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A general stochastic compartment model for cholera with higher-order perturbation is proposed, which incorporates direct and indirect transmission by contaminated water. Nonlinear incidence, multiple stages of infection, multiple states of pathogen, and second-order white-noises perturbation are introduced into the model, which includes and extends the existing cholera model. The existence and ergodicity of the stationary distribution for the cholera system are obtained by constructing a suitable Lyapunov function, which determines a sharp critical value R 0 s corresponding to the basic productive number R 0 of the ordinary differential equation. The results show that, if R 0 s > 1 , the system has a unique and ergodic stationary distribution, which implies the persistence of the diseases. Our general results are applied to a cholera system with a Holling type-II functional response. [ABSTRACT FROM AUTHOR]

The saturation incidence is of vital importance in the spread of large-scale epidemics. Although many achievements have been concerning deterministic epidemic models with saturated incidence, little is known about a stochastic version with a saturation incidence under semi-Markovian switching. Based on the above, this paper aims to propose such a stochastic model. Via constructing suitable Lyapunov functions, we prove the existence and uniqueness of a positive solution and provide a threshold value for disease tending to be extinct. Further, the bounded and persistent properties are provided for the model. Under sufficient conditions, the stochastic model has a unique ergodic stationary distribution. As illustrations, several examples are presented to show the theoretical results. The results show that the density function of the solution depends on regime switching. As an application, we study COVID-19 data in Iceland. The SMC-ABC algorithm is used to estimate unknown parameters of the model with semi-Markovian switching. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a general class of susceptible, infected, quarantine, recovered (SIQR) epidemic models with spatial heterogeneity and imperfect quarantine, focusing on the role of quarantines in controlling infectious diseases. We first determine the expression of the basic reproduction number $ \mathcal{R}_0 $ for two infected classes and allowing the transmission between them. Then, the stability of the steady states is investigated depending on $ \mathcal{R}_0 $. Moreover, we analyze the impact of small/large diffusion rates of susceptible and infected individuals depending on the quarantine rates. Finally, we provide a list of recommendations to contain the spread of a disease. [ABSTRACT FROM AUTHOR]

Brucellosis is a zoonotic disease that poses a huge economic burden to China's livestock industry. A S E I 1 I 2 R V dynamics model of brucellosis with nonlinear incidence is proposed to study the transmission of brucellosis. The model focuses on the acute and chronic phases of infection and the impact of control measures. We introduce the basic reproduction number R 0 . The threshold dynamics of the model are demonstrated through the Jacobian matrix, the Hurwitz criterion, constructing the Lyapunov function, and the second additive composite matrix. Perform sensitivity analysis on threshold parameter R 0 , obtain parameters that have a significant impact on R 0 , for example, β , γ 1 , γ 2 etc. Propose control measures based on them, and thus introduce the control system of the model. The optimal control strategy was obtained according to the principle of Pontryagin maximum, and the optimal control group was solved by the gradient descent method. Numerical simulations verify the theoretical results of threshold dynamics and sensitivity analysis. Simulate the effectiveness of the optimal system. We propose control measures for 11 different strategy combinations. Confirm that the simultaneous implementation of preventive and therapeutic methods can significantly inhibit the spread of sheep brucellosis. It is also suggested that when the optimal combination has been implemented, the total number of infected sheep can still be reduced by increasing the migration rate from the acute to the chronic stage of infection. [ABSTRACT FROM AUTHOR]

This study develops a modified SIR model (Susceptible–Infected–Recovered) to analyze the dynamics of the COVID-19 pandemic. In this model, infected individuals are categorized into the following two classes: I a , representing asymptomatic individuals, and I s , representing symptomatic individuals. Moreover, accounting for the psychological impacts of COVID-19, the incidence function is nonlinear and expressed as S g (I a , I s) = β S (I a + I s) 1 + α (I a + I s) . Additionally, the model is based on a symmetry hypothesis, according to which individuals within the same compartment share common characteristics, and an asymmetry hypothesis, which highlights the diversity of symptoms and the possibility that some individuals may remain asymptomatic after exposure. Subsequently, using the next-generation matrix method, we compute the threshold value ( R 0 ), which estimates contagiousness. We establish local stability through the Routh–Hurwitz criterion for both disease-free and endemic equilibria. Furthermore, we demonstrate global stability in these equilibria by employing the direct Lyapunov method and La-Salle's invariance principle. The sensitivity index is calculated to assess the variation of R 0 with respect to the key parameters of the model. Finally, numerical simulations are conducted to illustrate and validate the analytical findings. [ABSTRACT FROM AUTHOR]

