Forward Bifurcation and Stability Analysis

We propose a stage-structured model of childhood infectious disease transmission dynamics, with the population demographics dynamics governed by a certain family and population planning strategy giving rise to nonlinear feedback delayed effects on the reproduction ageing and rate. We first describe the long-term aging-profile of the population by describing the pattern and stability of equilibrium of the demographic model. We also investigate the disease transmission dynamics, using the epidemic model when the population reaches the positive equilibrium (limiting equation). We establish conditions for the existence, uniqueness and global stability of the disease endemic equilibrium. We then prove the global stability of the endemic equilibrium for the original epidemic model with varying population demographics. The global stability of the endemic equilibrium allows us to examine the effects of reproduction ageing and rate, under different family planning strategies, on the childhood infectious disease transmission dynamics. We also examine demographic distribution, diseases reproductive number, infant disease rate and age distribution of disease, and as such, the work can be potentially used to inform targeted age group for optimal vaccine booster programs.

This article discusses the mathematical model of the spread of COVID-19 by considering vaccination and population migration. The former model is analyzed by determining the equilibrium point, basic reproduction number, analyzing the stability of the equilibrium point, sensitivity analysis, and accompanied by numerical simulation. Analysis of the stability of disease-free and endemic equilibrium points using the Routh-Hurwitz Criteria and the Castillo-Chaves and Song theorems. The results of the analysis show that there are two equilibrium points, namely a disease-free equilibrium point (T1), which is locally asymptotically stable when R0 1, and an endemic equilibrium point (T2), which is locally asymptotically stable when R0 1. Furthermore, the sensitivity analysis showed that the most sensitive parameters to changes in the basic reproduction number were the emigration rate parameter (m2) and the infection probability parameter after contact between infected and susceptible individuals without vaccination (h). In addition, the numerical simulation results show that the sensitive parameter values, namely m2, h, zse, g, and # have a significant effect on the basic reproduction numbers. Suppressing the chance of infection in susceptible individuals and the rate of contact between susceptible and exposed individuals, as well as increasing the number of individuals who emigrate and who are vaccinated, can reduce the transmission of COVID-19.

A model for the dynamic transmission of SEITR pandemics is proposed, incorporating awareness and a mass action incidence function. The stability of both endemic and disease-free equilibrium points was examined, revealing that the disease-free equilibrium is locally and globally asymptotically stable when the Basic Reproduction Number is less than 1, while the endemic equilibrium is stable when . Additionally, it's shown that the Basic Reproduction Number (without awareness) is greater than (with awareness), this means that when , then . A Lyapunov function is constructed to demonstrate the global stability of both disease-free equilibrium when and the endemic equilibrium when . Numerical simulations illustrate the impact of awareness on disease spread, highlighting the importance of awareness in informing and educating individuals about infection risks during outbreaks, thereby reducing the health burden of the epidemic by lowering its peak.

We present in this research work, mathematical modeling of the transmission dynamics of measles using treatment as a control measure. We determined the Disease Free Equilibrium (DFE) point of the model after which we obtained the Basic Reproduction Number ( R0 ) of the model using the next generation approach. The model Endemic Equilibrium (EE) point was also determined after which we performed Local Stability Analysis(LAS) of the Disease Free Equilibrium point and result shows that the Disease Free Equilibrium point of the model would be stable if ( R0 <1). Global Stability Analysis (GAS) result shows that, ( R0 ≤ 1) remains the necessary and sufficient condition for the infection to go into extinction from a population. We carried out Sensitivity Analysis of the model using the Basic Reproduction Number and we discovered that ( δ , μ, ν , θ ) are sensitive parameters that should be targeted towards control intervention strategy as an increase in these values can reduce the value of ( R0 ) to a value less than unity and such can reduce the spread of measles in a population. Model simulation was carried out using mat lab software to support our analytical results.

