Endemic Equilibrium Research Articles

Evaluating the risk of malaria transmission within the Central African Republic with the goal of stabilising and eliminating the infection

In this paper, we formulate a multi-compartmental mathematical model for humans and mosquitoes. We construct the system of differential equations for an SEITVR for the human compartment and an SEI for the mosquito compartment. We investigate the outbreak of malaria and its effect on the Central African Republic. The analysis of the compartmental model is carried out using stability analysis and Routh Hurwitz Criterion technique is used to indicate the major impact of the model and to improve the model through minor modifications in the transformation of disease in the population. Our model exhibits two equilibrium points, disease free equilibrium points and endemic equilibrium points. The next generation matrix is used to determine the basic reproduction number R 0. A new compartmental model was framed and estimated the malaria spread after 2023 in the Central African Republic, which is the novelty of this research. Our main motivation is to make the Central African Republic a malaria free country. A Numerical example are provided to validate our results for both the disease free state and endemic state of each model. We believe that this investigation will be more effective in reducing malaria infection and stop spreading.

Mathematical analysis of the effects of precarious health system on COVID-19 transmission dynamics: case of Burkina Faso

n the early moments of the pandemic of coronavirus disease 2019 (COVID-19), the level of effectiveness in treatment and management of patients was limited due to the lack of effective medications and adapted infrastructures in several countries especially in most developing countries. In this study, we propose an SIR-pattern model to investigate the epidemiological effect of delay in hospitalization and case managements. After computing the basic reproduction number thanks to the Next Generation Matrix method, we proved that the model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number is less than one, and condition which ensure the appearance of two endemic equilibria indicating the possibility of having backward bifurcation is determined. When the basic reproduction number exceeds one, Lyapunov functional technique is used to establish the global stability of the unique endemic equilibrium point. Also, we presented a local sensitivity analysis that gives better understanding of the role of each model parameter appearing in the basic reproduction number. Numerical simulation of the model is also performed to corroborate the analytical results. Finally, we fitted our model by using reported data for Burkina Faso, and the results showed that it can be used for predicting the disease outcomes.

A comparison of stochastic and deterministic dynamics of tuberculosis model

Tuberculosis is a deadly infectious disease leading to a major health concern in Southern India while there are major obstacles to controlling its spread. Here, we present a mathematical model with five compartments. We categorize the infected compartment into two subcategories: latent TB-infected individuals and active TB-infected individuals. By introducing white noise in a deterministic system, we formulate a stochastic system. We investigate the model in deterministic and stochastic framework, followed by data calibration using TB infection data from Kanyakumari District in South India from 2019 to 2023. In the deterministic model, we derive the disease-free and endemic equilibrium points, compute the basic reproduction number, and examine their stability. Moreover, we perform sensitivity analysis to evaluate how variations in model parameters affect TB prevalence. In addition, we study the uniqueness of solutions of stochastic model. After that we derive the conditions for disease extinction and stochastic permanence, and execute extensive simulations to capture the variability and randomness in TB transmission dynamics. This study signifies that stochastic dynamics is richer than deterministic one in the way of mitigation TB transmission.

Stationary Distribution and Density Function for a High-Dimensional Stochastic SIS Epidemic Model with Mean-Reverting Stochastic Process

This paper explores a high-dimensional stochastic SIS epidemic model characterized by a mean-reverting, stochastic process. Firstly, we establish the existence and uniqueness of a global solution to the stochastic system. Additionally, by constructing a series of appropriate Lyapunov functions, we confirm the presence of a stationary distribution of the solution under R0s>1. Taking 3D as an example, we analyze the local stability of the endemic equilibrium in the stochastic SIS epidemic model. We introduce a quasi-endemic equilibrium associated with the endemic equilibrium of the deterministic system. The exact probability density function around the quasi-stable equilibrium is determined by solving the corresponding Fokker–Planck equation. Finally, we conduct several numerical simulations and parameter analyses to demonstrate the theoretical findings and elucidate the impact of stochastic perturbations on disease transmission.

The impact of obesity on chronic diseases: type 2 diabetes, heart disease, and high blood pressure

Obesity is a risk factor for several chronic diseases, including type 2 diabetes, heart disease, and high blood pressure. Developing models of obesity and its complications allows for close monitoring of these epidemics, informing the creation of effective control strategies. This research focuses on presenting an endemic model of obesity and its health complications, taking into account, social factors and relationships among people in society. The term epidemic is related to the influence of the health lifestyle of obese people on healthy individuals. The transmission of obesity and its complications is through peer pressure to consume fast food and acquire bad eating habits from relatives and friends. So, in our model, the effect of these factors is expressed with parameters β i . We demonstrate theoretically the asymptotic local stability of disease-free equilibrium and the asymptotic local stability of endemic equilibrium, proving the existence and uniqueness of the solution to the obesity model and its complications. To confirm the model's theoretical results, we conduct numerical simulations to study the impact of variation in social factor parameters on the diseases studied.