Symmetry in mathematical models often refers to invariance under certain transformations. In stochastic models, symmetry considerations must also account for the probabilistic nature of inter- actions and events. In this paper, a stochastic vector-borne model with plant virus disease resistance and nonlinear incidence is investigated. By constructing suitable stochastic Lyapunov functions, we show that if the related threshold R 0 s < 1 , then the disease will be extinct. By using the reproduction number R 0 , we establish sufficient conditions for the existence of ergodic stationary distribution to the stochastic model. Furthermore, we explore the results graphically in numerical section and find that random fluctuations introduced in the stochastic model can suppress the spread of the disease, except for increasing plant virus disease resistance and decreasing the contact rate between infected plants and susceptible vectors. The results reveal the correlation between symmetry and stochastic vector-borne models and can provide deeper insights into the dynamics of disease spread and control, potentially leading to more effective and efficient management strategies. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new viral infection model by incorporating a new compartment for follicular dendritic cell (FDC), nonlinear incidence, CTL immune response, and two intracellular delays. The main purpose of the paper is to make an improvement and supplement to the global dynamics of the model proposed by Callaway and Perelson (2002), in which global stability has not been studied. The global stabilities of equilibria are established by constructing corresponding Lyapunov functionals in terms of two threshold parameters, $ \mathfrak{R}_0 $ and $ \mathfrak{R}_1 $. The obtained results imply that both nonlinear incidence and intracellular time delays have no impact on the stability of the model.

With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when $ R_0 = 1, $ we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction.

In this paper, a SIS epidemic model with nonlinear incidence and ratio dependent pulse control is proposed and analyzed. Firstly, for the system that ignores the effect of pulses, the basic reproductive number R 0 is derived using the next-generation matrix method, and the stability of the equilibria of the system is analyzed. And then the dynamics of the system containing pulse effects was investigated. The existence of periodic solutions has been proven by constructing appropriate Poincaré mappings and using the fixed point theorem. We found that pulses have a significant impact on system dynamics. Under the influence of pulses, the system trajectory undergoes periodic oscillations, which are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when R 0 = 1 , we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction. With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction. [ABSTRACT FROM AUTHOR]

In this paper, taking into account the inevitable impact of environmental perturbations on disease transmission, we primarily investigate a stochastic model for measles infection with nonlinear incidence. The transmission rate in this model follows a logarithmic normal distribution influenced by an Ornstein--Uhlenbeck (OU) process. To analyze the dynamic properties of the stochastic model, our first step is to establish the existence and uniqueness of a global solution for the stochastic equations. Next, by constructing appropriate Lyapunov functions and utilizing the ergodicity of the OU process, we establish sufficient conditions for the existence of a stationary distribution, indicating the prevalence of the disease. Furthermore, we provide sufficient conditions for disease elimination. These conditions are derived by considering the interplay between the model parameters and the stochastic dynamics. Finally, we validate the theoretical conclusions through numerical simulations, which allow us to assess the practical implications of the established conditions and observe the dynamics of the stochastic model in action. By combining theoretical analysis and numerical simulations, we gain a comprehensive understanding of the stochastic model's behavior, contributing to the broader understanding of measles transmission dynamics and the development of effective control strategies. [ABSTRACT FROM AUTHOR]

In this article, we propose a cholera model to study the effects of multiple transmission pathways, imperfect vaccine, nonlinear incidences, and differential infectivity of vibrios. The expression of the basic reproductive number R 0 is derived. There is only the disease-free equilibrium E 0 if R 0 ≤ 1 , while, besides E 0 , there is also a unique endemic equilibrium E ∗ if R 0 > 1 . When R 0 < 1 , E 0 is globally asymptotically stable by using the technique of linearization and the fluctuation lemma. When R 0 > 1 , E ∗ is globally asymptotically stable by the Lyapunov direct method. These theoretical results are supported with numerical simulations for the case with Beddington-DeAngelis incidences. We further perform the sensitivity analyses of R 0 and the infection level at E ∗ to determine the significant parameters affecting disease outbreak and severity, respectively. Influences of the vaccination rate ϕ and the waning rate of vaccine η on the dynamical behaviors of the model are also discussed. [ABSTRACT FROM AUTHOR]

