point and two boundary endemic equilibrium points. In addition, there is one implicit coinfection equilibrium point. The basic reproduction number is obtained along with the stability of disease-free equilibrium. Condition for the instability of the two boundary equilibria is obtained. This condition allows for a bifurcation point in the numerical exploration of the coinfection equilibrium point and its stability properties. The numerical simulation shows that the coinfection point appears when the two boundary equilibria are unstable. In addition, two Hopf bifurcation points are found with stable limit cycles between them.

STABILITY ANALYSIS OF MATHEMATICAL MODEL AND OPTIMAL CONTROL STRATEGIES FOR REDUCING THE COVID-19 SPREAD

The respiratory infectious disease Coronavirus Disease 2019 caused by the SARS-CoV-2 virus has become a world pandemic. This paper aims to develop a mathematical model of the spread of Covid-19 SEIQR (Susceptible-Exposed-Infected-Quarantine-Recovered), then analyse the stability of the model and its optimal control. Based on the local stability analysis using the Routh-Hurwitz criteria obtained two equilibrium points, namely the local asymptotically stable disease-free equilibrium points if reproduction number less than one and the local asymptotically stable endemic equilibrium point if reproduction number more than one. Next, to analyze global stability at the equilibrium points is used the Lyapunov method. Further, four controls, namely vaccination, physical distancing, treatment of infected individuals, and self-prevention (wearing masks and hand sanitisers) are applied to reduce the spread of covid-19. We applied Pontryagin’s Maximum Principle to obtain the optimal solution for the control. Finally, based on numerical result is found that the value of reproduction number is 9,44183. By using data from province of East Java, Indonesia, it is obtained that the physical distancing control has a better level effectiveness than the others three controls.

Mathematical modeling for the control of fly-borne mastitis disease in cattle

Several diseases cause losses in cattle farming, especially in the dairy industry among which mastitis disease (Bovine mastitis) is the leading cause of health and economic damages globally as it results in animals' ill health and reduced quality and quantity of milk produced by infected cows. Some mathematical studies have been conducted that focused on mastitis transmission from one udder-quarter to another in an infected cow, even though clinical studies established the cow–cow and flies–cow transmissions. The present study, therefore, proposed a mathematical model for the control of mastitis disease in cattle in the presence of flies as vectors. The formulated model was shown to have non-negative solutions in feasible regions for both cattle and flies populations. Furthermore, the model has a stable disease-free equilibrium if the sum of the effective reproduction numbers for cattle–cattle and fly–cattle transmissions (ℜch and ℜch) is less than unity, and there is a possibility of multiple endemic equilibria if otherwise. The numerical results indicated that the infectious populations can be reduced if the rates of the control parameters are increased, thereby curtailing or eradicating mastitis in the cattle population.

Dispersal engendered synchronization and stability of mediated infectious diseases in the patchy environment using mean-field diffusive coupling

Mobility of humans and mediating agents significantly affects the scope of mediated infectious diseases. The spatial transmission of mediated infectious diseases ultimately depends on the movement of hosts and mediators. To comprehend the causes, predict, evaluate, and restrict epidemic transmission, it is crucial to investigate the dynamics of mediated infectious diseases in the presence of mobility. The network of mediators and the host metapopulation are related to the spatial coupling for disease dissemination. Thus, in this article, the metapopulation dynamics of the mediated infectious disease model is studied in a patchy environment where the hosts and mediators population is divided into sub-population. Two networks are used to represent the patchy environment: the hosts or humans network and the mediators network. Mean field diffusive coupling is used to connect the network patches. Both homogeneous and heterogeneous networks are investigated in terms of dynamics. Dispersal engenders the patches of corresponding networks to synchronize and reach bistable states of non-trivial amplitude death. Numerical simulations are performed to demonstrate the change from one state to another state, caused by transcritical bifurcation. The stability of disease free equilibrium and endemic equilibrium is also determined by the reproduction number R0.

