Endemic Equilibrium Research Articles

Analysis of a patch epidemic model incorporating population migration and entry–exit screening

This paper presents an SIQR patch model that combines population migration and entry–exit screening. The threshold for disease extinction is determined using the next-generation matrix method. By constructing the Lyapunov function, the global asymptotic stability of the disease-free equilibrium is demonstrated when R0 < 1. The local asymptotic stability of the endemic equilibrium is shown using the Hurwitz criterion, and it is found that the disease is uniformly persistent when R0 > 1. The influence of screening and migration on disease dynamics is discussed via numerical simulations. Our findings highlight the significance of the detection rate as a vital index in disease transmission and emphasize the effectiveness of screening strategies in preventing outbreaks. Therefore, during an outbreak, it is recommended to establish checkpoints in regions with high mobility to identify and isolate potentially infected individuals, thereby reducing the widespread dissemination of the pandemic.

Co-feeding transmission leads to bi-stability of tick-borne disease spread dynamics

Considering that co-feeding transmission depends on the loads of infected ticks on each host, we develop a tick-borne disease dynamics model with co-feeding transmission probability peaking at an intermediate level of infected tick loads. We stratify tick and host population by their infection status and divide the vector population in terms of infection status and post-egg stages (larvae, nymphs and adults). We use the tick population dynamics and disease spread basic reproduction numbers and co-feeding transmission characteristics to describe the disease endemic structure, and show, for the first time, that density-dependent co-feeding transmission provides a novel mechanism for bi-stability. Numerical simulations based on parameters from laboratory and fields data confirm the possibility of bi-stability in biologically realistic settings, and sensitivity analyses show that the nymphal tick load value at which the co-feeding transmission probability reaches the maximum impacts most significantly on the stable endemic equilibrium value.

A mathematical model for varicella-zoster and HIV co-dynamic supported by numerical simulations.

The prevalence of the varicella-zoster virus (VZV) and its correlation underscore its impact on a significant segment of the population. Notably contagious, VZV serves as a risk factor for the manifestation of HIV/AIDS, with its reactivation often signaling the onset of immunodeficiency. Recognizing the concurrent existence of these two diseases, this study focuses on the co-infection dynamics through a deterministic mathematical model. The population is categorized into seven exclusive groups, considering the complexities arising from the interplay of HIV and Zoster. We establish the non-negativity and boundedness of solutions, examine equilibrium points, calculate basic reproduction numbers via the next-generation matrix approach, and analyze the existence and local stabilities of equilibriums using the Routh-Hurwitz stability criteria. The numerical simulations reveal that the model converges to an endemic equilibrium point when the reproduction number exceeds unity. The primary objectives of this study are to comprehensively understand the transmission dynamics of HIV and Zoster in a co-infected population and to provide valuable insights for developing effective intervention strategies. The findings emphasize the importance of addressing these co-infections to mitigate their impact on public health.

STUDY ON DYNAMIC BEHAVIORS OF TUBERCULOSIS MODEL WITH GENDER AND AGE-STRUCTURE

This paper deals with the dynamics of tuberculosis transmission model with different genders and age structures. The basic reproduction numbers [Formula: see text] are defined, where [Formula: see text] and [Formula: see text] are the basic reproduction number of tuberculosis transmission in female and male populations, respectively. The global stability of the disease-free equilibrium was discussed when [Formula: see text]; The existence, uniqueness and global stability of the endemic equilibrium, and the persistence of the system both were proved when [Formula: see text]. The findings indicate that the tuberculosis dynamical model with different genders and age structures will experience the classical threshold dynamic behaviors, that is, if [Formula: see text], the disease will disappear; while if [Formula: see text], the disease will spread within the population, and eventually lead to a endemic disease. Finally, the theoretical analysis of the model is verified by numerical simulation and we proposed that male infected individuals should be focused on in the prevention and control of tuberculosis in the future.

