Saturated Incidence Research Articles

Global Dynamics in an Alcoholism Epidemic Model with Saturation Incidence Rate and Two Distributed Delays

In this study, considering the delays for a susceptible individual becoming an alcoholic and the relapse of a recovered individual back into being an alcoholic, we formulate an epidemic model for alcoholism with distributed delays and relapse. The basic reproduction number R0 is calculated, and the threshold property of R0 is established. By analyzing the characteristic equation, we demonstrate the local asymptotic stability of the different equilibria under various conditions: when R0<1, the alcoholism-free equilibrium is locally asymptotically stable; when R0>1, the alcoholism equilibrium exists and is locally asymptotically stable. Furthermore, we demonstrate the global asymptotic stability at each equilibrium using a suitable Lyapunov function. Specifically, when R0<1, the alcoholism-free equilibrium is globally asymptotically stable; when R0>1, the alcoholism equilibrium is globally asymptotically stable. The sensitivity analysis of R0 shows that reducing exposure is more effective than treatment in controlling alcoholism. Interestingly, we found that extending the latency delay h1 and relapse delay h2 also effectively contribute to the control of the spread of alcoholism. Numerical simulations are also provided to support our theoretical results.

ANALISIS KESTABILAN MODEL SI UNTUK PENYAKIT MENULAR DENGAN ADANYA TRANSMISI VERTIKAL DAN TINGKAT KEJADIAN JENUH

The transmission of infectious diseases can occur through two pathways: horizontal and vertical. Horizontal transmission occurs through direct or indirect physical contact with the infectious agent, while vertical transmission takes place when an infected mother transmits the disease to a fetus or a newborn. Within the context of disease transmission models, a critical feature is the saturation incidence rate, which refers to the impact of interventions that can reduce the rate of disease transmission among susceptible and infected individuals. This research aims to elucidate the formation of a model, determine equilibrium points, and calculate the basic reproduction number using the Next Generation Matrix method. The analysis involves assessing local stability through linearization methods and global stability using Lyapunov functions. Sensitivity analysis is conducted on the basic reproduction number, and numerical simulations are performed using the fourth-order Runge-Kutta method. The research findings indicate the establishment of an SIS (Susceptible-Infected) model for infectious diseases with vertical transmission and saturation incidence. This model depicts the spread of the disease in a population, where individuals can exist in susceptible or infected conditions. Equilibrium points include a disease-free equilibrium that is locally and globally stable when the basic reproduction number is less than one, and an endemic equilibrium that is locally and globally stable when the basic reproduction number exceeds one. Sensitivity analysis reveals that each parameter has varying influences on the basic reproduction number. An increase in the saturation incidence rate leads to a decrease in the number of infected subpopulations, while an increase in the vertical transmission rate results in a similar decline. Numerical simulations support stability analyses at equilibrium points. These findings provide a deeper understanding of the factors influencing the spread of diseases within a population.

Dynamic modeling and analysis of acute and chronic hepatitis B epidemic with saturation incidence

The new stochastic and deterministic hepatitis B epidemic models are established. The models consist of four types: susceptible individuals, acutely infected hepatitis B individuals, chronically infected hepatitis B individuals and recovered individuals. This study focuses on the transmission dynamics of acute and chronic hepatitis B epidemics problems and the development of optimal control strategies to control the transmission of hepatitis B in the population. To this end, we first calculate the equilibrium point and basic reproduction number of the deterministic hepatitis B model to study the stability of the above model at the equilibrium point. Secondly, we give the basic reproduction number of the stochastic model of hepatitis B virus. A suitable Lyapunov function is constructed, and the solvability of the stochastic model is confirmed using the Itô formula. By using a series of stochastic inequalities and strong number theorems, the extinction and stationary distribution of hepatitis B in this stochastic model are obtained. Finally, the optimal control theory is used to develop an optimal control strategy for eliminating hepatitis B virus transmission. The Runge–Kutta method is used to simulate the above models to verify the rationality of our main theoretical results.

