Disease-free Equilibrium Point Research Articles

Modeling, analyzing and simulating the dynamics of Tuberculosis-Covid-19 co-infection

Covid-19 and tuberculosis (TB) are two significant infectious illnesses that may pose significant threats to the public health and their co-infection aggravates the issue. Through the use of a set of nonlinear ordinary differential equations and contaminated surfaces from SARS-CoV-2 in the environment, we developed and investigated a mathematical model in this study for the dynamics of the co-infection of Covid-19 and TB transmission. The free equilibrium point of the disease was determined and its stability was investigated. The next generation matrix approach is used to determine the model’s basic reproduction number. The disease-free equilibrium point is not globally asymptotically stable, but it is locally asymptotically stable if R0 < L We found key model parameters for disease dynamics spread using normalized forward sensitivity analysis. From the numerical simulation results, we conclude that government stakeholders should work in reducing the transmission mechanisms of both diseases. In addition, they should look on mechanisms for removing SARS-CoV-2 contaminated surfaces from the community.

Analysis of the COVID-19 model with self-protection and isolation measures affected by the environment

<abstract><p>Since the global outbreak of COVID-19, the virus has continuously mutated and can survive in the air for long periods of time. This paper establishes and analyzes a model of COVID-19 with self-protection and quarantine measures affected by viruses in the environment to investigate the influence of viruses in the environment on the spread of the outbreak, as well as to develop a rational prevention and control measure to control the spread of the outbreak. The basic reproduction number was calculated and Lyapunov functions were constructed to discuss the stability of the model equilibrium points. The disease-free equilibrium point was proven to be globally asymptotically stable when $ R_0 < 1 $, and the endemic equilibrium point was globally asymptotically stable when $ R_0 > 1 $. The model was fitted using data from COVID-19 cases in Chongqing between November 1 to November 25, 2022. Based on the numerical analysis, the following conclusion was obtained: clearing the virus in the environment and strengthening the isolation measures for infected people can control the epidemic to a certain extent, but enhancing the self-protection of individuals can be more effective in reducing the risk of being infected and controlling the transmission of the epidemic, which is more conducive to the practical application.</p></abstract>

Exploring Hopf-bifurcations and endemic bubbles in a tuberculosis model with behavioral changes and treatment saturation.

To manage risks and minimize the transmission of contagious diseases, individuals may reduce their contact with each other and take other precautions as much as possible in their daily lives and workplaces. As a result, the transmission of the infection reduces due to the behavioral changes. These behavioral changes are incorporated into models by introducing saturation in disease incidence. In this article, we propose and analyze a tuberculosis model that incorporates saturated exogenous reinfection and treatment. The stability analysis of the model's steady states is rigorously examined. We observe that the disease-free equilibrium point and the endemic equilibrium point (EEP) are globally asymptotically stable if the basic reproduction number (R0) is less than 1 and greater than 1, respectively, only when exogenous reinfection is not present (p=0) and when treatment is available for all (ω=0). However, even when R0 is less than 1, tuberculosis may persist at a specific level in the presence of exogenous reinfection and treatment saturation, leading to a backward bifurcation in the system. The existence and direction of Hopf-bifurcations are also discussed. Furthermore, we numerically validate our analytical results using different parameter sets. In the numerical examples, we study Hopf-bifurcations for parameters such as β, p, α, and ω. In one example, we observe that increasing β leads to the loss of stability of the unique EEP through a forward Hopf-bifurcation. If β is further increased, the unique EEP restores its stability, and the bifurcation diagram exhibits an interesting structure known as an endemic bubble. The existence of an endemic bubble for the saturation constant ω is also observed.

Dynamical properties of a new SIR epidemic model

Taking into account the depletion of food supply by all individuals, and the fact that chronic infectious diseases will not cause the infected individuals to lose their fertility completely, we first propose a new SIR epidemic model of ODE. For this model, we derive its basic reproduction number $ \mathcal{R}_0 $, and show that the disease-free equilibrium point is globally asymptotically stable when $ \mathcal{R}_0\le1 $, while the unique positive equilibrium point is globally asymptotically stable when $ \mathcal{R}_0>1 $. Then we incorporate the spatial dispersion and free boundary condition into this ODE model. The well-posedness and longtime behaviors are obtained. Particularly, we find a spreading-vanishing dichotomy in which the basic reproduction number $ \mathcal{R}_0 $ plays a crucial role.