Modeling the Impact of Vaccination on Epidemic Disease Variants with Hospitalization: A Case Study for the COVID-19 Pandemic in Turkey

The stability analysis of an epidemic model that takes into account the impact of vaccination and hospitalization is investigated in this study. Disease-free and endemic equilibrium points are obtained for the stability analysis. The necessary conditions for analyzing local stability at equilibrium points as well as global stability at the disease-free equilibrium point are also defined. Using data from three different periods corresponding to the emergence of three different variants of the COVID-19 outbreak in Turkey, the numerical simulation with graph fitting for the model is also taken into account. The analysis considers the efficacy of vaccination in restricting the virus's spread.

Modelling the Use of the Trace-Test-Isolate-Treat Strategy for Controlling the Spread of COVID-19

During the COVID-19 pandemic that ravaged the entire world between 2019 and 2021, the Trace-Test-Isolate-Treat Strategy was devised as an emergency way of managing the spread of the disease. As the name implies, the Trace-Test-Isolate-Treat Strategy involves identifying those who had contact with an infected person through contact tracing, and subsequent isolation and treatment if confirmed to be infected with the disease. This paper aims to model the transmission dynamics of COVID-19, with the Trace-Test-Isolate-Treat Strategy as a control strategy. To do this, we propose a simple nonlinear system of ordinary differential equations that models COVID-19 dynamics and incorporates the Trace-Test-Isolate-Treat strategy as a way of controlling the spread of the disease. The analysis of the model shows that the disease-free equilibrium is locally asymptotically stable if the reproduction number, R_eff is less than one. Furthermore, the model is shown to possess a unique and stable endemic equilibrium if, R_eff>1. This confirms the global asymptotic stability of the disease-free equilibrium and the absence of backward bifurcation in the model. Numerical plots show the effectiveness of isolation and treatment of infected persons in reducing the spread of the disease.

On the dynamics of a Zika disease model with vector-bias

In this paper, we propose a Zika transmission model which considers human-to-human sexual transmission, the extrinsic incubation period of mosquitoes, and the vector-bias effect. Firstly, the explicit expression of the basic reproduction number [Formula: see text] is given by using the next-generation operator method, and the global dynamics of the model are established by taking [Formula: see text] as the threshold condition, that is, if [Formula: see text], the disease-free equilibrium is globally asymptotically stable, if [Formula: see text], the model has a unique endemic equilibrium that is locally asymptotically stable and the disease persists. And when we ignore the vector-bias effect, the global asymptotic stability of the endemic equilibrium is proved by constructing a Lyapunov function. Then, we select the reported epidemic data from Brazil for fitting, which verifies the obtained theoretical results. Meanwhile, we study the impact of human-to-human sexual transmission rate and mosquito-to-human transmission rate on the spread and prevalence of Zika. In addition, we calculate the sensitivity indices of [Formula: see text] to the model parameters and provide effective measures to control Zika transmission. The simulation results indicate that extending the extrinsic incubation period of mosquitoes is beneficial for disease control while ignoring the vector-bias effect will underestimate the risk of Zika transmission.

A robust Study of the Treatment Delay Effect on the Dynamics of Epidemic Disease

A general treatment function with an epidemic model that involves the delay in the treatment period has been proposed and studied in this work. This model contains two compartments, namely susceptible denoted by and infected denoted by . The existence of all the fixed points has been determined. The system has two equilibrium points, namely the uninfected equilibrium point (UIEP) and the endemic equilibrium point (EEP). The conditions for local stability and Hopf bifurcation have been discussed. The stability of the periodic solutions and the direction of the Hopf bifurcation properties have been studied analytically and numerically.

Existence of Traveling Waves of a Diffusive Susceptible–Infected–Symptomatic–Recovered Epidemic Model with Temporal Delay

The aim of this article is to investigate the existence of traveling waves of a diffusive model that represents the transmission of a virus in a determined population composed of the following populations: susceptible (S), infected (I), asymptomatic (A), and recovered (R). An analytical study is performed, where the existence of solutions of traveling waves in a bounded domain is demonstrated. We use the upper and lower coupled solutions method to achieve this aim. The existence and local asymptotic stability of the endemic (Ee) and disease-free (E0) equilibrium states are also determined. The constructed model includes a discrete-time delay that is related to the incubation stage of a virus. We find the crucial basic reproduction number R0, which determines the local stability of the steady states. We perform numerical simulations of the model in order to provide additional support to the theoretical results and observe the traveling waves. The model can be used to study the dynamics of SARS-CoV-2 and other viruses where the disease evolution has a similar behavior.