Dynamical systems analysis of a reaction-diffusion SIRS model with optimal control for the COVID-19 spread

We examine an SIRS reaction-diffusion model with local dispersal and spatial heterogeneity to study COVID-19 dynamics. Using the operator semigroup approach, we establish the existence of disease-free equilibrium (DFE) and endemic equilibrium (EE), and derive the basic reproduction number, R 0 . Simulations show that without dispersal, reinfection and limited medical resources problems can cause a plateau in cases. Dispersal and spatial heterogeneity intensify localised outbreaks, while integrated control strategies (vaccination and treatment) effectively reduce infection numbers and epidemic duration. The possibility of reinfection demonstrates the need for adaptable health measures. These insights can guide optimised control strategies for enhanced public health preparedness.

Dynamic analysis and optimal control of HIV/AIDS model considering the first 95% target

Based on the level of awareness of the population, an HIV/AIDS model is developed, which focused on the first 95% plan developed by UNAIDS. The threshold of model and the expressions of the disease‐free equilibrium and the endemic equilibrium are calculated, proving the existence of backward bifurcation. Backward bifurcation is caused by the imperfect protection rate of susceptible population due to education. Using China's actual data for parameter fitting, it is found that new HIV infections are on an upward trend. In response to this phenomenon, publicity and education, condoms, screening, and treatment of infected populations are considered as control measures. It is concluded that publicity and education is the primary strategy. This measure can not only effectively reduce the number of infected populations but also effectively increase the awareness rate of HIV‐infected populations. It is recommended to use condoms and have fewer sexual partners during sexual contact. Numerical simulation verifies that early stage publicity and education are much more important than post‐infection screening and treatment measures.

The modeling and mathematical analysis of the fractional-order of Cholera disease: Dynamical and Simulation

In this study, a cholera model with asymptomatic carriers was examined. A Holling type-II functional response function was used to describe disease transmission. For analyzing the dynamical behavior of cholera disease, a fractional-order model was developed. First, the positivity and boundedness of the system’s solutions were established. The local stability of the equilibrium points was also analyzed. Second, a Lyapunov function was used to construct the global asymptotic stability of the system for both endemic and disease-free equilibrium points. Finally, numerical simulations and sensitivity analysis were carried out using matlab software to demonstrate the accuracy and validate the obtained results.

Analysis of COVID-19’s Dynamic Behavior Using a Modified SIR Model Characterized by a Nonlinear Function

This study develops a modified SIR model (Susceptible–Infected–Recovered) to analyze the dynamics of the COVID-19 pandemic. In this model, infected individuals are categorized into the following two classes: Ia, representing asymptomatic individuals, and Is, representing symptomatic individuals. Moreover, accounting for the psychological impacts of COVID-19, the incidence function is nonlinear and expressed as Sg(Ia,Is)=βS(Ia+Is)1+α(Ia+Is). Additionally, the model is based on a symmetry hypothesis, according to which individuals within the same compartment share common characteristics, and an asymmetry hypothesis, which highlights the diversity of symptoms and the possibility that some individuals may remain asymptomatic after exposure. Subsequently, using the next-generation matrix method, we compute the threshold value (R0), which estimates contagiousness. We establish local stability through the Routh–Hurwitz criterion for both disease-free and endemic equilibria. Furthermore, we demonstrate global stability in these equilibria by employing the direct Lyapunov method and La-Salle’s invariance principle. The sensitivity index is calculated to assess the variation of R0 with respect to the key parameters of the model. Finally, numerical simulations are conducted to illustrate and validate the analytical findings.

Estimation of Epidemiological Parameters for COVID-19 Cases in Burkina Faso Using African Vulture Optimization Algorithm (AVOA)

In this paper, we present a mathematical model that accounts for virus transmission through deceased individuals to simulate the dynamics of COVID-19 spread in Burkina Faso. The existence and uniqueness of the model’s solution have been proven. The basic reproduction number was calculated using the Jacobian determinant method. The stability of the disease-free and endemic equilibrium points was studied. To estimate the model parameters, we used African Vulture Optimization Algorithm (AVOA). We then used this algorithm to estimate the parameters of the developed model using daily reported COVID-19 cases in Burkina Faso from March 11 to April 20, 2020. The results obtained show that the proposed model is more realistic for simulating the spread of COVID-19 in Burkina Faso.

