Disease-free Equilibrium State Research Articles

A simplicial SSIS epidemic model with the outgoing pressure

A new simplicial epidemic model that considers the pressure of out-going is proposed to describe the characteristics of clustering on disease transmission more accurately. In addition, the probability evolution equations of nodes in each state are obtained by the quenched mean-field method. Furthermore, we analyze the conditions of the existence and the stability of the equilibrium points. Subsequently, the sensitivity analysis of the parameters is investigated, and it can be concluded that the degree about pairwise transmission rate has great impact on the propagation threshold. Our simulation results indicate that the system produces forward bifurcation or backward bifurcation via the one-parameter bifurcation diagram, and the bistable state of the system appears under certain conditions. Meanwhile, we obtain the transition conditions of the system from the disease-free equilibrium state to the bistable state through the divisional diagram. It is also noticed that the pressure of out-going plays a crucial role in the spreading process of diseases. On the one hand, the increasing of the pressure of out-going leads to the decreasing of the disease transmission threshold and a faster outbreak of disease. On the other hand, an increase in the individuals without the pressure of out-going causes the increasing of transmission threshold and a slower outbreak of disease.

Dynamical Analysis of Peptic Ulcer Disease Model in Nigeria with the Effect of Vaccination and Treatment Program

The study developed a non-linear deterministic mathematical model to investigate the transmission dynamics of peptic ulcer disease in human population by considering both direct and indirect contact transmission with vaccine and treatment as control. In developing the model, the population was compartmentalized into susceptible-vaccinated-exposed-infected-treated-helicobacter pylori concentration in the environment recovered. The model developed is a system of differential equations. The rate of change of the system, existence and uniqueness of solution, region of absolute stability and positivity of the solution was established. Existence of disease-free equilibrium state and basic reproduction number was also established. Mathematical analysis was determined by the basic reproduction number if , the disease-free equilibrium is locally asymptotically stable whereas if then the equilibrium is unstable whereas the numerical analysis for the estimated basic reproduction number of the model is greater than unity in perspective of Nigeria, for which the Herd Immunity Threshold indicated that vaccinating 99.47% of Nigeria population can control spreading of Peptic ulcer disease in the country. The paper recommended amongst other things that if significant changes concerning the issue of threshold target are observed to improve the rate at which peptic ulcer can be minimized from the population where Government and non-governmental organization should encourage the use of peptic ulcer drugs and vaccines among individuals.

Analyzing a Dynamical System with Harmonic Mean Incidence Rate Using Volterra–Lyapunov Matrices and Fractal-Fractional Operators

This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics.

A comprehensive analysis of COVID-19 nonlinear mathematical model by incorporating the environment and social distancing

This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. The study develops a sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V–L) matrices method. To understand the dynamic behavior of COVID-19, numerical simulations are essential. For this purpose, the study employs a robust numerical technique known as the non-standard finite difference (NSFD) method, introduced by Mickens. Various results are visually presented through graphical representations across different parameter values to illustrate the impact of environmental factors and social distancing measures.

Mathematical modeling of chickenpox transmission using the Laplace Adomian Decomposition Method

In this work, mathematical modeling of a nonlinear differential equation was studied to investigate the effect of vaccination on the spread of chickenpox. The proof of existence and uniqueness of the positive solution and invariant region showed that the model is epidemiologically sound. We established the disease-free and endemic equilibrium states and carried out a stability analysis of the disease-free and endemic equilibrium states of the model to gain insight into the dynamics of the model. The rate of vaccination and precaution for the spread of chickenpox was a factor that influenced the basic reproductive number, which was calculated using the next-generation matrix approach. Forecasts made via numerical simulation using the Laplace Adomian Decomposition method highlight the temporal impact of vaccination on curbing the chicken pox trend.