Modeling the Mathematical Transmission of a Pneumonia Epidemic Model with Awareness

A new model for the transmission of a pneumonia epidemic, considering awareness and a mass action incidence function, is presented. Stability analyses for disease-free and endemic equilibrium points are conducted. The Basic Reproduction Number (Ro) for pneumonia with awareness is defined and analyzed, showing stability when (Ro < 1) and transitioning to an endemic state when (Ro > 1) . Additionally, a special case is highlighted where the Basic Reproduction Number (Ro*) (without awareness) is greater than (Ro). Basic Reproduction Number with awareness, i.e. (Ro* = 1.0965 > Ro = 0.8772). Furthermore, a numerical simulation is provided to depict how awareness influences the dynamic management of the disease. The results underscore the crucial role of awareness in educating the public about infection risks, ultimately contributing to a decrease in the health burden by mitigating the epidemic peak.

Global stability and Sensitivity Analysis of Malaria, Dengue and Typhoid Triple Infection

This study investigates the global stability of the endemic equilibrium point of triple co-infection of malaria, dengue, and typhoid. By using an appropriate Lyapunov function, the results show that the model is globally asymptotically stable. This implies that the diseases can be eradicated or kept at low levels, regardless of the population. Sensitivity analysis was also conducted to identify the most sensitive parameter. The results indicate that strategies to reduce malaria and dengue fever vectors should be prioritized to curb the spread of the diseases. Additionally, minimizing exposure to contaminated water and food, as well as reducing the discharge of typhoid bacteria into the environment, can help to reduce or curb the spread of typhoid in the environment.

Analysis of a Delayed Multiscale AIDS/HIV-1 Model Coupling Between-Host and Within-Host Dynamics

Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R0h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R˜0h, is rigorously investigated. More specifically, if R˜0h<1, the disease-free equilibrium is globally asymptotically stable; if R˜0h>1, the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1.

Mathematical modeling and stability analysis of the novel fractional model in the Caputo derivative operator: A case study

The fundamental goal of this research is to suggest a novel mathematical operator for modeling visceral leishmaniasis, specifically the Caputo fractional-order derivative. By utilizing the Fractional Euler Method, we were able to simulate the dynamics of the fractional visceral leishmaniasis model, evaluate the stability of the equilibrium point, and devise a treatment strategy for the disease. The endemic and disease-free equilibrium points are studied as symmetrical components of the proposed dynamical model, together with their stabilities. It was shown that the fractional calculus model was more accurate in representing the situation under investigation than the classical framework at α = 0.99 and α = 0.98. We provide justification for the usage of fractional models in mathematical modeling by comparing results to real-world data and finding that the new fractional formalism more accurately mimics reality than did the classical framework. Additional research in the future into the fractional model and the impact of vaccinations and medications is necessary to discover the most effective methods of disease control.

Asymptotic profiles of a spatial vector‐borne disease model with Fokker–Planck‐type diffusion

AbstractThis paper is concerned with a spatially heterogeneous vector‐borne disease model that follows the Fokker–Planck‐type diffusion law. One of the significant features in our model is that Fokker–Planck‐type diffusion is used to characterize individual movement, which poses new challenges to theoretical analysis. We derive for the first time the variational characterization of basic reproduction ratio for the model under certain conditions and investigate its asymptotic profiles with respect to the diffusion rates. Furthermore, via overcoming the difficulty of the associated elliptic eigenvalue problem, the asymptotic behaviors of endemic equilibrium for the model are discussed. Our results imply that whether rapid or slow movement of susceptible and infected individuals are conducive to disease control depends on the degree of disease risk in the habitat. Numerically, we verify the theoretical results and detect that Fokker–Planck‐type diffusion may amplify the scale of disease infection, which in turn increases the complexity of disease transmission by comparing the impacts of distinct dispersal types on disease dynamics.

