Endemic Equilibrium Research Articles

Dynamical behavior of a hepatitis B epidemic model and its NSFD scheme

AbstractHepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number $$\mathcal {R}_0$$ R 0 , stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when $$\mathcal {R}_0=1$$ R 0 = 1 . It is concluded that the endemic equilibrium is globally asymptotically stable when $$\mathcal {R}_0>1$$ R 0 > 1 , by utilizing Dulac’s criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained.

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.

Corruption dynamics: a mathematical model and analysis

This study proposes and analyzes a deterministic mathematical model to describe the dynamics of corruption transmission. We began by proving that the solution to the model is bounded and positive. The next-generation matrix approach is used to compute the basic reproduction number (R0) in relation to corruption-free equilibrium. The Jacobian and Lyapunov functions are used to show that corruption-free equilibrium is asymptotically stable in both locally and globally when R0<1, and otherwise, an endemic corruption equilibrium develops. Furthermore, the sensitivity of the model's parameters was investigated. The findings demonstrate that religious precepts govern public education. The two sectors most susceptible to corruption control are education and corrections. The study recommends investing more in the provision of public education to citizens by creating awareness among all and including it in the education curriculum and religious leaders to teach their followers seriously about the impact of corruption as well as the use of jail as punishment. The numerical simulation results agreed with the analytical results.

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

Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic.

Study on SEAI Model of COVID-19 Based on Asymptomatic Infection

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when R0<1, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R0>1, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease.

Modelling the impact of precaution on disease dynamics and its evolution.

In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction number . For , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics.

Modeling the Transmission Dynamics of Tuberculosis in Humans

Tuberculosis (TB) is a major threat to human health particularly in most of developing countries. In this article, we formulate and analyze a deterministic model for the transmission dynamics of pulmonary and extra-pulmonary tuberculosis. The next generation method is employed to find the basic reproduction number R0 which helps to determine whether TB clears or persists in the human population. Global stability of model equilibria is done through Lyapunov functions whereas the normalized forward sensitivity index method is adopted to determine parameters that drive tuberculosis. Analysis shows that both TB free and endemic equilibria exist. The TB free equilibrium is globally asymptotically stable whenever the basic reproduction number R0 < 1 whereas the endemic equilibrium is globally asymptotically stable whenever R0 > 1. Sensitivity analysis shows that the TB infection rate, the fraction of individuals who progress to pulmonary tuberculosis and its induced death drive TB. Numerical results indicate that when there are no interventions, susceptible humans decline significantly with time until when they are attracted to the steady state whereas latently infected, pulmonary and extra-pulmonary TB individuals increase until when they settle at the equilibrium states supporting the analytical results for existence of the endemic equilibrium. In light of these findings, we recommend treating humans infected with pulmonary TB who are carriers of the disease.

Stability analysis of a delayed SEIRQ epidemic model with diffusion

In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SEIRQ epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SEIRQ model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value $R_0$ of the delayed SEIRQ model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that the unique disease-free equilibrium is asymptotically stable , what is proven by the numericals scchema. Moreover,we search under their condition the endemic equilibrium is asymptotically stable .

Analysis of mathematical model for asthma caused by the effects of environmental pollution

Asthma is a condition in which your airways narrow and swell and may produce extra mucus. In this study, an improved mathematical model of asthma which includes infected class not detected, infected detected class and recovered (relieved) class was formulated. The model exhibits two equilibrium, namely disease free equilibrium and endemic equilibrium. The analysis of positivity and boundedness of solutions showed that the model is epidemiologically well pose d. Simulation results showed that when pollutants are discharged into the environment at a constant rate, the asthma disease cases also increase in the population due to the fact that the interaction rate of both susceptible and infected undetected with pollutants in creases.Finally, asthma can be control by restricting the smokers from the population and the rate of release of pollutants into the environment.