Fractional-Order Modeling of COVID-19 Transmission Dynamics: A Study on Vaccine Immunization Failure

COVID-19 is an enveloped virus with a single-stranded RNA genome. The surface of the virus contains spike proteins, which enable the virus to attach to host cells and enter the interior of the cells. After entering the cell, the virus exploits the host cell’s mechanisms for replication and dissemination. Since the end of 2019, COVID-19 has spread rapidly around the world, leading to a large-scale epidemic. In response to the COVID-19 pandemic, the global scientific community quickly launched vaccine research and development. Vaccination is regarded as a crucial strategy for controlling viral transmission and mitigating severe cases. In this paper, we propose a novel mathematical model for COVID-19 infection incorporating vaccine-induced immunization failure. As a cornerstone of infectious disease prevention measures, vaccination stands as the most effective and efficient strategy for curtailing disease transmission. Nevertheless, even with vaccination, the occurrence of vaccine immunization failure is not uncommon. This necessitates a comprehensive understanding and consideration of vaccine effectiveness in epidemiological models and public health strategies. In this paper, the basic regeneration number is calculated by the next generation matrix method, and the local and global asymptotic stability of disease-free equilibrium point and endemic equilibrium point are proven by methods such as the Routh–Hurwitz criterion and Lyapunov functions. Additionally, we conduct fractional-order numerical simulations to verify that order 0.86 provides the best fit with COVID-19 data. This study sheds light on the roles of immunization failure and fractional-order control.

EFFECTS OF PUBLIC AWARENESS AND VACCINATION IN MODELING THE DYNAMICS OF MONKEY-POX TRANSMISSION DISEASE IN NIGERIA

Monkey-pox disease is recognized as pathogens, disturbing animals and humans, it is among the family of orthopox virus and the disease causes lymph nodes to swell. In this paper, we developed a deterministic model system for Monkey pox infection by incorporating public awareness parameter and vaccination individual. The study verified the feasible region of the system equations and non-negativity of the solutions is achieved. The disease free and endemic equilibrium states have been obtained. The study computed and analyzed the reproduction number, of the system equations. The study presented and analysed, the global stability of disease free equilibrium and endemic equilibrium state, it has been found that when there is no transmission between human and non-human (), then meaning it is globally asymptotically stable (GAS) at DFE and using nonlinear lyapunov function, the study shows that the endemic equilibrium state is GAS if and unstable if by comparison method of lyapunov functions. Numerical Simulations were done, it was found that the effective reproduction number decreases as vaccination of individual increases, varying the public awareness, the effective reproduction number reduces to zero and becomes stable as public awareness increases. It was discovered that effective reproduction number decreases as public awareness increases.

A dynamical optimal control theory and cost-effectiveness analyses of the HBV and HIV/AIDS co-infection model.

Studies have shown that the co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B Virus (HBV) poses a major threat to the public health due to their combined negative impacts on health and increased risk of complications. Even though, some scholars formulated and analyzed the HBV and HIV co-infection model they did not consider the compartment that contains protected individuals against both HBV and HIV infections. They incorporated the optimal control theory and cost-effectiveness analysis simultaneously. With this in mind, we are motivated to formulate and analyze the HBV and HIV co-infection model, considering the protected group and incorporating optimal control theory and cost-effectiveness. In this study, we have theoretically computed all of the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points. The two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. We reformulated the optimal control problem by incorporating five time-dependent control measures and conducted its theoretical analysis by utilizing the Pontryagin's maximum principle. Using the fourth order Runge-Kutta numerical method and MATLAB ODE45, we performed the numerical simulations with various combinations of control efforts to verify the theoretical results and investigate the impacts of the suggested protection and treatment control strategies for both the HBV and HIV diseases. Also, we carried out a cost-effectiveness analysis of the proposed control strategies. Eventually, we compared our model results with other researcher similar model results whenever cost-effectiveness analysis is not carried out the findings of this particular study suggest that implementing each of the proposed control strategies simultaneously has a high potential to reduce and control the spread of HBV and HIV co-infections in the community. According to the cost-effectiveness analysis, implementing the HBV treatment and the HIV and HBV co-infection treatment measures has a high potential effect on reducing and controlling the HBV and HIV co-infection transmission problem in the community.

Modelling Of The Weed Invasion And Control Strategies For Mesquite Plants

The new study presents an innovative model for examining the spread of the mesquite species, which is well-known for its extensive presence around the world thanks to water dispersion in various areas. The goal is to assess the stability of the system locally and globally using epidemiological modeling techniques. The approach involved analyzing the basic reproduction number (R₀) using the next-generation matrix to find the coefficients for the Lyapunov function. The results indicated that, the absence of mesquite remains stable on both local and global scale when R₀ is less than 1, but when R₀ exceeds 1, an endemic equilibrium is observed. Numerical simulations using Runge Kutta ode45 algorithm implemented in MATLAB. The study demonstrated that, an increase in the number of water bodies flowing on land, leads to a faster spread and invasion of mesquite. The study suggests the use of this model in areas where the mesquite species is native to raise awareness, inform decision-making, and educate communities, ultimately aiding efforts to manage and eliminate the species.