Stability Analysis of Covid-19 Model Based on Compliance and Carrier Transmission

The Covid-19 pandemic has officially ended with the lifting of the Public Health Emergency of International Concern (PHEIC) status by the World Health Organization (WHO). The world has begun the transition from a pandemic to an endemic period through policy updates such as healthy living habits, wearing masks if sick, vaccination, self-quarantine, contact tracing or testing, increasing understanding or awareness of diseases and treatment. This research aims to analyze the role of individuals in the pandemic transition period and the addition of the Carrier subpopulation to the COVID-19 model. This model produces two equilibrium points: a disease-free equilibrium points and an endemic equilibrium point. Furthermore, stability analysis was carried out around the equilibrium point and obtained three basic reproduction numbers that became the threshold for the spread of disease around the equilibrium point, namely R0 less than one (R0 < 1) and greater than one (R0 > 1). This shows that increasing policies such as disease awareness or understanding, healthy living habits, and vaccination can prevent the spread of COVID-19 so that the pandemic period does not occur and the disease will disappear over time.

Qualitative behavior of a highly non-linear Cutaneous Leishmania epidemic model under convex incidence rate with real data.

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.

Mathematical modeling and dynamics of immunological exhaustion caused by measles transmissibility interaction with HIV host.

This paper mainly addressed the study of the transmission dynamics of infectious diseases and analysed the effect of two different types of viruses simultaneously that cause immunodeficiency in the host. The two infectious diseases that often spread in the populace are HIV and measles. The interaction between measles and HIV can cause severe illness and even fatal patient cases. The effects of the measles virus on the host with HIV infection are studied using a mathematical model and their dynamics. Analysing the dynamics of infectious diseases in communities requires the use of mathematical models. Decisions about public health policy are influenced by mathematical modeling, which sheds light on the efficacy of various control measures, immunization plans, and interventions. We build a mathematical model for disease spread through vertical and horizontal human population transmission, including six coupled nonlinear differential equations with logistic growth. The fundamental reproduction number is examined, which serves as a cutoff point for determining the degree to which a disease will persist or die. We look at the various disease equilibrium points and investigate the regional stability of the disease-free and endemic equilibrium points in the feasible region of the epidemic model. Concurrently, the global stability of the equilibrium points is investigated using the Lyapunov functional approach. Finally, the Runge-Kutta method is utilised for numerical simulation, and graphic illustrations are used to evaluate the impact of different factors on the spread of the illness. Critical factors that effect the dynamics of disease transmission and greatly affect the rate and range of the disease's spread in the population have been determined through a thorough analysis. These factors are crucial in determining the expansion of the disease.

Transmission dynamics of symptom-dependent HIV/AIDS models.

In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0^s $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.

Dynamics in a delayed rumor propagation model with logistic growth and saturation incidence

<abstract><p>This paper studies a delayed rumor propagation model with logistic growth and saturation incidence. The next generation matrix method, some inequality techniques, the Lyapunov-LaSalle invariance principle, and the Lyapunov method are used in this paper. Our results indicate that if the basic regeneration number (which is analogous to the basic reproduction number in disease transmission models) is less than 1, the rumor-free equilibrium point (which is analogous to the disease-free equilibrium point in disease transmission models) is globally stable. If the basic regeneration number is greater than 1, then the rumor is permanent, and some sufficient conditions are obtained for local and global asymptotic stability of the rumor prevailing equilibrium point (which is analogous to the endemic equilibrium point in disease transmission models). Finally, three examples with numerical simulations are presented to illustrate the obtained theoretical results.</p></abstract>

A study of periodic solutions of several types of nonlinear models in biomathematics

Abstract Biomathematics is a cross-discipline formed by the interpenetration of mathematics with life sciences, biology, and other disciplines, and biomathematical models provide an effective tool for solving problems in the above application areas. Our aim in this paper is to combine mathematical analytical tools and numerical simulation methods to investigate the existence and steady state of periodic solutions in different nonlinear models. Time lags with both discrete and distributed characteristics are introduced into the Lotka-Volterra predator-feeder system, and based on the discussion of the central manifold theorem and canonical type theory, it is proved that the branching periodic solution exists when the discrete time lag parameter τ > τ 0. In the SEIRS infectious disease model with nonlinear incidence term and vertical transmission, the global stability of the disease-free equilibrium point and the local asymptotic stability of the endemic equilibrium point are analyzed through the computation and discussion of the fundamental regeneration number R 0 (p, q). A class of convergence-growth models with nonlinear sensitivity functions is studied, and the global boundedness of classical solutions and their conditions are demonstrated based on global dynamics. A mathematical generalization of the muscular vascular model is made by introducing a centralized parameter, the relationship between periodic solutions and chaotic phenomena is explored utilizing a systematic equivalence transformation, and the equation of the homoscedastic orbitals is deduced to be z 2 = x 2 ( A - 1 2 x 2 ) {z^2} = {x^2}\left( {A - {1 \over 2}{x^2}} \right) .