Mathematical analysis of the impact of community ignorance on the population dynamics of dengue

This study proposes a dengue spread model that considers the nonlinear transmission rate to address the level of human ignorance of dengue in their environment. The SIR − UV model has been proposed, where SIR denotes the classification of the human population and UV denotes the classification of the mosquito population. Assuming that the total human population is constant, and the mosquito population is already in its steady-state condition, using the Quasi-Steady State Approximation (QSSA) method, we reduce our SIR − UV model into a more simple IR-model. Our analytical result shows that a stable disease-free equilibrium exists when the basic reproduction number is <1. Furthermore, our model also shows the possibility of a backward bifurcation. The more ignorant the society is about dengue, the higher the possibility that backward bifurcation phenomena may appear. As a result, the condition of the basic reproduction number being <1 is insufficient to guarantee the extinction of dengue in a population. Furthermore, we found that increasing the recovery rate, reducing the waning immunity rate, and mosquito life expectancy can reduce the possibility of backward bifurcation phenomena. We use dengue incidence data from Jakarta to calibrate the parameters in our model. Through the fast Fourier transform analysis, it was found that dengue incidence in Jakarta has a periodicity of 52.4, 73.4, and 146.8 weeks. This result indicates that dengue will periodically appear at least every year in Jakarta. Parameter estimation for our model parameters was carried out by assuming the infection rate of humans as a sinusoidal function by determining the three most dominant frequencies. Numerical and sensitivity analyses were conducted to observe the impact of community ignorance on dengue endemicity. From the sensitivity analysis, we found that, although a larger community ignorance can trigger a backward bifurcation, this threshold can be minimized by increasing the recovery rate, prolonging the temporal immunity, or reducing the mosquito population. Therefore, to control dengue transmission more effectively, media campaigns undertaken by the government to reduce community ignorance should be accompanied by other interventions, such as a good treatment in the hospital or vector control programs. With this combination of interventions, it will be easier to achieve a condition of dengue-free population when the basic reproduction number is less than one.

Backward Bifurcation and Hysteresis in a Mathematical Model of COVID19 with Imperfect Vaccine

Vaccination has been used as strategy to eradicate the spread of COVID-19. But imperfect vaccine has been reported to induce backward bifurcation and hysteresis in mathematical models of disease transmission. Backward bifurcation is a phenomenon whereby a stable endemic equilibrium exists contemporaneously with a stable disease-free equilibrium when the basic reproduction number is less than 1. This situation can cause difficulty in controlling an epidemic because the basic reproduction is no longer the only means of eradicating the disease. In this paper, we propose a mathematical model for the transmission of disease which includes imperfect vaccination. We show that our model is capable of capturing backward bifurcation under certain conditions. By using parameters that are relevant to COVID-19 transmission in Malaysia, our numerical analysis shows that low vaccine efficacy can trigger backward bifurcation.

Determining Sensitive Parameters in the Transmission Dynamics of Malaria Disease Using Mathematical Model Approach

The challenge posed by malaria disease calls for an urgent need for a better understanding of important parameters in the disease transmission and development of prevention and control of the spread of malaria disease. In this work, a mathematical model for the dynamics of malaria disease is developed and analyzed. There is existence of disease free equilibrium and endemic equilibrium point of the model, the local stability of disease free equilibrium is obtained using Jacobian matrix which is locally asymptotically stable whenever the basic reproduction number is less than unity. Finally, the results obtained in Table 2, Figure 6 and Figure 8 from sensitivity analysis reveal that malaria disease can be controlled if the biting rate of mosquito is eliminated in the population.

On the Optimal Control of HIV-TB Co-Infection and Improvement of Workplace Productivity

Human immunodeficiency virus (HIV) and tuberculosis (TB) have long been known to have a synergistic relationship. This is a result of each of the diseases impacting negatively on the immune system of the infected persons. The impact of these diseases on workforce productivity is studied in this paper from the viewpoint of dynamical systems. In this paper, we present a nonlinear ordinary differential equation model to study the dynamics of HIV-TB co-infection and its effect on workforce productivity. The main model is first decoupled into two basic submodels of HIV-only and TB-only models, whose qualitative properties are presented before the qualitative properties of the main model are studied. While the HIV-only model is shown to have a globally asymptotically stable disease-free equilibrium whenever its basic reproduction number is less than unity, the TB-only model is shown to exhibit backward bifurcation under some conditions. To investigate the impact of various intervention strategies on the control of the co-infection and improvement of workforce productivity, five time-dependent controls (involving transmission prevention for the two diseases, therapy for the two diseases, and capacity building for improved workforce productivity) are incorporated into the basic model to form an optimal control problem, which is qualitatively analyzed using Pontryagin’s maximum principle and numerically simulated. Incremental cost-effectiveness analysis is conducted with the results of the numerical simulations. It is observed that the most cost-effective strategy for fighting the spread of the co-infection with enhanced productivity is that of combining both preventative and curative measures along with skills training.

