Stability Of Endemic Equilibrium Point Research Articles

Stability and Hopf bifurcation analysis for fractional-order SVEIR computer virus propagation model with nonlinear incident rate and two delays

This paper is concerned with the stability and Hopf bifurcation for a fractional-order Susceptible-Vaccinated-Exposed-Infectious-Recovered (SVEIR) computer virus propagation model with a nonlinear incident rate. By employing the linearization technique and Routh-Hurwitz method, the sufficient criterion is established for the locally asymptotic stability of endemic equilibrium point. The Hopf bifurcation is also studied for the SVEIR computer virus model by taking time delay as the bifurcation parameter. The research results show that the stability and Hopf bifurcation of proposed model are significantly affected by both time delay and the order of the fractional derivative. Examples with proper parameters and several simulations are given to illustrate the validity of the theoretical results.

Stability and bifurcation analysis of a contaminated sir model with saturated type incidence rate and Holling type-III treatment function

The paper deals with an epidemic model which is contaminated. Logistic growth input of vulnerable is taken in our SIR model. Saturated type incidence rate alongside Holling type III treatment function and time delay in growth component has been considered. The stability analysis of equilibrium points is done and the basic reproduction number (R 0 ) is obtained. Using (R 0 ), Local Stability around disease-free equilibrium point has been acquired. When R 0 is less than one, it is locally stable which means disease doesn’t exist in the environment anymore and when R0 is greater than one, it is unstable which means disease still exists in the environment. We have shown that at R 0 =1 the model exhibits forward bifurcation. The conditions for the local stability of endemic equilibrium point have also been obtained. Conditions for the existence of Hopf bifurcation along with its direction and stability has been determined. Numerical simulations have been performed utilizing MATLAB to confirm the logical outcomes acquired. We have seen that to diminish the effect of illness, contamination ought to be decreased. Also, suitable strategies should be chosen to expand the capacity of treatment to decrease the effect of infection.

Model of transmission of three types Leprosy Disease with treatment

Leprosy disease is caused by infection of Mycobacterium leprae that affects the skin and nerve. The treatment duration for leprosy takes 6 to 24 months with multi-drug therapy. In this article, a 6-d mathematical model for transmission of three types leprosy disease with treatment was constructed. Analysis and numerical simulation were done to explore the existence and stability of equilibrium points and basic reproduction number R0. The local stability of endemic equilibrium point was done by numerical simulation. Numerical simulation also confirmed that the disease-free equilibrium is locally asymptotically stable when R0 < 1 and the endemic equilibrium point locally asymptotically stable when R0 > 1. Numerical simulation was also done to show the sensitivity of R0 with respect to some contact parameters. The result showed that the change of contact rate between susceptible people and lepromatous type infected hosts has more effect to change the value of R0. The reduction of contact rate between susceptible people and lepromatous type infected hosts is more effective to control prevention leprosy disease compared with tuberculoid and borderline types.

Global stability and optimal control of melioidosis transmission model with hygiene care and treatment in human and animal populations

A compartmental deterministic mathematical model for melioidosis transmission is proposed. It involves human and animal population and bacteria. We present two main equilibrium points (disease-free and endemic) with the analysis of their local stability. The basic reproduction number and its sensitivity index to the parameters in the model are calculated. Lyapunov's direct method is used to analyse the global stability of endemic equilibrium point. Further, by using Pontryagin's minimum principle, the optimal control problem is constructed with two controls, i.e., hygiene care and treatment controls for human population. Finally, the numerical simulations are established and our results show that a combination of both controls give more impact in reducing the number of infected human and bacteria than in reducing the number of infected animals. Because melioidosis prevalence is still increasing, more effort in sharing knowledge and controlling the spreading of this disease is still much required.

Mathematical Epidemiology Model Analysis on the Dynamics of COVID-19 Pandemic

In the present work, Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) mathematical model for COVID-19 Pandemic is formulated and analyzed. The positivity, boundedness, and existence of the solutions of the model are proved. The Disease-free equilibrium point and endemic equilibrium points are identified. Local Stability of disease-free Equilibrium point is checked with the help of Next generation matrix. Global stability of endemic equilibrium point is proved using the Concept of Liapunove function. The basic reproduction number for Novel Corona virus pandemic is computed as R₀ = (αβΛ) ⁄ [(δ + μ) (β + δ + μ) (γ + δ + μ)] which depend on six different parameters. It is observed that if basic reproduction number is less than one, then number of cases decrease over time and eventually the disease dies out, and if the basic reproduction number is equals to one, then number of cases are stable. On the other hand, if the basic reproduction number is greater than one, then the number of cases increase over time gets worth. Sensitivity analysis of the basic reproduction number is done with respect to each parameter. It is observed that only some parameters Λ, α, β have high impact on the basic reproduction number. Consequently, with real data on the parameter it is helpful to predict the disease persistence or decline in the present situation. Lastly, numerical simulations are given using DEDiscover software to illustrate analytical results.

Global stability and optimal control of melioidosis transmission model with hygiene care and treatment in human and animal populations

A compartmental deterministic mathematical model for melioidosis transmission is proposed. It involves human and animal population and bacteria. We present two main equilibrium points (disease-free and endemic) with the analysis of their local stability. The basic reproduction number and its sensitivity index to the parameters in the model are calculated. Lyapunov's direct method is used to analyse the global stability of endemic equilibrium point. Further, by using Pontryagin's minimum principle, the optimal control problem is constructed with two controls, i.e., hygiene care and treatment controls for human population. Finally, the numerical simulations are established and our results show that a combination of both controls give more impact in reducing the number of infected human and bacteria than in reducing the number of infected animals. Because melioidosis prevalence is still increasing, more effort in sharing knowledge and controlling the spreading of this disease is still much required.

Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order

In this paper, we propose and analyze a fractional order model of viral kinetics for primary infection of HIV-1 in presence of immune control with treatment, as the classical model of target-cell-limited model is unable to predict long term viral kinetics unless a delayed immune effect is assumed. Further, the reverse transcriptase treatment can be incorporated by reducing the target cell infection rate. The existence of equilibria and their asymptotical stability results using fractional Routh–Hurwitz stability criterion will be discussed. A threshold value ′R0′ known as basic reproduction number has been found to ensure the extinction or persistence of the infection. The numerical solution using generalized Adams–Bashforth–Moulton method to the proposed HIV-1 fractional model is obtained. Also, the sufficient conditions that guarantee the asymptotic stability of endemic equilibrium point is presented. Meanwhile, global asymptotic stability of the endemic equilibrium point is investigated by constructing a suitable Lyapunov functions. The fractional derivative is described in Caputo sense. The obtained numerical results of the proposed model show the effectiveness and strength of the Adams–Bashforth–Moulton method. Some numerical simulations are given to illustrate the analytical results.

Effect of vaccination and treatment on the MSEICR model of the transmission of hepatitis B virus

This article studies a development of SEIR standard model for the spread of hepatitis B virus. The model is developed by considering immunized and carrier compartments of population. The model includes immunized, suspected, exposed, infected, carrier, and recovered compartments and written as MSEICR. Some of new born and the suspected are given vaccines and the infected is given a treatment. These strategies aim to reduce transmissions of hepatitis B virus in the population. The existence and stability of endemic and non-endemic equilibrium points are analysed via basic reproduction number (ℛ0) which is derived from the next generation matrix method. The results showed that the endemic equilibrium point does not exist when ℛ0 < 1. The endemic will appear when the value of ℛ0 > 1. Sensitivity analyses showed that vaccination and treatment may reduce the spread of hepatitis B virus and also eliminate endemic condition. Some simulations were conducted to visualize the effects of vaccination and treatment on the existence and stability of endemic equilibrium point.

Modeling and analysis of plant disease with delay and logistic growth of insect vector

Plants are essential for the survival of human beings. Plants can be subjected to diseases. Plant diseases are caused by pathogens such as fungi, bacteria and viruses. Most of these pathogens are transmitted by insect vectors. In this paper we formulate and analyze a delay differential equation model for plant disease by incorporating the incubation delay which is the time taken for a plant to become infected. The mathematical model is formulated by considering both the plant and the insect vector populations. The total plant population is taken as a constant and the insect vector population is taken as variable. It is assumed that the insect vector population is growing logistically in the environment. The existence and stability of equilibria of the model are discussed in detail. The basic reproduction number $R_0$ of the model is computed and it is observed that the disease-free equilibrium point is stable for all delay whenever $R_0 1$ the endemic equilibrium point is stable in the absence of delay. We have estimated the length of delay which preserves the stability of endemic equilibrium point. So when the delay is less than a threshold value, the endemic equilibrium point is stable. At that threshold value we get Hopf bifurcation and system shows oscillatory behaviour. Here numerical simulation is also performed to support the analytical results.

Numerical solution of compartmental models by meshless and finite difference methods

In this paper, an operator splitting method based on meshless and finite difference procedures, is being considered for numerical solution of compartmental epidemiological population models with and without diffusion. A one step explicit meshless procedure is also applied for the numerical solution of the nonlinear model. The compartmental model contains susceptible, vaccinated, exposed, infected, and recovered (SVEIR) classes of the population. Effects of the diffusion on the simulation results of the model are being studied. Stability of endemic equilibrium point along with bifurcation analysis has also been investigated. Due to non-availability of the exact solution, the numerical results obtained are mutually compared and their correctness is being verified by the theoretical results as well.

Model Matematika Penyebaran HIV/AIDS dengan Edukasi Kesehatan

In this research has been carried out the stability analysis of HIV/AIDS epidemic model with a public health educational through the expansion of the SI (susceptible-infected) model. In modeling of HIV/AIDS epidemic, the population is divided into six subpopulations: uneducated susceptible individuals, educated susceptible individuals, uneducated infected individuals without AIDS symptoms, educated infected individuals with AIDS symptoms, uneducated infected individuals with AIDS symptoms and educated infected individuals with AIDS symptoms. The disease-free equilibrium point of the HIV transmission model with education program is locally asymptotically stable if the basic reproduction number is less than unity and unstable if the basic reproduction number is greater than unity. The endemic equilibrium point is exist if the effective reproduction number is greater than unity and stability of endemic equilibrium point has been determined using the Center manifold theory. The center manifold theory can be used to analyze the stability near the disease-free equilibrium point (the effective reproduction number is equal to unity). The impact of a public health education on the spread of HIV/AIDS, the sensitivity analysis on effective reproduction numbers respect to all the parameters which drive the disease dynamics.

Global stability of endemic equilibrium point of basic virus infection model with application to HBV infection

In this paper, the authors first show that if R0 ≤ 1, the infection free steady state is globally attractive by using approaches different from those given by Min, et al.(2008). Then the authors prove that if R0 > 1, the endemic steady state is also globally attractive. Finally, based on a patient’s clinical HBV DNA data of anti-HBV infection with drug lamivudine, the authors establish an ABVIM. The numerical simulations of the ABVIM are good in agreement with the clinical data.