Dynamics of Hepatitis B Virus Disease with Infectious Latent and Vertical Transmission

Hepatitis B has become a major health threat because it is a life-threatening liver disease with an estimated 0.25 billion people suffering from this infectious disease worldwide. This paper presents a SLITR (Susceptible-Latent-Infectious-Treatment-Recovery) mathematical model that combines both vaccination and treatment as a means of controlling the hepatitis B virus (HBV). The nonlinear ordinary differential equations for the HBV transmission capacities were resolved and the basic reproduction number R0 computed using the next generation matrix method and simulated numerically using the Runge-Kutta fourth order scheme implemented using MatLab. The stability points for disease-free equilibrium state (DFE), endemic equilibrium state (EE), and basic reproduction number R0 were obtained and the results show that the disease-free equilibrium was both locally and globally asymptotically stable (R0<1) . Similarly, treatment or vaccine administered was effective in alleviating the spread of HBV disease, and when both control strategies are combined, the diseases are quickly controlled and eventually eradicated.

Analyzing Global Stability of M-Pox Disease Dynamics: Mathematical Insights into Detection and Treatment Strategies

In this research, a mathematical model is constructed to scrutinize the transmission patterns of monkeypox (mpox), with a specific emphasis on integrating the early detection of infected undetected individuals to curb its transmission. This research takes into account a range of factors influencing the propagation of monkeypox, encompassing population demographics, contact dynamics, and the efficacy of early detection of unidentified infected individuals. Employing the next-generation matrix method, the basic reproduction number (R0) is computed, revealing that the disease-free equilibrium state is locally asymptotically stable when (R0<1). This suggests that containment of monkeypox is achievable within a human populace where (R0) remains below one (1), yet it transitions to an endemic state when (R0) exceeds this critical value one (1). Furthermore, a sensitivity analysis is conducted to evaluate the robustness of our findings to variations in model parameters. Utilizing numerical simulations conducted via MAPLE 18, we demonstrate the significance of prompt identification and immediate intervention for infected individuals who may otherwise go undetected, in effectively diminishing the dynamic propagation of monkeypox. The results underscore the pivotal role of early detection in mitigating monkeypox outbreaks and curtailing transmission rates.

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.

Analysis of HIV/AIDS Model with Nonlinear Incidence Function

Human Immunodeficiency Virus – Acquired Immune Deficiency Syndrome HIV/AIDS stands as one of the most prevalent sexually transmitted disease globally and is regarded as one of the deadliest epidemic in human history. This study presents a mathematical model for understanding the dynamics of HIV/AIDS transmission, incorporating a saturated incidence rate. The model employs a system of ordinary differential equations, comprising various group of individuals including susceptible, asymptomatic infective, symptomatic infective, treated and AIDS class. The validity of the solution states affirms that the model is well-defined and holds epidemiological significance. The disease-free and endemic equilibrium states are identified, and their stability is analyzed using Routh Hurwitz criteria. Sensitivity analysis was carried out using normalized forward sensitivity index and result showed that the contact rate is the most sensitive parameter. However, it is observed from the numerical simulation that screening and treatment of the infective play a significant role in reducing the transmission of the disease. The outcome of the stability analysis for both disease-free and endemics equilibrium states indicates the potential for HIV/AIDS control.

Modeling dynamics and stability analysis of pneumonia disease infection with parameters uncertainties control

In this work, a mathematical model of six compartments is formulated, showing the dynamic flow of pneumonia disease in the human population with treatment and vaccination interventions. Equilibria points and stability analyses were carried out using the Lyapunov function approach. Analytically, it is found that at the disease-free equilibrium state, local and global asymptotic stability behaviors are achieved when [Formula: see text] with instability if [Formula: see text]. However, at the endemic equilibrium point, asymptotic stability is attainable if [Formula: see text] and instability otherwise. The study indicates that pneumonia disease infection is successfully reduced when treatment and vaccination interventions are administered to the patients. The work also proposes an adaptive sliding mode control approach with a closed-loop control system to manage pneumonia epidemic model uncertainties. This approach intends to reduce disease transmission and infection through successful tracking of defined trajectories and managing uncertainties. For the control rates [Formula: see text], the technique managed to track the disease carriers and infectious agents accurately even in the presence of parameter uncertainties. In conclusion, an increase in the control rates [Formula: see text] in the existence of parameter uncertainty control systems significantly reduces the number of disease transmitters and infectious agents quicker than in their absence. Hence, this study signifies the pivotal role of treatment and vaccination in the control of pneumonia infection as well as the control of parameter uncertainties by the proposed method.