Throughout the progress of epidemic scenarios, individuals in different health classes are expected to have different average daily contact behavior. This contact heterogeneity has been studied in recent adaptive models and allows us to capture the inherent differences across health statuses better. Diseases with reinfection bring out more complex scenarios and offer an important application to consider contact disaggregation. Therefore, we developed a nonlinear differential equation model to explore the dynamics of relapse phenomena and contact differences across health statuses. Our incidence rate function is formulated, taking inspiration from recent adaptive algorithms. It incorporates contact behavior for individuals in each health class. We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios. The relationship between the different contact rates heavily influences these conditions. Numerical examples highlight the effect of temporarily recovered individuals and initial conditions on infected population persistence.

The presence of information asymmetry can hinder the public's ability to make well-informed decisions, resulting in unwarranted suspicion and the widespread dissemination of rumors. Therefore, it is crucial to provide individuals with consistent and dependable scientific education. Regular popular science education is considered a periodic impulsive intervention to mitigate the impact of information asymmetry and promote a more informed and discerning public. Drawing on these findings, this paper proposes a susceptible-hesitant-infected-refuting-recovered (SHIDR) rumor-spreading model to explain the spread of rumors. The model incorporates elements such as time delay, nonlinear incidence, and refuting individuals. Firstly, by applying the comparison theorem of an impulsive differential equation, we calculate two thresholds for rumor propagation. Additionally, we analyze the conditions of global attractiveness of the rumor-free periodic solution. Furthermore, we consider the condition for the rumor's permanence. Finally, numerical simulations are conducted to validate the accuracy of our findings. The results suggest that increasing the proportion of impulsive vaccination, reducing the impulsive period, or prolonging the delay time can effectively suppress rumors. [ABSTRACT FROM AUTHOR]

A vector-borne disease model with spatial diffusion with time delays and a general incidence function is studied. We derived conditions under which the system exhibits threshold behavior. The stability of the disease-free equilibrium and the endemic equilibrium are analyzed by using the linearization method and constructing appropriate Lyapunov functionals. It is shown that the given conditions are satisfied by at least two common forms of the incidence function. [ABSTRACT FROM AUTHOR]

In this paper, we propose a SVEIR-I epidemic model with media coverage in a spatially heterogeneous environment, and study the role of media coverage in the spread of diseases in a spatially heterogeneous environment. In a spatially heterogeneous environment, we first set up the well-posedness of the model. Then, we define the basic reproduction number $ R_0 $ of the model and establish the global dynamic threshold criteria: when $ R_0 < 1 $, disease-free steady state is globally asymptotically stable, while when $ R_0 > 1 $, the model is uniformly persistent. In addition, the existence and uniqueness of the equilibrium state of endemic diseases were obtained when $ R_0 > 1 $ in homogeneous space and heterogeneous diffusion environment. Further, by constructing appropriate Lyapunov functions, the global asymptotic stability of disease-free and positive steady states was established. Finally, through numerical simulations, it is shown that spatial heterogeneity can increase the risk of disease transmission, and can even change the threshold for disease transmission; media coverage can make people more widely understand disease information, and then reduce the effective contact rate to control the spread of disease.

In this article, we are concerned with a diffusive SVEIR epidemic model with nonlinear incidences. We first obtain the well-posedness of solutions for the model. Then, the basic reproduction number R 0 and the local basic reproduction number R ¯ 0 (x) are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between R 0 and R ¯ 0 (x) as well as the asymptotic properties of R 0 when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators L 1 and L 2 . Using the theory of monotone dynamical systems and the persistence theory of dynamical systems, we show that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , while the disease is uniformly persistent when R 0 > 1 . Furthermore, in the spatially homogeneous case, by using the Lyapunov functions method and LaSalle's invariance principle, we completely obtain that the disease-free equilibrium is globally asymptotically stable if R 0 ≤ 1 , and the endemic equilibrium is globally asymptotically stable if R 0 > 1 and an additional condition is satisfied. [ABSTRACT FROM AUTHOR]