Stochastic dual epidemic hypothesis model with Ornstein-Uhlenbeck process: Analysis and numerical simulations with SARS-CoV-2 variants

As it is widely known, the spread of infectious diseases can result in significant socioeconomic consequences and pose a threat to public health. However, biologically plausible models that incorporate stochastic interference and dual epidemic hypotheses have received limited attention. This paper aims to bridge this gap by examining a stochastic dual epidemic hypothesis model that incorporates Ornstein-Uhlenbeck processes to perturb the nonlinear incidence rate. We provide a rigorous analysis of the model, first proving the existence and uniqueness of global solution. We also analyze sufficient conditions for the extinction or persistence of each disease. Additionally, we establish the existence of an ergodic stationary distribution and derive expressions for the normal distribution followed by the global solution around the endemic equilibrium. Finally, using several numerical experimental examples, we validate our theoretical results with the case of dual variants of SARS-CoV-2 that can simultaneously infect humans. This paper contributes to the understanding of stochastic dual epidemic hypotheses and provides a foundation for future research in this field.

Mathematical Modelling and Computational Dynamics of the impact of Quasi-Lockdown Policy in Control of COVID-19 in Akwa Ibom State, Nigeria

Determining the impact of a local control strategy, implemented to curtail the spread of COVID-19 has become obligatory to enable policy makers to combat the so-called long covid. This research used a deterministic SEIAHR-model, incorporating a logistic function to investigate the effect of quasi-lockdown policy during reemergence of corona virus pandemic. The notion of uniform convergence and fixed point theory were used to establish positive invariant region, ultimate boundedness, and existence of unique solution of the model. Nonlinear dynamical behaviours of the model such as stability and oscillating flows occurred at the disease free, and endemic equilibrium points. Using centre manifold theory, a trans-critical bifurcation with hysteresis effect were established, when quasi-lockdown policy was used as a control parameter in the model. A forward sensitivity analysis was conducted to unfold parameters that contributed significantly towards the spread and control of the infection. These parameters were estimated and fitted using least square technique, comparative to the observed real datasets adapted from the Nigeria Centre for Disease Control (NCDC). The polynomial regression model fitted the observed datasets with an average coefficient of determination given as R2 = 0.94. Simulations in form of phase space diagrams were used to validate the theoretic results. Epidemiologically, the model demonstrated that quasi-lockdown policy was necessarily a strategic policy to curb the spread of the infection, but not sufficient to eradicate the disease as the disease persisted endemically. It was shown that the basic reproduction number of the infection was within the critical threshold (R0 \(\le\) 1), when the lockdown policy was enforced strictly, but otherwise on relaxation of the lockdown policy (R0 \(\ge\) 1) Thus a relaxation of the quasi-lockdown policy leads to increase in susceptibility of the populace as the infection risk ratio increases in the model. Similarly, the model predicted that a pre-mature lifting of the lockdown policy could have led to high infectivity on the susceptible class and disastrous to healthy living of the citizens.

Modelling the preventive treatment under media impact on tuberculosis: A comparison in four regions of China

Preventive treatment for people with latent Tuberculosis infection (LTBI) has aroused our great interest. In this paper, we propose and analyze a novel mathematical model of TB considering preventive treatment with media impact. The basic reproduction number R0 is defined by the next generation matrix method. In the case without media impact, we prove that the disease-free equilibrium is globally asymptotically stable (unstable) if R0<1(R0>1). Furthermore, we obtain that a unique endemic equilibrium exists when R0>1, which is globally asymptotically stable in the case of permanent immunity and no media impact. We fit the model to the newly reported TB cases data from 2009 to 2019 of four regions in China and estimate the parameters. And we estimated R0=0.5013<1 in Hubei indicating that TB in Hubei will be eliminated in the future. However, the estimated R0=1.015>1 in Henan, R0=1.282>1 in Jiangxi and R0=1.930>1 in Xinjiang imply that TB will continue to persist in these three regions without further prevention and control measures. Besides, sensitivity analysis is carried out to illustrate the role of model parameters for TB control. Our finding reveals that appropriately improving the rate of timely treatment for actively infected people and increasing the rate of individuals with LTBI seeking preventive treatment could achieve the goal of TB elimination. In addition, another interesting finding shows that media impact can only reduce the number of active infections to a limited extent, but cannot change the prevalence of TB.

