Stability Of Disease-free Equilibrium Research Articles

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.

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

Modeling the Transmission Dynamics of the Ebola Virus: Effects of Quarantine and Vaccination

Quarantine and vaccination of individuals suspected of exposure to infectious agents are fundamental public health strategies that have historically been employed to mitigate the transmission of contagious diseases within human populations. This study introduced a modified SEIVQRD deterministic model to evaluate the population-level effects of quarantine and vaccination on individuals potentially exposed to the Ebola virus. The study showed that the Model exhibits backward bifurcation when . This implies that even when the reproductive number An unstable endemic and a stable disease-free equilibrium can coexist in less than one. This phenomenon arises from imperfect quarantine and indicates that while is necessary for adequate infection control; it is no longer sufficient and creates additional challenges for effectively controlling Ebola. Furthermore, the sensitivity analysis revealed that the quarantine effectiveness parameter and the parameter related to the isolation of vulnerable individuals had less influence on the incidence of new Ebola cases. However, vaccinating non-quarantined susceptible individuals significantly affects the infection burden and can lower the reproductive value to less than one. Overall, the Model emphasizes the critical role of vaccination in reducing Ebola virus transmission. Although quarantine measures alone may not be sufficient, their combination with vaccination can significantly reduce infection rates.

Simplicial epidemic model with a threshold policy

We establish a simplicial empty-susceptible–infected (ESI) model with consideration of threshold policy to depict the network-based epidemic transmission, where the underlying propagation structures are expanded from edges to higher-order structures. To address the epidemic evolution in an explicit network, we formulate the quenched mean-field probability evolution about each site, which is composed of non-smooth differential equations based on network topology. Remarkably, under the combined action of non-smooth and high-order structures, a tristable state is observed in empirical social networks, which is consistent with the coexistence of three stable equilibria by analysis of the mean-field system. Moreover, we find that a sliding mode exists in empirical social networks, which is also indicated by the theoretical analysis of the mean-field probability equations. Finally, the system is divided into the free subsystem without the threshold policy and control subsystem with the threshold policy. Both subsystems admit a stable disease-free equilibrium and a stable endemic equilibrium, as well as coexistence of a stable disease-free equilibrium and a stable pseudo equilibrium in the system, thereby admitting three types of the bistable state under the policy with different critical levels.

Smart waterborne disease control for a scalable population using biodynamic model in IoT network

We propose a biodynamic model for managing waterborne diseases over an Internet of Things (IoT) network, leveraging the scalability of LoRa IoT technology to accommodate a growing human population. The model, based on fractional order derivatives (FOD), enables smart prediction and control of pathogens that cause waterborne diseases using IoT infrastructure. The human-pathogen-based biodynamic FOD model utilises epidemic parameters (SVIRT: susceptibility, vaccination, infection, recovery, and treatment) transmitted over the IoT network to predict pathogenic contamination in water reservoirs and dumpsites in Iji-Nike, Enugu, the study community in Nigeria. These pathogens contribute to person-to-person, water-to-person, and dumpsite-to-person transmission of disease vectors. Five control measures are proposed: potable water supply, treatment, vaccination, adequate sanitation, and health education campaigns. A stable disease-free equilibrium point is found when the effective reproduction number of the pathogens, R0eff<1 and unstable if R0eff>1. While other studies showed a 98.2% reduction in infections when using IoT alone, this paper demonstrates that combining the SVIRT epidemic control parameters (such as potable water supply and health education campaign) with IoT achieves a 99.89% reduction in infected human populations and a 99.56% reduction in pathogen populations in water reservoirs. Furthermore, integrating treatment with sanitation results in a 99.97% reduction in infected populations. Finally, combining these five control strategies nearly eliminates infection and pathogen populations, demonstrating the effectiveness of multifaceted approaches in public health and environmental management. This study provides a blueprint for governments to plan sustainable smart cities for a growing population, ensuring potable water free from pathogenic contamination,in line with the United Nations Sustainable Development Goals #6 (Clean Water and Sanitation) and #11 (Sustainable Cities and Communities).

An optimal control vaccine model of COVID-19 with cost-effective analysis

This study presents a comprehensive COVID-19 vaccination model, integrating breakthrough transmission, and evaluates its efficacy through various analytical techniques. To assess secondary infections, four effective reproduction rates are computed corresponding to different disease states. The sensitivity analysis of the reproduction rates is conducted using the PRCC to identify influential compartments and parameters. The model is extended to incorporate five time-dependent control interventions to reduce infected compartments, with optimisation performed using the Pontryagin maximum principle. A cost-effectiveness analysis of these strategies is conducted using the average cost-effectiveness ratio. Analysis shows that the model does not undergo backward bifurcation, meaning a stable disease-free equilibrium cannot coexist with a stable endemic equilibrium. The findings show that strict adherence to standard operating procedures is the most effective single intervention. In contrast, a combined approach incorporating all five control interventions is the most effective in averting infections with the lowest ACER.

Optimal control and bifurcation analysis of a delayed fractional-order epidemic model for the COVID-19 pandemic

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed.

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

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 &lt; 1 and an endemic equilibrium when R0 &gt; 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 &gt; 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 &gt; 1 in conditions 1 to 4 while in condition 5 produces R0 &lt; 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 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 &lt; 1 and an asymptotic stable EE whenever Rog &gt; 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.

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 &amp;lt;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 &amp;lt;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.