We propose a generalization of a model with age of infection in a heterogeneous environment. Firstly, we give the well‐posedness of the model and prove that the solutions are bounded and positive. The difficult mathematical issue in this research is that the model is partially degenerate, and the solution map is not compact. In addition, we construct a global attractor of a bounded set to establish the existence of total trajectory. Moreover, we define the principal eigenvalue associated with a principal eigenvalue problem to give a relation with the basic reproduction number R0$$ {R}_0 $$ and this value. By assuming that R0<1$$ {R}_0<1 $$, then the infection‐free steady‐states E0$$ {E}^0 $$ is globally asymptotically stable. Furthermore, for R0>1$$ {R}_0>1 $$ and by using the persistence results, we prove the existence of endemic steady‐states E∗$$ {E}^{\ast } $$, and by constructing an appropriate Lyapunov function, we show that E∗$$ {E}^{\ast } $$ is globally asymptotically stable. Lastly, we validate our theoretical analysis by giving some numerical graphics. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed. The wellposedness of the solution is firstly established. Then the basic reproduction number R0 is defined and a threshold dynamics is obtained. That is, when R0 < 1, the disease-free steady state is locally stable, which implies that the disease is extinct, when R0 > 1, the disease is permanent, and there exists at least one positive steady state solution. Finally, the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated. Furthermore, as an application of theoretical analysis, a numerical example involving the spread of influenza is discussed. Based on the numerical simulations, we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation, and the increase of diffusion rate, saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation. Therefore, we propose to reduce the flow of people to lower the effect of spatial heterogeneity, increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate, and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation. [ABSTRACT FROM AUTHOR]

Throughout the progress of epidemic scenarios, individuals in different health classes are expected to have different average daily contact behavior. This contact heterogeneity has been studied in recent adaptive models and allows us to capture the inherent differences across health statuses better. Diseases with reinfection bring out more complex scenarios and offer an important application to consider contact disaggregation. Therefore, we developed a nonlinear differential equation model to explore the dynamics of relapse phenomena and contact differences across health statuses. Our incidence rate function is formulated, taking inspiration from recent adaptive algorithms. It incorporates contact behavior for individuals in each health class. We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios. The relationship between the different contact rates heavily influences these conditions. Numerical examples highlight the effect of temporarily recovered individuals and initial conditions on infected population persistence. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of age-structured cholera model with general infection rates. We first explore the existence and uniqueness, dissipativeness and persistence of the solutions, and the existence of the global attractor by verifying the asymptotical smoothness of the orbits. We then give mathematical analysis on the existence and local stability of the positive equilibrium. Based on the preparation, we further investigate the global behavior of the cholera infection model. Corresponding numerical simulations have been presented. Our results improve and generalize some known results on cholera models.

The paper mainly investigates a stochastic SIRS epidemic model with Logistic birth and nonlinear incidence. We obtain a new threshold value ( R 0 m ) through the Stratonovich stochastic differential equation, different from the usual basic reproduction number. If R 0 m < 1 , the disease-free equilibrium of the illness is globally asymptotically stable in probability one. If R 0 m > 1 , the disease is permanent in the mean with probability one and has an endemic stationary distribution. Numerical simulations are given to illustrate the theoretical results. Interestingly, we discovered that random fluctuations can suppress outbreaks and control the disease. [ABSTRACT FROM AUTHOR]

Mosquito bites from the genus Aedes spread the Zika virus to humans, which can be transmitted through sexual contact and blood transfusions. This study formulated and analyzed a mathematical model for the virus in human and mosquito populations. Based on nonlinear incidence, the infected population is divided into two, namely symptomatic and asymptomatic. The existence and stability of the model equilibriums are based on the reproduction ratio. Furthermore, the stable local endemic and non-endemic equilibrium point is R0 < 1 and R0 > 1, respectively. The significant parameter affects the number of symptomatic and asymptomatic infections. It was determined using sensitivity analysis. Also, control efforts were made to reduce transmission rates by eradicating mosquito populations using insecticides, reducing direct contact with mosquitoes, and direct routine health checks. The Pontryagin Maximum Principle showed that the three control strategies can significantly reduce the number of infected individuals. [ABSTRACT FROM AUTHOR]

ln this paper, we extend the classical SEIRS model within the framework of the fractional model in the sense of Caputo–Fabrizio derivative modified with an auxiliary parameter. This model describes the transmission of the influenza virus and shows the influence of the disease resistance and the form of the non-linearity of the incidence rate on the spread of the infectious disease. The existence and uniqueness of the variables of the given model are examined using the fixed point theorems. A detailed analysis of the stability of the disease-free equilibrium point was presented. Finally, using the three-step fractional Adams–Bashforth scheme, the profile of each state variable is depicted for different values of the fractional-order α and compared with the first-order derivative curves. [ABSTRACT FROM AUTHOR]