Atherosclerosis is a non-communicable disease (NCDs) which appears when the blood vessels in the human body become thick and stiff. The symptoms range from chest pain, sudden numbness in the arms or legs, temporary loss of vision in one eye, or even kidney failure, which may lead to death. Treatment in cases with severe symptoms requires surgery, in which the number of doctors or hospitals is limited in some countries, especially countries with low health levels. This article aims to propose a mathematical model to understand the impact of limited hospital resources on the success of the control program of atherosclerosis spreads. The model was constructed based on a deterministic model, where the hospitalization rate is defined as a time-dependent saturated function concerning the number of infected individuals. The existence and stability of all possible equilibrium points were shown analytically and numerically, along with the basic reproduction number. Our analysis indicates that our model may exhibit various types of bifurcation phenomena, such as forward bifurcation, backward bifurcation, or a forward bifurcation with hysteresis depending on the value of hospitalization saturation parameter and the infection rate for treated infected individuals. These phenomenon triggers a complex and tricky control program of atherosclerosis. A forward bifurcation with hysteresis auses a possible condition of having more than one stable endemic equilibrium when the basic reproduction number is larger than one, but close to one. The more significant value of hospitalization saturation rate or the infection rate for treated infected individuals increases the possibility of the stable endemic equilibrium point even though the disease-free equilibrium is stable. Furthermore, the Pontryagin Maximum Principle was used to characterize the optimal control problem for our model. Based on the results of our analysis, we conclude that atherosclerosis control interventions should prioritize prevention efforts over endemic reduction scenarios to avoid high intervention costs. In addition, the government also needs to pay great attention to the availability of hospital services for this disease to avoid the dynamic complexity of the spread of atherosclerosis in the field.

<abstract><p>Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.</p></abstract>

A new mathematical model for the transmission dynamics of herpes simplex virus type 2 (HSV-2), which takes into account disease transmission by infected individuals in the quiescent state and an imperfect HSV-2 vaccine, is designed and qualitatively analysed. In the absence of vaccination, it is shown that the model has a globally asymptotically stable (GAS) disease-free equilibrium (DFE) point whenever an epidemiological threshold, known as the ‘basic reproduction number’, is less than unity. Further, this model has a unique endemic equilibrium whenever the reproduction number exceeds unity. Using a non-linear Lyapunov function, it is shown that the unique endemic equilibrium is GAS (for a special case) when the associated reproduction threshold is greater than unity. On the other hand, the model with vaccination undergoes a vaccine-induced backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium when the reproduction threshold is less than unity. Threshold analysis of the vaccination model reveals that the use of an imperfect HSV-2 vaccine could have positive or negative population-level impact (in reducing disease burden). Simulations of the vaccination model show that an HSV-2 vaccine could lead to effective disease control or elimination if the vaccine efficacy and the fraction of susceptible individuals vaccinated at steady state are high enough (at least 80% each).

Dengue is one of the major health problems in Bangladesh and many people are died in recent years due to the severity of this disease. Therefore, in this paper, a SIRS model for the human and SI model for vector population with saturated incidence rate and constant treatment function has been presented to describe the transmission of dengue. The equilibrium points and the basic reproduction number have been computed. The conditions which lead the disease free equilibrium and the endemic equilibrium have been determined. The local stability for the equilibrium points has been established based on the eigenvalues of the Jacobian matrix and the global stability has been analyzed by using the Lyapunov function theory. It is found that the stability of equilibrium points can be controlled by the reproduction number. In order to calculate the infection rate, data for infected human populations have been collected from several health institutions of Bangladesh. Numerical simulations of various compartments have been generated using MATLAB to investigate the influence of the key parameters for the transmission of the disease and to support the analytical results. The effect of treatment function over the infected compartment has been illustrated. The sensitivity of the reproduction number concerning the parameters of the model has been analyzed. Finally, the most sensitive parameter that has the highest effect over reproduction number has been identified.

The transmission dynamics of COVID-19 is investigated through the prism of the Atangana-Baleanu fractional model with acquired immunity. Harmonic incidence mean-type aims to drive exposed and infected populations towards extinction in a finite time frame. The reproduction number is calculated based on the next-generation matrix. A disease-free equilibrium point can be achieved globally using the Castillo-Chavez approach. Using the additive compound matrix approach, the global stability of endemic equilibrium can be demonstrated. Utilizing Pontryagin’s maximum principle, we introduce three control variables to obtain the optimal control strategies. Laplace transform allows simulating the fractional-order derivatives analytically. Analysis of the graphical results led to a better understanding of the transmission dynamics.

In this paper we have presented a deterministic model for pneumonia transmission and we have used the model to avail the potential impact of therapy. The model is based on the vaccinated-susceptible-carrier-infected-recovered-susceptible compartmental structure and their possible interventions with the possibility of infected individual recovery from natural immunity. Here, we have modeled Pneumonia considering vaccination, screening and treatment with a system of nonlinear ordinary differential equation. The model reproduction number R0 is derived and the stability of the equilibria are derived. The stability of equilibrium points is analyzed. The results shows that there exists a locally stable disease free equilibrium points, E0 when R0<1 and a unique endemic equilibrium E1, when R0>1. Infection free point was found to be locally stable and if reproduction number is greater than unity, then there is unique endemic equilibrium point and if it is less than unity, the endemic equilibrium point is globally asymptotically stable and pneumonia will be eliminated.