Traveling waves for a nonlocal dispersal SIRS epidemic model with age structure

<abstract><p>This paper focuses on a SIRS infectious model of nonlocal dispersal adopted with age structure. We primarily investigate the existence and nonexistence of traveling wave solutions connecting the disease-free equilibrium state and the endemic equilibrium state. To be more precise, we obtain the existence of traveling wave solutions by constructing suitable upper and lower solutions and then applying Schauder's fixed point theorem when $ R_0 > 1 $ and $ c > c^* $. In addition, we prove the nonexistence of traveling wave solutions by applying the Laplace transform for $ R_0 > 1 $ and $ 0 < c < c^* $.</p></abstract>

A MATHEMATICAL MODEL FOR TUBERCULOSIS INFECTION TRANSMISSION DYNAMICS IN THE PRESENCE OF TESTING AND THERAPY, ISOLATION AND TREATMENT

In this study, a mathematical model for Tuberculosis infection transmission dynamics is developed by incorporating testing and therapy of latent individuals, the isolation of infectious individuals and the treatment of the isolated individuals. The basic reproduction number was computed using the next generation matrix method. Analysis of the model at the disease-free equilibrium state and the endemic equilibrium states shows that it is locally and globally asymptomatically stable whenever the basic reproduction number is less than unity at the disease -free equilibrium state and locally and globally asymptotically stable whenever the basic reproduction number is greater than unity. The result from the sensitivity index of show that the infection transmission parameter and other control parameters such as early detection and therapy, the isolation of infected individuals and treatment are crucial parameters to tuberculosis management. It is shown from numerical simulations that the early detection and therapy, isolation and treatment of infected individuals will reduce the infection transmission. Further numerical results show that the combination of early detection and therapy, isolation and treatment of infectious individuals will decrease the infection transmission and its eventual eradication from the human population.

Analysis of Stability, Sensitivity Index and Hopf Bifurcation of Eco-Epidemiological SIR Model under Pesticide Application

In this paper, a deterministic SIR plant mathematical model is proposed and analysed with the application of pesticides as a control measure. The primary purpose of this model is to study the role of pesticides in controlling disease prevalence in plant populations. The total plant population is subdivided into three categories: susceptible, infected, and recovered. Pesticides are considered to be applied to both susceptible and infected populations to prevent the spread of infection to unaffected plant populations. It is considered that plant populations can be recovered only through the use of pesticides. To ensure the biological validity and well-defined nature of the model, the positivity, boundedness, uniqueness and existence of solutions are analysed. The basic reproduction number (R0) of the infection is determined and observed that the disease-free equilibrium state is locally asymptotically stable whenever (R0) is less than unity and unstable otherwise. The sensitivity analysis of the basic reproduction number is carried out, and it is observed that the value of R0 decreases as the value of the death rate and the recovery rate of plants increases. Moreover, it is revealed that above a critical parameter value of the infective induce rate, the population starts oscillating periodically, and the endemic equilibrium state becomes unstable. Finally, numerical simulations are conducted in MATLAB software to compare the analytical findings. Overall, the results obtained from this analysis are both novel and significant, making them an intriguing and potentially valuable contribution to the field of theoretical ecology.

Global dynamics of a tuberculosis model with age-dependent latency and time delays in treatment.

Since there exists heterogeneity in incubation periods of tuberculosis and a time lag between treatment and recovery. In this study, we develop a tuberculosis model that takes into account age-dependent latency and time delays in treatment to describe the transmission of tuberculosis. We first show that the solution semi-flow of the model is well-posed and has a global attractor [Formula: see text] within an infinite dimensional space [Formula: see text]. Then we define the basic reproduction number [Formula: see text] and prove that it determines the global dynamics of the model. If [Formula: see text], the global attractor [Formula: see text] reduces to the disease-free equilibrium state, indicating that the disease-free equilibrium state is globally asymptotically stable. When [Formula: see text], the semi-flow generated by the model is uniformly persistent, and there exists an interior global attractor [Formula: see text] for this uniformly persistent model. By constructing a suitable Lyapunov function and applying LaSalle's Invariance Principle, we show that the global attractor [Formula: see text] is reduced to the endemic equilibrium state, which means that the endemic equilibrium state is globally asymptotically stable. Based on the tuberculosis data in China from 2007 to 2020, we simulate the parameters and initial values of the proposed model. Furthermore, we calculate the sensitivity of [Formula: see text] to the parameters and find the most sensitive parameters to [Formula: see text]. Finally, we present an improved strategy to achieve the WHO's goal of reducing the incidence of tuberculosis by 90% by 2035 compared to 2015.