In this paper, based on Euler-Marryama method and theory of stochastic processes, a stochastic discrete SIVS epidemic model with general nonlinear incidence and vaccination is proposed by adding random perturbation and then discretizing the corresponding stochastic differential equation model. Firstly, the basic properties of continuous and discrete deterministic SIVS epidemic models are obtained. Then a criterion on the asymptotic mean-square stability of zero solution for a general linear stochastic difference system is established. As the applications of this criterion, the sufficient conditions on the stability in probability of the disease-free and endemic equilibria for the stochastic discrete SIVS epidemic model are obtained. The numerical simulations are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the spread of infectious disease using a fractional-order model with Caputo's fractional derivative operator. The considered model includes an infectious disease that includes two types of infected class, the first shows the presence of symptoms (symptomatic infected persons), and the second class does not show any symptoms (asymptomatic infected persons). Further, we considered a nonlinear incidence function, where it is obtained that the investigated fractional system shows some important results. In fact, different types of bifurcation are obtained, as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation, where it is discussed in detail through the research. For the numerical part, a proper numerical scheme is used for the graphical representation of the solutions. The mathematical findings are checked numerically.

In this paper, based on Euler–Marryama method and theory of stochastic processes, a stochastic discrete SIVS epidemic model with general nonlinear incidence and vaccination is proposed by adding random perturbation and then discretizing the corresponding stochastic differential equation model. Firstly, the basic properties of continuous and discrete deterministic SIVS epidemic models are obtained. Then a criterion on the asymptotic mean-square stability of zero solution for a general linear stochastic difference system is established. As the applications of this criterion, the sufficient conditions on the stability in probability of the disease-free and endemic equilibria for the stochastic discrete SIVS epidemic model are obtained. The numerical simulations are given to illustrate the theoretical results.

In this paper we propose and study an epidemic model with nonlinear incidence rate, describing some factors effect (protection, exposure, immigration, social distancing, vaccination) on the spread of certain diseases on the community like the novel coronavirus COVID-19. The dynamical behavior of the proposed model is examined. We investigate the existence and stability of the disease-free equilibrium and the endemic equilibrium. The existence of a limit cycle is studied. Simulations of the model are performed to illustrate and support the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract We introduce a diffusive SEIR model with nonlocal delayed transmission between the infected subpopulation and the susceptible subpopulation with a general nonlinear incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number R 0 = ∂ I F ( S 0 , 0 ) / γ $R_{0}=\partial _{I}F(S_{0},0)/\gamma $ of the corresponding ordinary differential equations and the minimal wave speed c ∗ $c^{*}$ . The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. In the present paper, we overcome these difficulties to obtain the threshold dynamics. In view of the numerical simulations, we also obtain that the minimal wave speed is explicitly determined by the time delay and nonlocality in disease transmission and by the spatial movement pattern of the exposed and infected individuals.

This paper is about the existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. Because of the introduction of nonlocal dispersal and the generality of incidence function, it is difficult to investigate the existence of critical traveling waves. To this end, we construct an auxiliary system and show the existence of traveling waves for the auxiliary system. Employing the results for the auxiliary system, we obtain the existence of traveling waves for the delayed nonlocal dispersal SIR epidemic model with the critical wave speed under mild conditions.

In this paper, we study a diffusive HCV infections epidemic model with nonlinear incidence rate and analyze the stability of the two kinds of equilibria. By constructing various Lyapunov functions, we prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R 0 < 1 and the endemic equilibrium is globally asymptotically stable when the basic reproduction number R 0 > 1 . Finally, some numerical simulations are given to confirm the theoretical analysis. The results show that when other parameters are the same, the linear infection rate and the non-linear infection rate have different effects on disease spread. [ABSTRACT FROM AUTHOR]

In order to reduce the great harm of infectious diarrhoeal disease caused by Norovirus infection to human health,based on the transmission characteristics of Norovirus,the transmission dynamics behavior of Norovirus was studied.Taking into account the characteristics that the latent infected with Norovirus can also transmit the disease,a dynamic model of Norovirus transmission with nonlinear incidence was established.The basic reproduction number R0of the model was calculated and then the stability of the disease-free equilibrium point and the endemic equilibrium point were proved by using the Lyapunov function and the geometric method.The results show that when R0≤1,the disease-free equilibrium point is globally asymptotically stable and the disease disappears; when R0>1,under certain conditions,the endemic equilibrium point is global asymptotically stable.The theoretical results are verified by numerical simulation.The research results have enriched the theory of infectious virus transmission and provide a reference for the study of virus transmission mechanism.