Education and Mathematics Models (A Case Study of Epidemiology of Virus Spread)

Viruses are the cause of diseases that affect the human body and can lead to pandemics and epidemics in various countries. No new content has been added beyond the original text. Viruses are the cause of diseases that affect the human body and can lead to pandemics and epidemics in various countries. The language used is clear, objective, and value-neutral, with a formal register and precise word choice. The text adheres to conventional structure and format, with consistent citation and footnote style. The text is free from grammatical errors, spelling mistakes, and punctuation errors. The purpose of this research is to determine the SIS model for the spread of viral diseases, such as Covid-19 and bird flu, and their resolution behavior. The model is formed by creating a flow diagram of the disease spread using the SIS (Susceptible, Infected, Susceptible) model. The sentences and paragraphs create a logical flow of information with causal connections between statements. The study revealed two equilibrium points: the disease-free equilibrium point and the endemic equilibrium point. To analyze the stability of the disease-free equilibrium point, linearization around the equilibrium point was used. The disease-free equilibrium point is asymptotically stable if the basic reproduction number is less than one, indicating that the disease will disappear after a certain period of time. Numerical simulations were conducted to analyze the behavior of the disease model.
 Keywords: Education Models, Mathematics Models, SIS (Susceptible, Infected, Susceptible)

Dynamics of an eco-epidemic model with Allee effect in prey and disease in predator

Abstract In this work, the dynamics of a food chain model with disease in the predator and the Allee effect in the prey have been investigated. The model also incorporates a Holling type-III functional response, accounting for both disease transmission and predation. The existence of equilibria and their stability in the model have also been investigated. The primary objective of this research is to examine the effects of the Allee parameter. Hopf bifurcations are explored about the interior and disease-free equilibrium point, where the Allee is taken as a bifurcation point. In numerical simulation, phase portraits have been used to look into the existence of equilibrium points and their stability. The bifurcation diagrams that have been drawn clearly demonstrate the presence of significant local bifurcations, including Hopf, transcritical, and saddle-node bifurcations. Through the phase portrait, limit cycle, and time series, the stability and oscillatory behaviour of the equilibrium point of the model are investigated. The numerical simulation has been done using MATLAB and Matcont.

Dynamics and Event-Triggered Impulsive Control of a Fractional-Order Epidemic Model with Time Delay

Due to the lack of timely protection measures against infectious diseases, or based on the particularity of the transmission of some infectious diseases and the time-varying connections between people, the transmission dynamics of infectious diseases in the information society are becoming more and more complex and changeable. A fractional-order epidemic mathematical model with network weighting and latency is proposed in this paper, and the stability near the disease-free equilibrium point and endemic equilibrium point are discussed separately. Subsequently, an event-triggered impulsive control strategy based on an infection rate threshold is put forward. By selecting the appropriate control gain, the Zeno phenomenon can be eliminated on the premise of ensuring the stability of the control error system. Finally, the theoretical results were validated numerically and some conclusions are presented. These findings contribute to future research on the limited-time event-triggered impulsive control of infectious diseases.

Optimal control strategies on HIV/AIDS and pneumonia co-infection with mathematical modelling approach

In this paper, a compartmental model on the co-infection of pneumonia and HIV/AIDS with optimal control strategies was formulated using the system of ordinary differential equations. Using qualitative methods, we have analysed the mono-infection and HIV/AIDS and pneumonia co-infection models. We have computed effective reproduction numbers by applying the next-generation matrix method, applying Castillo Chavez criteria the models disease-free equilibrium points global stabilities were shown, while we have used the Centre manifold criteria to determine that the pneumonia infection and pneumonia and HIV/AIDS co-infection exhibit the phenomenon of backward bifurcation whenever the corresponding effective reproduction number is less than unity. We carried out the numerical simulations to investigate the behaviour of the co-infection model solutions. Furthermore, we have investigated various optimal control strategies to predict the best control strategy to minimize and possibly to eradicate the HIV/AIDS and pneumonia co-infection from the community.

Algorithmic Approach for a Unique Definition of the Next-Generation Matrix

The basic reproduction number R0 is a concept which originated in population dynamics, mathematical epidemiology, and ecology and is closely related to the mean number of children in branching processes (reflecting the fact that the phenomena of interest are well approximated via branching processes, at their inception). Despite the very extensive literature around R0 for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that R0 is not a function of the original ODE model, unless we also include in it a certain (F,V) gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the (F,V) gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a (F,V) gradient decomposition (and hence of the resulting R0). (2) A fixed point equation for the extinction probabilities of a stochastic model associated to a deterministic ODE model, which may be expressed in terms of the (F,V) decomposition. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable (F,V) decompositions are available as well.