Investigation of an optimal control strategy for a cholera disease transmission model with programs

Cholera is a disease of poverty affecting people with inadequate access to safe water and basic sanitation. Conflict, unplanned urbanization and climate change all increase the risk of cholera. In this article, an optimal control deterministic mathematical model of cholera disease with cost-effectiveness analysis is developed and analyzed considering both direct and indirect contact transmission pathways. The model qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction number of the model is obtained. We also performed sensitivity analysis of the basic parameters of the model. Then an optimal control problem is designed with a control functional having five controls: vaccination, treatment, environment sanitation and personal hygiene, and water quality improvement program. We examined the existence and uniqueness of the optimal controls of the system. Through the implementation of Pontryagin's maximum principle, the characterization of the optimal controls optimality system is established. The numerical simulation results the integrated control strategies demonstrated that strategy 2, 7, and 12 are effective programs to combat cholera disease from the community. Based on the local circumstances, available funds, and resources, it is recommended to the government stakeholders and policymakers to execute any one of the three integrated intervention programs.

Rich dynamics of a hepatitis C virus infection model with logistic proliferation and time delays

AbstractIn this paper, we study a dynamic model of hepatitis C virus (HCV) infection with density‐dependent proliferation of uninfected and infected hepatocytes and two time delays, which is derived from a three‐dimensional model by the quasi‐steady‐state approximation. The model can exhibit forward bifurcation or backward bifurcation, and an explicit control threshold parameter is obtained for the case of backward bifurcation. It is shown that if the proliferation rate of infected hepatocytes is greater than the proliferation rate of uninfected hepatocytes by a certain amount, it becomes more difficult for the virus to be removed. The model has rich dynamical properties: (i) In some parameter regions, bistability can occur; (ii) both time delays (virus‐to‐cell delay) and (cell‐to‐cell delay) can lead to Hopf bifurcations; (iii) same length of time delays and can lead to at most one stability switch, but different time delays can lead to multiple stability switches. Several sufficient conditions for the global stability of the disease‐free equilibrium and the endemic equilibrium are obtained for both forward and backward bifurcation scenarios. In particular, several sharp results on global stability are obtained. Theoretical and numerical results portray the complexity of viral evolutionary dynamics in chronic HCV‐infected patients.

Spatiotemporal dynamics in a fractional diffusive SIS epidemic model with mass action infection mechanism.

This paper is concerned with spatiotemporal dynamics of a fractional diffusive susceptible-infected-susceptible (SIS) epidemic model with mass action infection mechanism. Concretely, we first focus on the existence and stability of the disease-free and endemic equilibria. Then, we give the asymptotic profiles of the endemic equilibrium on small and large diffusion rates, which can reveal the impact of dispersal rates and fractional powers simultaneously. It is worth noting that we have some counter-intuitive findings: controlling the flow of infected individuals will not eradicate the disease, but restricting the movement of susceptible individuals will make the disease disappear.

Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns.

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

Mathematical Model for Antimicrobial Resistance Transmission Dynamics in Hospitals

In this study, we formulate and analyze a mathematical model to describe the possible transmission routes of Antimicrobial Resistance (A.M.R) in a hospital setting. We have examined the significant means of transmission of resistance in hospitals and found that the significant means of transmission is the use of antibiotics and through the contamination by health care workers. It has been shown that the resistance free equilibrium point is locally asymptotically stable. We have also shown that the model has a unique positive endemic equilibrium point which is locally asymptotically stable.

Numerical Simulation of SEIR Model for COVID-19 Infection

One of the most basic models in epidemiology, the SEIR (Susceptible-Exposed-Infectious-Recovered) model explains how viral infections spread through communities. This study analyzes the SEIR model's differential equations to learn more about its dynamic behaviors. This study discusses the 2019 corona virus disease (COVID-19) epidemic model using numerical methods of susceptible exposed infected recovered (SEIR). A numerical description of the notion is provided using two numerical approaches, including forward Euler and RK-4 approaches. The subplots of SEIR model are drawn by ode 45 commands. The stability of disease free equilibrium and Endemic equilibrium points are depicted. The significance of comprehending the interaction between epidemiological variables and population dynamics in developing efficient public health treatments is brought to light in this study, which investigates the consequences of these equations on the transmission and management of infectious diseases. Insights into the dynamics of illness transmission and potential measures for mitigating its effects are presented through the use of mathematical analysis and computational simulations. Jagannath University Journal of Science, Volume 11, Number 1, June 2024, pp. 172−185