Tuberculosis is an infectious disease caused by Mycobacterium tuberculosis. The disease is considered dangerous because it infects the lungs and other organs of the body and can lead to death. This study discusses a mathematical model for the spread of tuberculosis with two treatment sites as an effort to reduce the transmission rate of TB cases. Treatment for TB patients can be done at home and in hospitals. The purpose of this study was to construct a mathematical model and analyze the qualitative behavior of the TB spread model. The construction of the model uses the SEIR epidemic model which is divided into five subpopulations, namely susceptible subpopulations, latent subpopulations, infected subpopulations receiving treatment at home, and infected subpopulations receiving treatment at the hospital, and cured subpopulations. The analysis of qualitative behavior in the model includes determining the local and global equilibrium and stability points. The results of the analysis shows that the model has two equilibrium points, namely a disease-free equilibrium point and the endemic equilibrium point. The existence of endemic equilibrium point and the local and global stability of the two equilibrium points depend on the basic reproduction number denoted by . If , there is only disease-free equilibrium point. If , there are two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. Stability analysis shows that the disease-free equilibrium point is locally and globally asymptotically stable if . While, if , the endemic equilibrium point will be asymptotically stable locally and globally.

Global stability of endemic equilibrium of an epidemic model with birth and death on complex networks

Wheat is one of the most significant crops grown worldwide and serves as both an important source of food for consumers and a means of income for farmers. However, the fungus that causes wheat yellow rust (WYR) seriously damages the world’s wheat production, especially in developing countries. In this study, we proposed and analyzed a compartmental mathematical model to describe the transmission dynamics of WYR disease on wheat crops. For the proposed model, the existence and uniqueness, the positivity and boundedness of the solutions are explained. The two equilibrium points, namely disease-free and endemic equilibrium points are demonstrated, and the basic reproduction number is evaluated using the next-generation matrix approach. The Routh Hurwitz stability criteria and center manifold theory are used to analyze the local stability of the equilibrium points and the existence of forward bifurcation, respectively. The global stability of the equilibrium points are also established using Lyapunov’s approach. A sensitivity analysis of the model is also carried out. To validate our analytical results, MATLAB software is utilized to conduct several numerical simulations and generate graphical representations of the results. The findings of this study suggest that effective use of fungicides can significantly prevent the spread of WYR disease.

Psittacosis is a disease in human beings that is commonly associated with pet birds such as cockatiels and parrots, and among poultry such as ducks and turkeys. This paper proposed and developed a deterministic epidemiological model that explains the transmission dynamics of Psittacosis infection in humans and poultry. The Psittacosis deterministic model was analyzed to determine positivity of the solution set, the invariant feasible region, the basic reproduction number, the disease free equilibrium points, the endemic equilibrium points and the corresponding stability of each of the equilibrium points. The basic reproduction number is calculated using the next generation matrix and it was found to be entirely dependent on the poultry population parameters. The study established that whenever R0 < 1, Psittacosis dies out from the population and when R0 > 1, Psittacosis keeps persisting in the population. Sensitivity analysis was conducted to determine the contribution of each parameter to the basic reproduction number. The more sensitive parameters were found to be responsible for the further propagation of Psittacosis while the less sensitive parameters rarely contributed to the spread. Stability analysis of both the disease free equilibrium and the Endemic equilibrium were conducted. Lastly, numerical simulation was conducted to justify quantitative analysis of the dynamics of the transmission of Psittacosis.

In this work, we investigated the consequences of an inadequate vaccination along with other exogenous variables such as treatment-associated re-infection and re-infection among former patients. Our models represent six stages: Susceptible population, Vaccinated population, latently infected population, Rate of induced population, Quarantine population and Rate of recover population. In this model, we find out disease-free equilibrium points, reproduction number R_0 is used to control the re-infection rate in the population. Furthermore, the results suggest that even a subpar Mycobacterium tuberculosis vaccine is always successful at reducing the spread of infectious diseases among the population, even though its general effect improves with greater efficacy and penetration. It is demonstrated that a small percentage of the overall population receives immunization in a stable state and vaccine efficacy plays an equal part in decreasing the burden of disease. Based on the numerical simulation, the establishment of the TB disease is decrease and control by the vaccination coverage and efficacy of the treatment. Using an imperfect vaccine can control mycobacterium in the population. The next-generation matrix approach was utilized to determine the effective reproduction number of the proposed system. The utilization of the exponential theory which provided the disease-free equilibrium is stable and coexists endemic equilibrium point. The Lyapunov function were used to determine the disease is globally asymptotically stable. We find out the non-negative (in variant region). We used random data in MATLAB to figure out the result of the model.