A mathematical model with fractional order for obesity with positive and negative interactions and its impact on the diagnosis of diabetes

Overweight and obesity are current problems humankind faces and have serious health consequences because they contribute to diseases such as heart diseases and diabetes. In this paper, we present a mathematical model for the study of overweight and obesity in a population and its impact on the growth of the number of diabetics. For the construction of the model, we take into account social factors and the interactions between different elements of society. We use fractional-order derivatives in the Caputo sense because of the advantages of this type of technique with respect to the memory effect, and it shows different behaviors depending on the fractional order. We find the basic reproduction number and prove the local and global stability of the disease-free equilibrium points. We study the sensitivity index with respect to the basic reproduction number for parameters associated with weight gain due to social pressure and the rate of diagnosis of diabetes not associated with body weight. To validate the model, we perform computational simulations with data extracted from the literature. We conclude that for higher fractional orders a higher value of the basic reproduction number was reached. We show that at the end of the study for different fractional orders that normal-weight individuals are decreasing, and overweight, obese, and diabetic people are increasing.

Analyzing Dynamical Behavior of Tuberculosis Transmission

In this article, we investigated the dynamical behavior of a fuzzy mathematical model with tuberculosis (TB) treatment function. The considered model analyzes fuzzy basic reproduction numbers and stability analysis for both local and global stability conditions around disease-free equilibrium points, as well as its local stability conditions around endemic equilibrium points. We examine the TB model’s sensitivity analysis with a focus on the variation of each of its parameters. We propose predicting the extension of the virus load in TB using fuzzy bifurcation. At the end, we have given a numerical simulation to illustrate the outcomes by considering some special types of treatment functions.

MATHEMATICAL MODELS OF DENGUE TRANSMISSION DYNAMICS WITH VACCINATION AND WOLBACHIA PARAMETERS AND SEASONAL ASPECTS

The Aedes aegypti mosquito is the main carrier of dengue virus transmission to humans. In this study, a mathematical model for the transmission of the dengue virus is constructed using vaccination and Wolbachia parameters in an attempt to control the virus's spread. Furthermore, the fundamental reproduction number is set as a parameter of the infection threshold. Based on the stability of the equilibrium point analysis, it is found that the disease-free equilibrium point is locally asymptotically stable if . Then, a mathematical model of dengue was created by examining the seasonal aspect and adding a periodic term to the mosquito birth rate. Dengue virus transmission in mosquito populations is controlled by air temperature in addition to seasonal variables. In this study, three weather scenarios were simulated: scenario 1 for cold weather (air temperature 14 °C), scenario 2 for hot weather (air temperature 26 °C), and scenario 3 for moderate weather (air temperature between 14 and 26 °C).

ANALYSIS OF THE COVID-19 EPIDEMIC MODEL WITH SELF-ISOLATION AND HOSPITAL ISOLATION

This research developed the SIR model with self-isolation and hospital isolation. The analysis is carried out through the disease-free and endemic equilibrium point analysis and the sensitivity analysis of the basic reproduction number. Based on the disease-free equilibrium point analysis, for a certain period of time the population will be free from COVID-19 if the basic reproduction number is less than 1. If the basic reproduction number is more than 1, the disease will persist in the population, this will lead to an endemic equilibrium point. Based on the sensitivity analysis of parameter values on the basic reproduction number, the parameter for the isolation rate of individually infected individuals in hospitals is -0.4615166040, and the self-isolation rate at home is -0.01853667767. This indicates that isolation in hospitals is more effective than self-isolation in suppressing the spread of COVID-19.

A FRACTIONAL DIFFERENTIAL EQUATION MODEL FOR THE SPREAD OF POTATO LEAF ROLL VIRUS (PLRV) ON POTATOES

Potatoes infected with the PLRV virus will experience a decrease in production up to 90%. In this paper, The PLRV distribution fractional differential equation model with potato and vector populations is reformulated by adding one new parameter, namely the rate of vector death due to predators. The model is divided into susceptible and infected classes. The PLRV dispersion model was developed and converted to a fractional order form for 0<σ ≤ 1. Next, the invariant region, positive solutions, basic reproduction number, equilibrium point, and stability were determined. Based on the stability analysis, it is shown that the stability of the disease-free equilibrium point is locally stable and globally stable if the basic reproduction number (R0)<1, and the stability of the endemic equilibrium point is globally stable if the basic reproduction number (R0)>1. Numerical solutions were also carried out to determine the effect of several parameters on the PLRV distribution model on potatoes. The numerical solution results show that the elimination rate of infected potatoes and the infection rate of potatoes have a significant role in controlling the spread of PLRV in potatoes.