Exploring the complex dynamics of a diffusive epidemic model: Stability and bifurcation analysis.

The recent pandemic has highlighted the need to understand how we resist infections and their causes, which may differ from the ways we often think about treating epidemic diseases. The current study presents an improved version of the susceptible-infected-recovered (SIR) epidemic model, to better comprehend the community's overall dynamics of diseases, involving numerous infectious agents. The model deals with a non-monotone incidence rate that exhibits psychological or inhibitory influence and a saturation treatment rate. It has been identified that depending on the measure of medical resources and the effectiveness of their supply, the model exposes both forward and backward bifurcations where two endemic equilibria coexist with infection-free equilibrium. The model also experiences local and global bifurcations of codimension two, including saddle-node, Hopf, and Bogdanov-Takens bifurcations. Additionally, the stability of equilibrium points is investigated. For a spatially extended SIR model system, we have shown that cross-diffusion allows S and I populations to coexist in a habitat. Also, the Turing instability requirements and Turing bifurcation regime are derived. The relationship between distinct role-playing model parameters and various pattern formations like spot and stripe patterns is validated by carrying out in-depth numerical simulations. The findings in the vicinity of the endemic equilibrium solution demonstrate the significance of positive and negative valued cross-diffusion coefficients in regulating the genesis of spatial patterns in susceptible as well as diseased individuals. The discussion of the findings of epidemiological ramifications concludes the manuscript.

Stability and control of a plant epidemic model with pesticide intervention

This paper introduces a model for studying plant epidemics that applies pesticides to control disease spread among two types of plant populations: those that are susceptible and those that are already infected. The model uses non-linear ordinary differential equations and the Holling type II response function to depict how disease spreads based on the number of susceptible plants available. The model is carefully checked for biological accuracy, ensuring characteristics such as positivity and boundedness. It defines points of equilibrium where the numbers of susceptible and infected plants stabilize. The study looks at scenarios with no infected plants (disease-free equilibrium) and scenarios where the disease continues to exist within the plant population (endemic equilibrium). The basic reproduction number, R0, is calculated to assess the system's stability. If R0 is less than 1, the disease is unlikely to spread widely, and the system is likely to return to being disease-free, both locally and globally, over time. However, if R0 is greater than 1, it indicates that the disease will persist in the population. This endemic state has also been shown to be stable both locally and globally. A sensitivity analysis helps identify key factors that affect disease spread and assists in forming strategies to manage the disease. Finally, numerical simulations are used to support the findings of the analysis.

Mathematical Analysis of Tuberculosis Transmission Model with Multidrug and Extensively Drug-resistant Incorporating Chemoprophylaxis Treatment

Tuberculosis has remained the principal cause of mortality worldwide, and one of the major sources of concern is drug-resistant TB. The increasing emergence of extensively drug-resistant and multidrug-resistant TB has further increased the TB epidemic. In this current work, we suggest a model to study the transmission of TB with extensively drug-resistant and multidrug-resistant compartments, incorporating chemoprophylaxis treatment. In the theoretical analysis, the concept of the next-generation matrix and the Jacobian method are applied to obtain a formula that states the reproductive number. The existence of endemic and disease-free equilibrium points was checked, and their stability has been analyzed using the Lyapunov method. The qualitative-based analysis indicated the local asymptotic stability of the disease-free-state for R0 1, whereas the endemic state is globally asymptotically stable if R0 1. Moreover, sensitivity analysis was carefully done using normalized forward sensitivity, and numerical simulation was carried out. Based on the results of numerical simulation and sensitivity analysis, chemoprophylaxis treatment was found to drastically minimize the progression of exposed individuals to infectious classes and also reduce the progression to extensively drug-resistant and multidrug-resistant classes, which decreases disease transmission.