Mathematical Modeling and Optimal Control of Intervention Strategies for Covid-19 Disease

This research work used mathematical modeling in understanding the dynamics of covid-19 disease. We modified the work of Chen et al. (2020) by incorporating vaccination and pets (spread agents) compartments, making the model a ten (10) compartmental model, and also augmenting four controls to it namely: vaccination, use of face mask and physical distancing, sanitation, and treatment. We developed from the model, a system of non-linear Ordinary Differential Equations from which the positivity of solution was proven. We established the equilibrium states, determined the reproduction number which was utilized to predict the disease's transmission dynamics, hence establishing the conditions for local and global stability of the disease free- equilibrium using Routh- Hurwitz criterion and the Castillo-Chavez technique, respectively. The outcome of the investigation of the stability of the disease-free equilibrium state that covid-19 disease transmission can be significantly degraded and eliminated if the secondary infection’s rate is maintained at a value less than unity. We also used Pontryagin's Maximum Principle to establish the optimality system. The optimality system was numerically solved in Matlab to establish the best strategy in controlling the transmission of covid-19 disease in the population. The graphical solutions revealed that the most effective strategy is the combination of vaccination, use of face mask and physical distancing, sanitation, and treatment of infected individuals in the population.

On the Optimal Control of Intervention Strategies for Hepatitis B Model

Hepatitis B is one of the leading causes of morbidity and mortality, affecting hundreds of millions of people worldwide. Thus, this paper focuses on three control measures as the best way to intervene against the hepatitis B viral infection. These measures are condom use, testing and treatment, and vaccination to stop the disease from spreading over a community. The model comprises seven (7) compartments that include susceptible individuals, latent individuals, acute-infected individuals, chronic-infected individuals, infected by carrier individuals, recovered individuals from the disease, and the vaccinated population. We mathematically established a nonlinear differential equation to study the dynamics of the model. The disease-free equilibrium (DFE) and endemic equilibrium (EE) are reached. The basic reproduction numbers, R 0 A , R 0 H , and R 0 C , determine the transmission of the disease and thus are gotten. We perform sensitivity analysis on the reproduction numbers to identify the factors that affect the reproduction numbers. The results of the sensitivity analysis paved a way for introducing a controlled system which was solved using Pontryagin’s maximum principle (PMP) and the optimality system got. The optimality system was then solved numerically using the forward and backward sweep approach, and graphs were generated, establishing the conditions for local and global stability of the disease-free equilibrium using the Routh-Hurwitz criterion and Castillo-Chavez approach, respectively. We also used Pontryagin’s maximum principle to determine the optimality system. The result of the analysis of the stability of the disease-free equilibrium states that hepatitis B virus can be completely wiped out if the rate of infection is kept at a number less than unity. A numerical simulation of the model was carried out and showed that hepatitis B virus transmission can best be controlled when condom use, testing and treatment, and vaccination are implemented.

Transmission dynamics of a novel HIV/AIDS model through a higher-order Galerkin time discretization scheme

There are numerous contagious diseases caused by pathogenic microorganisms, including bacteria, viruses, fungi, and parasites, that have the propensity to culminate in fatal consequences. A communicable disease is an illness caused by a contagion agent or its toxins and spread directly or indirectly to a susceptible animal or human host by an infected person, animal, vector, or immaterial environment. Human immunodeficiency virus (HIV) infection, hepatitis A, B, and C, and measles are all examples of communicable diseases. Acquired immunodeficiency syndrome (AIDS) is a communicable disease caused by HIV infection that has become the most severe issue facing humanity. The research work in this paper is to numerically explore a mathematical model and demonstrate the dynamics of HIV/AIDS disease transmission using a continuous Galerkin–Petrov time discretization of a higher-order scheme, specifically the cGP(2)-scheme. Depict a graphical and tabular comparison between the outcomes of the mentioned scheme and those obtained through other classical schemes that exist in the literature. Further, a comparison is performed relative to the well-known fourth-order Ruge–Kutta (RK4) method with different step sizes. By contrast, the suggested approach provided more accurate results with a larger step size than RK4 with a smaller step size. After validation and confirmation of the suggested scheme and code, we implement the method to the extended model by introducing a treatment rate and show the impact of various non-linear source terms for the generation of new cells. We also determined the basic reproduction number and use the Routh-Hurwitz criterion to assess the stability of disease-free and unique endemic equilibrium states of the HIV model.