Fractional optimal control strategies for hepatitis B virus infection with cost-effectiveness analysis

Hepatitis B disease is a communicable disease that is caused by the hepatitis B virus and has become a significant health problem in the world. It is a contagious disease that is transmittable from person to person either horizontally or vertically. This current study is aimed at sensitivity analysis and optimal control strategies for a fractional hepatitis B epidemic model with a saturated incidence rate in the sense of the Caputo order fractional derivative approach. Fundamental properties of the proposed fractional order model are obtained and discussed. A detailed analysis of disease-free equilibrium and endemic equilibrium points is given by applying fractional calculus theory, which is a generalized version of classical calculus. Sensitivity indexes are calculated for the classical order model. Illustrative graphics that show the dependence of the sensitivity index on fractional order derivative for alpha in (0,1) are provided. Based on the results of the sensitivity analysis and using Pontryagin’s Maximum Principle, optimal control strategies for preventing hepatitis B infection with vaccination and treatment are considered. Fractional Euler’s method is used to carry out the numerical simulation for the proposed fractional optimal control system and the obtained results are analyzed. The results of the analysis reveal that hepatitis B disease can be prevented if necessary precautionary is taken or effective vaccination and treatment control measures are applied. The analysis of cost-effectiveness is also conducted and discussed.

A stochastic co-infection model for HIV-1 and HIV-2 epidemic incorporating drug resistance and dual saturated incidence rates

This study investigates a mathematical model encompassing both HIV-1 and HIV-2 epidemics, with a focus on drug resistance and dual saturated incidence rates. We begin by analyzing the local stability of equilibrium points in the deterministic system using Routh-Hurwitz and Jacobian matrix approaches. Furthermore, we establish the existence of a global positive solution for our model. We identify specific parameter conditions that lead to disease extinction. Additionally, we explore the persistence of HIV-1 and HIV-2 infections by examining their mean values. To assess the practical implications of our theoretical analysis, we conduct graphical evaluations under various noise disturbances, employing Milstein's Higher Order Method. This research contributes to a deeper understanding of the dynamics of HIV-1 and HIV-2 co-infection, with potential implications for treatment strategies and disease management.

Fractional perspective evaluation of chikungunya infection with saturated incidence functions

It is significant to investigate the transmission dynamics of vector-borne infection because it has a global impact, can help predict and prevent future outbreaks, is important for understanding the impact of climate change on public health, can lead to more effective control strategies, and can improve our comprehension of the dynamics of these infections. Our paper presents a new model for chikungunya virus infection, which considers treatment and vaccination, using the Atangana-Baleanu derivative within the framework of the Caputo definition. First of all, we examine the positivity and uniqueness of the solution for the model. Then, we find out the fundamental results of the suggested model, such as the steady-state without infection and the R0 value that indicates endemicity pertaining to the system. Furthermore, we verify the local asymptotic stability of the infection-free steady-state. Through the application of fixed-point theory, we demonstrate the existence of the solution and propose a numerical methodology to investigate the dynamic behavior of the model. Finally, we performed simulations to demonstrate the effects of vaccination, transmission probability, treatment, and index of memory on the system's solution pathways. The findings of this research suggest the most sensitive parameters of the system responsible for the control and prevention of the infection. Emphasizing the significance of the memory index as an attractive parameter, we propose its consideration to the policy makers as a control parameter for the prevention of the infection.

Sliding dynamics and optimal control of avian influenza model with saturated incidence rate

AbstractIn this paper, we develop a susceptible‐infective‐susceptible‐infected‐recovered avian influenza model with saturated incidence rate and three control measures, which is proposed to evaluate the comprehensive effects of different control measures on the spread of avian influenza. Based on the threshold policy, the threshold level is used as the reference index of whether to take control measures. No control measures will be taken when the total number of infected birds and humans is less than the threshold level. Otherwise, the three control measures will be implemented simultaneously to mitigate the spread of the disease. Furthermore, by using the Utkin equivalent control method, the existence of sliding mode equation and its dynamics are investigated. Additionally, the method of dynamic programming is introduced to study the optimal control for the model. Through constructing and proving the existence of viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation, the optimal control pair is further obtained. Finally, numerical simulations are performed to illustrate the corresponding theoretical analysis results.

Stability and bifurcation control of a delayed fractional eco-epidemiological system with saturated incidence

This paper proposes a delayed feedback control strategy for a fractional ecological infectious disease system with infected predators. Firstly, for the uncontrolled system, the stability of each equilibrium is discussed. Choosing the gestational delay as the branching parameter, the conditions of Hopf bifurcation are given at the positive equilibrium. The results show that the stable domain for the fractional system is larger than that for the integer-order system. Secondly, for the controlled system, the effects of gestational delay and feedback delay on the induction of the Hopf bifurcation are investigated, respectively. In addition, we find that the feedback gain coefficient also changes the critical value at which Hopf bifurcation occurs. Selecting a suitable feedback controller can effectively extend the stable domain of the system compared to the control-free system. Finally, the accuracy and effectiveness of the delay feedback controller in controlling the bifurcation are verified through numerical simulations.