SENSITIVITY ANALYSIS AND IMPACT OF AN IMPERFECT VACCINE OF TWO STRAINS OF HEPATITIS B VIRUS INFECTION

A mathematical model considering two strains of hepatitis B virus (HBV) chronic carriers, to assess the impact of dose-structured imperfect vaccine, in a population, is designed and analyzed. The model is shown to have a locally and globally asymptotically stable disease-free equilibrium (DFE) whenever its associated reproduction number is numerically less than unity. Numerical analysis of the model shows that with the expected 50% minimum efficacy of the first vaccine dose, vaccinating 55% of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of hepatitis B infection. Such effective control can also be achieved if 50% of the first vaccine dose recipients take the second dose. Threshold analysis reveals that an imperfect HBV vaccine should have positive or negative population-level effect. Latin hypercube sampling–PRCC analysis illustrates that disease transmission rate, birth rate, natural death rate and proportion of children born with maternal immunity are most influential parameters in the disease dynamics. In this paper, the sensitivity analysis based on mathematical and in addition statistical techniques have been performed to determine the significance of the model parameters. It is observed that a number of the parameters play an important role to determine the magnitude of the basic reproduction number. Sensitivity analysis is achieved to determine model parameters’ importance in disease dynamics. It is observed that the reproduction number is the most responsive quantity to the potent transmission rate of HBV and in addition also vital to control the spread of the disease.

Optimal Control and Bifurcation Analysis of HIV Model.

In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R 0 is computed using the next-generation matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R 0 ≤ 1, the disease-free equilibrium is stable both locally and globally whereas if R 0 > 1, based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R 0 = 1, the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin's maximum principle is applied to form an optimality system. Further, forward fourth-order Runge-Kutta's method is applied to obtain the solution of state variables whereas Runge-Kutta's fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population's dynamic behavior.

Modeling and analysis of tuberculosis and pneumonia co-infection dynamics with cost-effective strategies

Tuberculosis and Pneumonia are among the major airborne diseases that shares common clinical traits with high human morbidity and mortality impacts. Thus, the co-infection of the two diseases becomes unavoidable in a community of comorbid populations. Due to scare resources, the huge disease burden co-imposed by these endemics requires an effective intervention to cushion the effect. To achieve this goal, cost–benefit analysis for the optimal controls is deemed necessary. In this study, an SEIR model for the co-dynamics of tuberculosis and pneumonia is formulated. The model was shown to have a globally stable disease-free (endemic) equilibrium where the reproduction number is less (greater) than unity. Sensitivity analysis by partial rank correlation coefficients shows that increasing infection recovery is only one among the three most influential parameters that reduces the magnitude of the diseases. Through Pontryagin′s maximum principle, the number of infected individuals and cost of controls (public health campaign, chemoprophylaxis, oral fluid intake and treatment) was minimized. In implementing the cost–benefit analysis, we employ three techniques: the Infection Averted Ratio (IAR), the Average Cost-effectiveness Ratio (ACER) and the Incremental Cost-effectiveness Ratio (ICER). On comparing the strategies considering cases (A–E), the ICER results showed that, strategy with treating tuberculosis alone is the most effective and cost-saving strategy. But in terms of infections averted, strategy with all the control measures is the best; however, unaffordable since it is highly expensive. We hope that, the findings of this study should help policy-makers when dealing with the co-existence of tuberculosis and pneumonia.

MATHEMATICAL ANALYSIS OF TRANSMISSION DYNAMICS OF HIV/ AIDS IN RISK STRUCTURED POPULATION

A model for assessing the impact of key population at risk in spreading HIV/ AIDS is designed. The model is shown to have a globally asymptotically stable (GAS) disease-free equilibrium whenever the associated reproduction number is less than unity. It has a unique GAS endemic equilibrium whenever reproduction number exceeds unity, if there is no back and forth movement in susceptible classes. It is shown that if infected individuals with high risk abandon their risky behavioral practices, then the disease burden will reduce. Furthermore, if infected individuals with low risk do not adopt risky behaviors then the transmission of the disease will decrease. Partial Rank Correlation Coefficient (PRCC) indices are assessed using all the infection-related classes and basic reproduction number of the model. Uncertainty and sensitivity analysis results show that controlling the effective contact rate, disease-induced death rate in AIDS stage with low and high risk classes will help controlling the transmission of the disease.