Modeling and analysis of novel COVID-19 outbreak under fractal-fractional derivative in Caputo sense with power-law: a case study of Pakistan.

In this paper, a five-compartment model is used to explore the dynamics of the COVID-19 pandemic, taking the vaccination campaign into account. The present model consists of five components that lead to a system of five ordinary differential equations. In this paper, we examined the disease from the perspective of a fractal fractional derivative in the Caputo sense with a power law type kernal. The model is also fitted with real data for Pakistan between June 1, 2020, and March 8, 2021. The fundamental mathematical characteristics of the model have been investigated thoroughly. We have calculated the equilibrium points and the reproduction number for the model and obtained the feasible region for the system. The existence and stability criteria of the model have been validated using the Banach fixed point theory and the Picard successive approximation technique. Furthermore, we have conducted stability analysis for both the disease-free and endemic equilibrium states. On the basis of sensitivity analysis and the dynamics of the threshold parameter, we have estimated the effectiveness of vaccination and identified potential control strategies for the disease using the proposed model outbreaks. The stability of the concerned solution in Ulam-Hyers and Ulam-Hyers-Rassias sense is also investigated. For the proposed problem, some results regarding basic reproduction numbers and stability analysis for various parameters are represented graphically. Matlab software is used for numerical illustrations. Graphical representations are given for different fractional orders and for various parametric values.

Vector-borne diseases with nonstationary vector populations: The case of growing and decaying populations.

Since the last century, deterministic compartmental models have emerged as powerful tools to predict and control epidemic outbreaks, in many cases helping to mitigate their impacts. A key quantity for these models is the so-called basic reproduction number, R_{0}, that measures the number of secondary infections produced by an initial infected individual in a fully susceptible population. Some methods have been developed to allow the direct computation of this quantity provided that some conditions are fulfilled, such that the model has a prepandemic disease-free equilibrium state. This condition is fulfilled only when the populations are stationary. In the case of vector-borne diseases, this implies that the vector birth and death rates need to be balanced. This is not fulfilled in many realistic cases in which the vector population grows or decreases. Here we develop a vector-borne epidemic model with growing and decaying vector populations that in the long term converge to an asymptotic stationary state, and study the conditions under which the standard methods to compute R_{0} work and discuss an alternative when they fail. We also show that growing vector populations produce a delay in the epidemic dynamics when compared to the case of the stationary vector population. Finally, we discuss the conditions under which the model can be reduced to the Susceptible, Infectious, and/or Recovered (SIR) model with fewer compartments and parameters, which helps in solving the problem of parameter unidentifiability of many vector-borne epidemic models.

Mathematical Model on the Dynamics of Bacterial Blight of Rice in the Presence of Lysobacter antibioticus Considering Introduction at Different Stages

In this research we formulated the Plants diseases model with the aim of studying the dynamics of the use of lysobacter antibioticus for prevention and control of rice bacterial blight. The disease free equilibrium state of the models was also obtained by equating each of the equation of the modified model to zero and simplifying. The basic reproduction number for the model was derived using the next generation matrix approach. Numerical simulation was carried out using MATLAB2018a to virtualize the dynamics of the model. Five numerical experiment was carried out and it was shown that biocontrol help to reduce the population of the pathogen as well as act as treatment for those that are already exposed or infected with the disease. It was also observed that the biocontrol agent provide immunity to rice plants against been infected with the disease. Finally, we observed from the simulation that the earlier the control is introduced the more protection plants will receive.