This paper studies a delayed viral infection model with diffusion and a general incidence rate. A discrete-time model was derived by applying nonstandard finite difference scheme. The positivity and boundedness of solutions are presented. We established the global stability of equilibria in terms of $ \mathfrak {R}_0 $ by applying Lyapunov method. The results showed that if $ \mathfrak {R}_0 $ is less than 1, then the infection-free equilibrium $ E_0 $ is globally asymptotically stable. If $ \mathfrak {R}_0 $ is greater than 1, then the infection equilibrium $ E_* $ is globally asymptotically stable. Numerical experiments are carried out to illustrate the theoretical results.

Ever since the World Health Organization gave the name COVID-19 to the coronavirus pneumonia disease, much of the world has been severely impact by the pandemic socially and economically. In this paper, the mathematical modeling and stability analyses in terms of the susceptible–exposed–infected–removed (SEIR) model with a nonlinear incidence rate, along with media interaction effects, are presented. The sliding mode control methodology is used to design a robust closed loop control of the epidemiological system, where the property of symmetry in the Lyapunov function plays a vital role in achieving the global asymptotic stability in the output. Two policies are considered: the first considers only the governmental interaction, the second considers only the vaccination policy. Numerical simulations of the control algorithms are then evaluated. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the traveling waves in a nonlocal dispersal SIR epidemic model with nonlinear incidence and distributed latent delay. It is found that the traveling waves connecting the disease-free equilibrium with endemic equilibrium are determined by the basic reproduction number R 0 $\mathcal{R}_{0}$ and the minimal wave speed c ∗ $c^{*}$ . When R 0 > 1 $\mathcal{R}_{0}>1$ and c > c ∗ $c>c^{*}$ , the existence of traveling waves is established by using the upper-lower solutions, auxiliary system, constructing the solution map, and then the fixed point theorem, limiting argument, diagonal extraction method, and Lyapunov functions. When R 0 > 1 $\mathcal{R}_{0}>1$ and 0 < c < c ∗ $0< c< c^{*}$ , the nonexistence result is also obtained by using the reduction to absurdity and the theory of asymptotic spreading.

Abstract Dengue fever is a common disease which can cause shock, internal bleeding, and death in patients if a second infection is involved. In this paper, a multi-serotype dengue model with nonlinear incidence rate is formulated to study the transmission of two dengue serotypes. The dynamical behaviors of the proposed model depend on the threshold value R 0 n $R_{{0}}^{{n}}$ known as the reproductive number which depends on the associated reproductive numbers with serotype-1 and serotype-2. The value of R 0 n $R_{{0}}^{{n}}$ is used to reflect whether the disease dies out or becomes endemic. It is found that the proposed model has a globally stable disease-free equilibrium if R 0 n ≤ 1 $R_{{0}}^{{n}}\leq 1$ , which indicates that if public health measures that make (and keep) the threshold to a value less than unity are carried out, the strategy in disease control is effective in the sense that the number of infected human and mosquito populations in the community will be brought to zero irrespective of the initial sizes of sub-populations. When R 0 n > 1 $R_{{0}}^{{n}}>1$ , the endemic equilibria called the co-existence primary and secondary infection equilibria are locally asymptotically stable. The effects of cross immunity and nonlinear incidence rate are explored using data from Thailand to determine the effective strategy in controlling and preventing dengue transmission and reinfection.

