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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]

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.

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]

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]

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 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]

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.

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 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.

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]

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]

Infectious diseases kill millions of people each year, and they are the major public health problem in the world. This paper presents a modified Susceptible-Latent-Infected-Removed (SLIR) compartmental model of disease transmission with nonlinear incidence. We have obtained a threshold value of basic reproduction number (R0) and shown that only a disease-free equilibrium exists when R01. With the help of the Lyapunov-LaSalle Invariance Principle, we have shown that disease-free equilibrium and endemic equilibrium are both globally asymptotically stable. The study has also provided the model calibration to estimate parameters with month wise coronavirus (COVID-19) data, i.e. reported cases by worldometer from March 2020 to May 2021 and provides prediction until December 2021 in China. The Partial Rank Correlation Coefficient (PRCC) method was used to investigate how the model parameters’ variation impact the model outcomes. We observed that the most important parameter is transmission rate which had the most significant impact on COVID-19 cases. We also discuss the epidemiology of COVID-19 cases and several control policies and make recommendations for controlling this disease in China.

In this paper, we study a more general diffusive spatially dependent vaccination model for infectious disease. In our diffusive vaccination model, we consider both therapeutic impact and nonlinear incidence rate. Also, in this model, the number of compartments of susceptible, vaccinated and infectious individuals are considered to be functions of both time and location, where the set of locations (equivalently, spatial habitats) is a subset of $ \mathbb {R}^n $ with a smooth boundary. Both local and global stability of the model are studied. Our study shows that if the threshold level $ \mathcal {R}_0 \le 1, $ the disease-free equilibrium $ E_0 $ is globally asymptotically stable. On the other hand, if $ \mathcal {R}_0> 1 $ then there exists a unique stable disease equilibrium $ E^* $ . The existence of solutions of the model and uniform persistence results are studied. Finally, using finite difference scheme, we present a number of numerical examples to verify our analytical results. Our results indicate that the global dynamics of the model are completely determined by the threshold value $ \mathcal {R}_0 $ .

In this paper, we study a more general diffusive spatially dependent vaccination model for infectious disease. In our diffusive vaccination model, we consider both therapeutic impact and nonlinear incidence rate. Also, in this model, the number of compartments of susceptible, vaccinated and infectious individuals are considered to be functions of both time and location, where the set of locations (equivalently, spatial habitats) is a subset of R n with a smooth boundary. Both local and global stability of the model are studied. Our study shows that if the threshold level R 0 ≤ 1 , the disease-free equilibrium E 0 is globally asymptotically stable. On the other hand, if R 0 > 1 then there exists a unique stable disease equilibrium E ∗ . The existence of solutions of the model and uniform persistence results are studied. Finally, using finite difference scheme, we present a number of numerical examples to verify our analytical results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R 0 . [ABSTRACT FROM AUTHOR]

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In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number ${\cal R}_0$ of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for ${\cal R}_0 \lt 1$ and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever ${\cal R}_0 \gt 1$. Several numerical simulations are given to illustrate our results.

In this paper, we study the dynamics of a vector-borne disease model with two transmission paths: direct transmission through contact and indirect transmission through vector. The direct transmission is considered to be a nonmonotone incidence function to describe the psychological effect of some severe diseases among the population when the number of infected hosts is large and/or the disease possesses high case fatality rate. The system has a disease-free equilibrium which is locally asymptotically stable when the basic reproduction number ( R 0 ) is less than unity and may have up to four endemic equilibria. Analytical expression representing the epidemic growth rate is obtained for the system. Sensitivity of the two transmission pathways were compared with respect to the epidemic growth rate. We numerically find that the direct transmission coefficient is more sensitive than the indirect transmission coefficient with respect to R 0 and the epidemic growth rate. Local stability of endemic equilibrium is studied. Further, the global asymptotic stability of the endemic equilibrium is proved using Li and Muldowney geometric approach. The explicit condition for which the system undergoes backward bifurcation is obtained. The basic model also exhibits the hysteresis phenomenon which implies diseases will persist even when R 0 < 1 although the system undergoes a forward bifurcation and this phenomenon is rarely observed in disease models. Consequently, our analysis suggests that the diseases with multiple transmission routes exhibit bistable dynamics. However, efficient application of temporary control in bistable regions will curb the disease to lower endemicity. Additionally, numerical simulations reveal that the equilibrium level of infected hosts decreases as psychological effect increases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the traveling wave solutions of a discrete diffusive epidemic model with nonlinear incidence and time delay. Employing the method of upper and lower solutions, Schauder's fixed point theorem and a limiting approach, we prove the existence of bounded super-critical and critical traveling wave solutions. Moreover, we obtain the positiveness and asymptotic boundary of the traveling wave solutions, which guarantee that the traveling wave solutions are non-trivial. The existence results show that the traveling waves are mixed of front type and pulse type. By way of contradiction and two-sided Laplace transform, we derive the non-existence of non-trivial, positive and bounded traveling wave solutions. It is the first time to apply the method of upper and lower solutions together with Schauder's fixed point theorem and two-sided Laplace transform to investigate the existence and non-existence of traveling wave solutions for discrete diffusive epidemic models, respectively. [ABSTRACT FROM AUTHOR]

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 known as the reproductive number which depends on the associated reproductive numbers with serotype-1 and serotype-2. The value of 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 , 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 , 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. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a new SEIRS epidemic model describing nonlinear incidence with a more general form and the transmission of influenza virus with disease resistance. The basic reproductive number ℜ0 $\Re_{0}$ is obtained by using the method of next generating matrix. If ℜ01 $\Re_{0}>1$, by using the geometric method, we obtain some sufficient conditions for global stability of the unique endemic equilibrium. Finally, numerical simulations are provided to support our theoretical results.