Stability analysis of HCV dynamic model with saturation incidence, cellular immunity and interferon effect in intrahepatic and extrahepatic tissues

An HCV model with saturation incidence and cellular immunity is constructed and analyzed. The model takes account of both intrahepatic and extrahepatic tissues and the effect of antiviral drugs especially the interferon. The basic reproduction number R01 and CTL immune reproduction number R02 are calculated, respectively. The existence of three equilibria, namely infection-free, immune response-free, infection-immune coexistence equilibrium is analyzed with a novel method. Stability analysis is also given with corresponding reproduction number, and the results show that when R01<1, there are no virus and immune response existed in the end; when R02<1<R01, the virus survives, but immune response gradually disappears; when R02>1, the system will be globally stable with both virus and immune response existing. Numerical analysis is also given which is consistent with the theoretical analysis.

Dynamics of a nonlinear epidemic transmission model incorporating a class of hospitalized individuals: a qualitative analysis and simulation

This study aims to develop a novel mathematical epidemic compartmental model that includes a compartment or class for individuals who become infected and experience severe illness due to the infection. These individuals require hospitalization and the use of specialized medical equipment, such as ventilators, ICU beds, etc, during an outbreak. This compartment is referred to as the ‘hospitalized population compartment’ throughout this study. Additionally, the model incorporates a saturated incidence rate for new infection cases and the hospitalization rate for individuals severely affected by the infection, intending to create a more realistic scenario of the dynamics of disease transmission. The model is developed by integrating a compartment for hospitalized individuals into the standard susceptible-infected-recovered compartmental model and is subsequently mathematically analyzed for qualitative behavior. In this model, the saturated hospitalization rate reflects that the number of severely infected individuals who can be hospitalized is limited at any given time due to constraints in sufficient hospital infrastructure availability. The incidence rate of susceptibles becoming infected is modeled using the Holling Type II functional form, which incorporates inhibitory effects observed within the population. The study analyzes the mathematical model for two types of equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). To investigate the stability of both equilibria, the basic reproduction number, , is calculated using the next-generation matrix method. The findings indicate that when , the DFE is locally asymptotically stable. Conversely, when , the DFE becomes unstable, leading to the emergence of a positive EE. Additionally, the study explores the occurrence of forward and backward transcritical bifurcations under specific conditions when . Furthermore, the study delves into both the local and global stability behaviors of the EE. Numerical simulations of the model are also performed to support the theoretical findings.

Complex dynamics of a Caputo derivative-based fractional-order SIR model incorporating saturated incidence and recovery

Complex dynamics of a Caputo derivative-based fractional-order SIR model incorporating saturated incidence and recovery

Dynamics of a modified Leslie–Gower Holling-type II eco-epidemiological model on time scales

Migratory birds play an important role in the transmission of illnesses like West Nile Virus and Salmonella, etc. The long-time dynamic behaviors of a novel class of nonautonomous model for migratory birds with Leslie–Gower Holling-type II functional response and saturation incidence rate on time scales are investigated in this study. Within this scope, certain necessary conditions are found for the persistence and extinction of the illness under very weak assumptions. Furthermore, the model's global attractiveness is examined by creating a Lyapunov function. Additionally, included are numerical simulations ( T = R , Z ) that back up our theoretical research.

Mathematical analysis of HIV/HTLV-I co-infection model with saturated incidence rate

Mathematical analysis of HIV/HTLV-I co-infection model with saturated incidence rate

Global dynamic analysis of a nonlinear state-dependent feedback control SIR model with saturation incidence

Global dynamic analysis of a nonlinear state-dependent feedback control SIR model with saturation incidence

A viral co-infection model with general infection rate in deterministic and stochastic environments

In this paper, we present a model of infection in which a virus can simultaneously infect two types of target cells. The model has a general form of infection rates in deterministic and stochastic settings, incorporating bilinear infection rates, saturated incidences and half-saturated incidences. In the deterministic case, we investigate the existence and stability of infection-free steady state and infection steady state, respectively. Considering that the infection rate coefficient is affected by random noise, we built the corresponding stochastic model using the Ornstein–Uhlenbeck process. In the stochastic case, by constructing suitable Lyapunov functions, we established sufficient conditions for the stationary distribution and extinction of the model, respectively. In addition, the covariance matrix in the probability density function of the model near the infection steady state is determined. Finally, we consider the three most common forms of virus infection and perform comprehensive numerical simulations to support our theoretical results.