The existence of critical traveling wave solutions for a class of discrete diffusion SIR models with nonlinear incidence and time delay were studied. Under the condition that the total population is not a constant, the upper and lower solutions method and the Schauder fixed point theorem were used to prove the existence of the solution on a finite interval. Furthermore, the existence of critical traveling wave solutions was proved on the real number field through limit arguments. Finally, with the fluctuation lemma and the proof by contradiction, the asymptotic boundary of the critical traveling wave was obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study an SIRS epidemic model with ratio‐dependent incidence rate function describing the mechanisms of infectious disease transmission. Impacts of vaccination and treatment on the transmission dynamics of the disease have been explored. The treatment rate is constant when the number of infected individuals is greater than the maximal capacity of treatment and proportional to the number of infected individuals when the number of infected individuals is less than the maximal capacity of treatment. Analysis shows that (1) the sufficiently large value of the preventive vaccination rate can control the spread of disease, and (2) a threshold level of the psychological (or inhibitory) effects in the incidence rate function is enough to decrease the infective population. It is also obtained that model undergoes transcritical and saddle‐node bifurcations with respect to disease contact rate. Moreover, in the presence of treatment strategy, the model has multiple endemic equilibria and undergoes a backward bifurcation. The maximal capacity of treatment plays important roles on the disease dynamics of the model. The number of infected individuals decreases with respect to the maximal capacity of treatment, and the disease completely dies out from the system for the large capacity of the treatment when constant treatment strategy is applied. Further, it is also found that the spread of disease can be suppressed by increasing treatment rate. From sensitivity analysis, we have observed that the transmission and treatment rates are most sensitive parameters. The effects of different parameters on the disease dynamics have also been investigated via numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIRS epidemic model with nonlinear incidence and vaccination is formulated to investigate the transmission dynamics of infectious diseases. The model not only incorporates the white noise but also the external environmental noise which is described by semi-Markov process. We first derive the explicit expression for the basic reproduction number of the model. Then the global dynamics of the system is studied in terms of the basic reproduction number and the intensity of white noise, and sufficient conditions for the extinction and persistence of the disease are both provided. Furthermore, we explore the sensitivity analysis of Rs0 with each semi-Markov switching under different distribution functions. The results show that the dynamics of the entire system is not related to its switching law, but has a positive correlation to its mean sojourn time in each subsystem. The basic reproduction number we obtained in the paper can be applied to all piecewise-stochastic semi-Markov processes, and the results of the sensitivity analysis can be regarded as a prior work for optimal control. [ABSTRACT FROM AUTHOR]

This paper studies a delayed viral infection model with diffusion and a general incidence rate. A discrete-time model was derived by applying nonstandard finite difference scheme. The positivity and boundedness of solutions are presented. We established the global stability of equilibria in terms of R 0 by applying Lyapunov method. The results showed that if R 0 is less than 1, then the infection-free equilibrium E 0 is globally asymptotically stable. If R 0 is greater than 1, then the infection equilibrium E ∗ is globally asymptotically stable. Numerical experiments are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we establish stochastic differential equations on the basis of a nonlinear deterministic model and study the global dynamics. For the deterministic model, we show that the basic reproduction number ℜ0 determines whether there is an endemic outbreak or not: if ℜ0 ℜ0 < 1, the disease dies out; while if ℜ0 > 1, the disease persists. For the stochastic model, we provide analytic results regarding the stochastic boundedness, perturbation, permanence and extinction. Finally, some numerical examples are carried out to confirm the analytical results. One of the most interesting findings is that stochastic fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain an existence theorem of critical waves for a diffusive Susceptible‐Infected‐Removed (SIR) epidemic model with saturating incidence rate and infinitely distributed delay. Our results completely answer an aforementioned unsolved questions by using the Schauder's fixed point theorem and constructing a perfect pair of upper‐lower solutions. [ABSTRACT FROM AUTHOR]

We introduce a diffusive SEIR model with nonlocal delayed transmission between the infected subpopulation and the susceptible subpopulation with a general nonlinear incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number R 0 = ∂ I F (S 0 , 0) / γ of the corresponding ordinary differential equations and the minimal wave speed c ∗ . The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. In the present paper, we overcome these difficulties to obtain the threshold dynamics. In view of the numerical simulations, we also obtain that the minimal wave speed is explicitly determined by the time delay and nonlocality in disease transmission and by the spatial movement pattern of the exposed and infected individuals. [ABSTRACT FROM AUTHOR]

In this paper, for a class of nonlocal dispersal SIR epidemic models with nonlinear incidence, we study the existence of traveling waves connecting the disease-free equilibrium with endemic equilibrium. We obtain that the existence of traveling waves depends on the minimal wave speed c ∗ and basic reproduction number R 0 . That is, if R 0 > 1 and c > c ∗ then the model has a traveling wave connecting the disease-free equilibrium with endemic equilibrium. Otherwise, if R 0 > 1 and 0 < c < c ∗ , then there does not exist the traveling wave connecting the disease-free equilibrium with endemic equilibrium. The numerical simulations verify the theoretical results. Our results improve and generalize some known results. [ABSTRACT FROM AUTHOR]