In this paper, we propose an SIVS epidemic model with continuous age structures in both infected and vaccinated classes and with a general nonlinear incidence. Firstly, we provide some basic properties of the system including the existence, uniqueness and positivity of solutions. Furthermore, we show that the solution semiflow is asymptotic smooth. Secondly, we calculate the basic reproduction number $ \mathcal {R}_0 $ by employing the classical renewal process, which determines whether the disease persists or not. In the main part, we investigate the global stability of the equilibria by the approach of Lyanpunov functionals. Some numerical simulations are conducted to illustrate the theoretical results and to show the effect of the transmission rate and immunity waning rate on the disease prevalence.

Abstract An HIV/AIDS epidemic model with general nonlinear incidence rate and treatment is formulated. The basic reproductive number ℜ 0 $\Re_{0}$ is obtained by use of the method of the next generating matrix. By carrying out an analysis of the model, we study the stability of the disease-free equilibrium and the unique endemic equilibrium by using the geometric approach for ordinary differential equations. Numerical simulations are given to show the effectiveness of the main results.

We investigate a diffusive SEIR epidemic model with nonlinear incidence of the form I p S q for 0 < p ≤ 1 , which is used to describe the spread of an infection among susceptible and infected individuals. By constructing suitable Lyapunov functions, we study the global stability of the endemic equilibrium and the disease-free equilibrium. Our results reveal the important role of the parameter p in the persistence and extinction of the disease. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate a reaction–diffusion virus infection model with humoral immunity and nonlinear incidence. In spatially heterogeneous case, the basic reproduction number of virus infection R 0 is calculated, when R 0 ≤ 1 the global asymptotical stability of the infection-free steady state is established, and when R 0 > 1 the uniform persistence of infected cells and viruses, as well as the existence of antibody-free infection steady state are also obtained. In spatially homogeneous case, the antibody response basic reproduction number R 1 is calculated, by using the Lyapunov functions method and the persistence theory of dynamical systems we obtain that when R 0 > 1 and R 1 ≤ 1 the antibody-free infection equilibrium is globally asymptotically stable, and when R 0 > 1 and R 1 > 1 the model is uniformly persistent and the infection equilibrium exists and is also globally asymptotically stable. Finally, the numerical examples are presented in order to verify the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

Highlights • Propose and study an SIR model with a generalized incidence under preventive vaccination and treatment controls. • Treatment strategies are responsible for backward bifurcations and multiple endemic equilibria. • The infective population decreases with respect to the maximal capacity of treatment; and/or the preventive vaccination rate. • The infective population also decreases with respect to the psychological or inhibitory effects of infectives. Abstract In this paper, we propose and study an SIRS epidemic model that incorporates: a generalized incidence rate function describing mechanisms of the disease transmission; a preventive vaccination in the susceptible individuals; and different treatment control strategies depending on the infective population. We provide rigorous mathematical results combined with numerical simulations of the proposed model including: treatment control strategies can determine whether there is an endemic outbreak or not and the number of endemic equilibrium during endemic outbreaks, in addition to the effects of the basic reproduction number; the large value of the preventive vaccination rate can reduce or control the spread of disease; and the large value of the psychological or inhibitory effects in the incidence rate function can decrease the infective population. Some of our interesting findings are that the treatment strategies incorporated in our SIRS model are responsible for backward or forward bifurcations and multiple endemic equilibria; and the infective population decreases with respect to the maximal capacity of treatment. Our results may provide us useful biological insights on population managements for disease that can be modeled through SIRS compartments. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold $R_0$ which completely governs the disease dynamics: when $R_01$, the disease is permanent. It is interesting that the threshold value $R_0$ bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.

We consider two classes of infinitely dimensional epidemic models with nonlinear incidence, where one assumes that the rate of a vaccinated individual losing immunity depends on the vaccine-age and another assumes that, before the vaccine begins to wane, there is a period during which the vaccinated individuals have complete immunity against the infection. The first model is given by a coupled ordinary-hyperbolic differential system and the second class is described by a delay differential system. We calculate their respective basic reproduction numbers, and show they characterize the global dynamics by constructing the appropriate Lyapunov functionals.

This paper is concerned with the existence and non-existence of traveling wave solutions for a diffusive SIR model with delay and nonlinear incidence. First, we construct a pair of upper and lower solutions and a bounded cone. Then we prove the existence of traveling wave by using Schauder's fixed point theorem and constructing a suitable Lyapunov functional. The nonexistence of traveling wave is obtained by two-sided Laplace transform. Moreover, numerical simulations support the theoretical results. Finally, we also obtain that the minimal wave speed is decreasing with respect to the latent period and increasing with respect to the diffusion rate of infected individuals. [ABSTRACT FROM AUTHOR]

In this paper, we propose an SIVS epidemic model with continuous age structures in both infected and vaccinated classes and with a general nonlinear incidence. Firstly, we provide some basic properties of the system including the existence, uniqueness and positivity of solutions. Furthermore, we show that the solution semiflow is asymptotic smooth. Secondly, we calculate the basic reproduction number by employing the classical renewal process, which determines whether the disease persists or not. In the main part, we investigate the global stability of the equilibria by the approach of Lyanpunov functionals. Some numerical simulations are conducted to illustrate the theoretical results and to show the effect of the transmission rate and immunity waning rate on the disease prevalence. [ABSTRACT FROM AUTHOR]

A recent paper Xu et al. (2018) studied the global stability and traveling wave solution of a vaccination model with nonlinear incidence. Furthermore, in this paper, we first study the local stability of this model, and then establish the uniform persistence result. Our results are a supplement to above paper. [ABSTRACT FROM AUTHOR]