Stability analysis of fractional epidemic model for two infected classes incorporating hospitalization impact

Abstract This article presents an epidemic disease propagation mathematical model in fractional order. The epidemiological characteristics are presented based on the susceptible, exposed, unknown infected, known infected, hospitalized population and the population in the secure zone. Both the disease endemic equilibrium and the disease-free equilibrium's stability characteristics have been examined using the basic reproduction number. Variation of basic reproduction number based on the different sensitive parameters has been discussed. It has been disputed whether the fractional model provides a uniform, reliable solution. An analysis of the time history of unknown and known infected populations, hospitalized populations and recovered populations at different values of various sensitive parameters has been carried out. To support the key theoretical conclusions, some numerical simulations are completed using MATLAB. The impact of various populations on the propagation of the illness has also been investigated, as well as how specific state variables change over time for various fractional order values.

Wiener and Lévy processes to prevent disease outbreaks: Predictable vs stochastic analysis

The study considers stochastic behavior along with modeling, mathematical analysis, theory development, and numerical simulation of the COVID-19 virus. To evaluate current trends and make future projections regarding the basic reproduction number and infection case, we have taken the modified five-compartment SEIRD mathematical model. Since the basic reproduction number R0 is not sufficient to predict the outbreak, we applied the Itô stochastic differential equations (SDEs) along with the Weiner process and Lévy jump to investigate the nature of the disease outbreak. Since COVID-19 is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 “(SARS-CoV-2)”, that has common symptoms including fever, cough, and difficulty breathing, we have illustrated the boundedness and positivity of solutions and investigated the stability of endemic and disease-free equilibrium points. The findings show that the dynamics of disease epidemics are influenced by contact patterns. For all the models, the threshold quantity, R0 is calculated which is the key reason to prove the global and local stability analysis of disease-free equilibrium and endemic equilibrium points. Using the least squares method for data fitting, we have performed a case study on COVID-19 data in Italy for this article. A sensitivity analysis is part of our investigation to find important factors. This paper investigates a stochastic epidemic model (SDE) using Lévy jumps and Weiner processes. Local and global stability in a stochastic setting are examined in the analysis. To view the analytical study, a multitude of numerical results are achieved. The study is important to comprehend the stochastic behavior of the virus in the contagious model in order to prevent similar disease outbreaks in the future.

A deterministic compartment model for analyzing tuberculosis dynamics considering vaccination and reinfection

Tuberculosis is a pressing global health concern, particularly pervasive in many developing nations. This study investigates the influence of treatment failure on tuberculosis control strategies, incorporating vaccination interventions using a deterministic compartmental epidemiological model. Mathematical analysis unveils disease-free and endemic equilibrium points, with the control reproduction number determined using next-generation methods. Identifying endemic equilibrium points and determining the control reproduction number provide essential metrics for assessing the effectiveness of control strategies and guiding policy decisions. The model exhibits a backward bifurcation phenomenon, leading to multiple endemic equilibria despite a reproduction number below one due to reinfection. Sensitivity analysis using Latin Hypercube Sampling/Partial Rank Correlation Coefficient elucidates parameter impacts on the control reproduction number. Vaccination efficacy is crucial for quality and validity, with superior quality and longer validity yielding more significant effects. While reinfection may not directly affect the reproduction number, its influence is pivotal in determining tuberculosis persistence or extinction. This study underscores the intricate interplay of factors in tuberculosis control strategies, providing insights vital for effective interventions and policy formulation.

Modeling and analysis of a fractional order spatio-temporal SEIR model: Stability and prediction

This study introduces a novel fractional-order spatio-temporal SEIR model for epidemic modeling, providing an advanced approach to understanding disease dynamics. Our model, categorizing the population into Susceptible (S), Exposed (E), Infected (I), and Recovered (R), incorporates fractional calculus to accurately reflect the complex, non-linear nature of infectious diseases. Key findings include the confirmation of the existence and uniqueness of the model’s solutions, ensuring reliability for epidemiological predictions. Through rigorous stability analysis at both disease-free and endemic equilibrium points, we identified critical parameters influencing epidemic outcomes. Numerical simulations reveal that the fractional order significantly impacts disease progression, offering valuable insights for intervention strategies.