Stability and Optimal Control of a Fractional SEQIR Epidemic Model with Saturated Incidence Rate

The fractional differential equation has a memory property and is suitable for biomathematical modeling. In this paper, a fractional SEQIR epidemic model with saturated incidence and vaccination is constructed. Firstly, for the deterministic fractional system, the threshold conditions for the local and global asymptotic stability of the equilibrium point are obtained by using the stability theory of the fractional differential equation. If R0&lt;1, the disease-free equilibrium is asymptotically stable, and the disease is extinct; when R0&gt;1, the endemic equilibrium is asymptotically stable and the disease persists. Secondly, for the stochastic system of integer order, the stochastic stability near the positive equilibrium point is discussed. The results show that if the intensity of environmental noise is small enough, the system is stochastic stable, and the disease will persist. Thirdly, the control variables are coupled into the fractional differential equation to obtain the fractional control system, the objective function is constructed, and the optimal control solution is obtained by using the maximum principle. Finally, the correctness of the theoretical derivation is verified by numerical simulation.

Stability and Hopf bifurcation on an immunity delayed HBV/HCV model with intra- and extra-hepatic coinfection and saturation incidence

Stability and Hopf bifurcation on an immunity delayed HBV/HCV model with intra- and extra-hepatic coinfection and saturation incidence

Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments

In this article, we study the dynamical properties of susceptible-vaccinated-infected-susceptible (SVIS) epidemic system with saturated incidence rate and vaccination strategies. By constructing the suitable Lyapunov function, we examine the existence and uniqueness of the stochastic system. With the help of Khas'minskii theory, we set up a critical value [Formula: see text] with respect to the basic reproduction number [Formula: see text] of the deterministic system. A unique ergodic stationary distribution is investigated under the condition of [Formula: see text]. In the epidemiological study, the ergodic stationary distribution represents that the disease will persist for long-term behavior. We focus for developing the general three-dimensional Fokker-Planck equation using appropriate solving theories. Around the quasi-endemic equilibrium, the probability density function of the stochastic system is analyzed which is the main theme of our study. Under [Formula: see text], both the existence of ergodic stationary distribution and density function can elicit all the dynamical behavior of the disease persistence. The condition of disease extinction of the system is derived. For supporting theoretical study, we discuss the numerical results and the sensitivities of the biological parameters. Results and conclusions are highlighted.

Dynamical Analysis of Hyper-ILSR Rumor Propagation Model with Saturation Incidence Rate.

With the development of the Internet, it is more convenient for people to obtain information, which also facilitates the spread of rumors. It is imperative to study the mechanisms of rumor transmission to control the spread of rumors. The process of rumor propagation is often affected by the interaction of multiple nodes. To reflect higher-order interactions in rumor-spreading, hypergraph theories are introduced in a Hyper-ILSR (Hyper-Ignorant-Lurker-Spreader-Recover) rumor-spreading model with saturation incidence rate in this study. Firstly, the definition of hypergraph and hyperdegree is introduced to explain the construction of the model. Secondly, the existence of the threshold and equilibrium of the Hyper-ILSR model is revealed by discussing the model, which is used to judge the final state of rumor propagation. Next, the stability of equilibrium is studied by Lyapunov functions. Moreover, optimal control is put forward to suppress rumor propagation. Finally, the differences between the Hyper-ILSR model and the general ILSR model are shown in numerical simulations.

Stability analysis of an SIR model with alert class modified saturated incidence rate and Holling functional type-II treatment

Abstract This study discusses an SIR epidemic model with modified saturated incidence rates and Holling functional type-II therapy. In this study, we take the new alert compartment (A) in the SIR compartment model. Consider the modified non-linear incidence rate from the susceptible to the infected class and the second non-linear incidence rate from the alert to the infected class. Further, we investigate the elementary reproduction number, the equilibrium points of the model, and their stability. We apply manifold theory to discuss bifurcations of the disease-free equilibrium point. This study shows that the infected population decreases with the Holling functional type II treatment rate. It also shows that the number of infected people decreases when the psychological rate increases and the contact rate decreases.