Prediction of COVID-19 spread with models in different patterns: A case study of Russia

Abstract This study deals with a mathematical model that examines the spread of Coronavirus disease (COVID-19). This model has been handled with different processes such as deterministic, stochastic, and deterministic–stochastic. First of all, a detailed analysis is presented for the deterministic model, which includes the positivity of the solution, the basic reproduction number, the disease, and endemic equilibrium points. Then, for the stochastic model, we investigate under which conditions, the solution exists and is unique. Later, model is reconsidered with the help of the piecewise derivative, which can combine deterministic and stochastic processes. Numerical simulations are presented for all these processes. Finally, the model has been modified with the rate indicator function. The model presenting these four different situations is compared with the real data in Russia. According to the results obtained from these situations, the model that is obtained by adding the rate indicator function predicts the COVID-19 outbreak in Russia more accurately. Thus, it is concluded that the model with the rate indicator function presents more realistic approach than the previous ones.

Stability and codimension 2 bifurcations in an [formula omitted] model with incubation delay

The aim of this paper is to investigate the complex nonlinear dynamics, such as the existence and effect of various types of bifurcations, in an infectious disease model system. An SIR delay model that accounts for saturated incidence, saturated treatment, and self-protection in the susceptible population is considered. The model system has a disease free equilibrium, and its local stability is determined by the basic reproduction number for any delay value. The existence of multiple endemic equilibria is obtained, and these equilibria may switch stability due to variation of bifurcation parameters. When incubation delay is zero, Bogdanov–Takens bifurcation occurs, which implies the existence of Hopf bifurcation, saddle–node, and homoclinic bifurcations. Thus, the disease will persist within or on the homoclinic loop and otherwise dies out. We also find the existence of backward bifurcation and forward (transcritical) bifurcation. The delay-induced stability switches are observed due to Hopf bifurcations, leading to endemic bubbles. Incubation delay affects the stability of equilibria when the model system has multiple endemic equilibria, so multi-stability is observed. Due to incubation delay, the two frequency Hopf–Hopf bifurcation and Bogdanov–Takens bifurcation are found. Thus, both codimension 1 and codimension 2 bifurcations in the effect of incubation delay are exhibited in this study. Numerically, we notice that with increasing incubation delay, the endemic equilibrium either changes stability (from stable to unstable or unstable to stable) or remains unstable with different frequencies of oscillations.

Global stability for SIR models with nonlinear incidence and removal functions

In this work, we provide some sufficient conditions to study the global asymptotic stability of the endemic equilibrium for certain models in mathematical epidemiology with nonlinear incidence and removal functions. We also present numerical examples in order to illustrate our results.

Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási–Albert (BA), Erdös–Rényi (ER), and Watts–Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM<R0. For RM,R0>1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.

An infectious disease epidemic model with migration and stochastic transmission in deterministic and stochastic environments

Understanding population migration is essential for controlling highly infectious diseases. This paper studies the global dynamics of an infectious disease epidemic model incorporating population migration and a stochastic transmission rate. Our findings demonstrate that in deterministic and stochastic environments, the models exhibit global Lyapunov stability near the disease-free equilibrium point, determined by a threshold parameter. Furthermore, we analyze the effect of migration on infectious diseases. We discover that the number of infections and the peak value of the infection curve increase with a higher level of population migration. These results are supported by numerical illustrations that hold epidemiological relevance. Additionally, the disease-free equilibrium of the associated time delay model is linearly asymptotically stable, and the endemic equilibrium exhibits more bifurcation for larger time delay values.

Optimal control of a multi-scale immuno-influenza A transmission model with viral load-dependent infection

Influenza A is a highly contagious respiratory illness that spreads globally and results in millions of cases worldwide. The transmission rate and mortality rate of influenza in the population are determined by the load of the influenza A virus. In this study, we have developed a multi-scale immuno-influenza model considering incomplete vaccine immunity. We have calculated the basic reproduction numbers for both immunological and epidemiological models. Additionally, we have examined the stability of the disease-free equilibrium and the existence of the endemic equilibrium at the population level. To assess the cost-effectiveness of antiviral therapy at an individual level and vaccination at a population level, we have formulated an optimal control problem. The objective of this problem is to minimize both the number of influenza cases and the cost of vaccination. To solve this optimal control problem, we have employed the Ekeland variational principle. Furthermore, we have designed a global forward and backward sweep algorithm to estimate the impact of antiviral treatment on reducing influenza transmission among humans. Our findings indicate that the use of antiviral treatment significantly reduces viral loads in individuals. However, it may also increase the potential risk at the population level. Therefore, vaccination against influenza is the most effective option from a public health perspective.