Nonlinear Incidence Rate Research Articles

Complex dynamics of a discrete-time SIR model with nonlinear incidence and recovery rates

In this paper, a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method. The existence and stability of fixed points in this model are well studied. The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters. We discuss in detail some codimension-one bifurcations such as transcritical, period-doubling and Neimark–Sacker bifurcations, and a codimension-two bifurcation with 1:2 resonance. In addition, the phase portraits, bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis. It is found that the numerical results are consistent with the theoretical analysis. More interestingly, we also found other bifurcations in the model during the numerical simulation, such as codimension-two bifurcations with 1:1 resonance, 1:3 resonance and 1:4 resonance, generalized period-doubling and fold-flip bifurcations. The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model. Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases, but also make a great contribution to the prevention and control of infectious diseases.

Effect of a novel generalized incidence rate function in [formula omitted] model: Stability switches and bifurcations

When a disease outbreaks in a population, the incidence rate function may increase initially with an increase of infective and then decrease due to inhibitory effects before getting saturated. For this purpose, we propose a new generalized incidence rate function and demonstrate the impact of this incidence rate function’s non-monotonic feature on the stability of endemic equilibria. In this paper, an SIR compartmental model with this new nonlinear incidence rate function and saturated treatment rate function is proposed. We obtain a disease free equilibrium which is locally stable for R0<1 and unstable otherwise. The system may have multiple endemic equilibria which can be either anti-saddle (node, focus or center) or saddle. We discover that when the incidence rate function is increasing at an anti-saddle endemic equilibrium, the endemic equilibrium may become either stable, unstable or switch stability via a parameter change. However, when the incidence rate function is non-monotonic and if the incidence rate decreases at an anti-saddle endemic equilibrium, the endemic equilibrium is always stable. Due to the limited treatment, we find forward (transcritical) and backward bifurcations. We also establish Hopf bifurcation and saddle–node bifurcation in the model system. Further, we incorporate the incubation delay in the model system. We obtain the same number of equilibria in the delay model. The disease free equilibrium is locally asymptotically stable for all delay lengths for R0<1. The endemic equilibrium switches its stability due to change in delay. Due to delay variation, the increasing incidence rate at an anti-saddle endemic equilibrium causes the multiple stability switches of the equilibrium from both stable and unstable initial states. Further, when the incidence rate decreases at the stable anti-saddle endemic equilibrium, it switches its stability once or multiple times and finally becomes unstable due to delay variation. We explicitly, show multiple stability switches of an endemic equilibrium due to existence of multiple Hopf bifurcations. When we further analyzed model system numerically, we could establish the existence of Hopf–Hopf bifurcation of codimension 2. In this case, the increase in the delay parameter causes the existence of two oscillatory solutions around the unstable endemic equilibrium. Hence, our results show that the delay system either destabilize developing the same frequency oscillations via Hopf bifurcation or two different frequency oscillations via Hopf–Hopf bifurcation of codimension 2. The existence and exhibition of these two frequency oscillations are novel and have not been explored much in epidemiological models.

Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate

In this study, we developed an autonomous non-linear epidemic model for the transmission dynamics of susceptible, vaccinated, infected, and recovered individuals (SVIR model) with non-linear saturation incidence and vaccination rates. The non-linear saturation incidence rate significantly reduces the death ratio of infected individuals by increasing human immunity. We discuss a detailed explanation of the model equilibrium, its basic reproduction number R0, local stability, and global stability. The disease-free equilibrium is observed to be stable if R0&lt;1, while the endemic equilibrium exists and the disease exists permanently in the population if R0&gt;1. To approximate the solution of the model, the well-known Runge–Kutta (RK4) methodology is utilized. The implications of numerous parameters on the population dynamics of susceptible, vaccinated, infected, and recovered individuals are addressed. We discovered that increasing the value of the disease-included death rate ψ has a negative impact on those affected, while it has a positive impact on other populations. Furthermore, the value of interaction between vaccinated and infected λ2 has a decreasing impact on vulnerable and vaccinated people, while increasing in other populations. On the other hand, the model is solved using Euler and Euler-modified techniques, and the results are compared numerically and graphically. The quantitative computations demonstrate that the RK4 method provides very precise solutions compared to the other approaches. The results show that the suggested SVIR model that approximates the solution method is accurate and useful.

Asymptotic dynamics of a logistic SIS epidemic reaction-diffusion model with nonlinear incidence rate

A logistic SIS epidemic reaction-diffusion model with the nonlinear incidence rate SqIp in a spatially heterogeneous environment is concerned. The L∞-estimates of solutions are established under p,q>0 to explore the long-time behaviors of the model. In a special case, asymptotic dynamics and bifurcation behaviors under p=1 are analyzed in terms of the basic reproduction number. Asymptotic profiles of the endemic equilibrium for small and large diffusion rates of the susceptible and infected populations are investigated. It is found that the infectious disease is hard to control even under a policy of mobility regulation when 0<p<1 and q>1.

Global dynamics for an SVEIR epidemic model with diffusion and nonlinear incidence rate

In this paper, we investigate an SVEIR epidemic model with reaction–diffusion and nonlinear incidence. We establish the well-posedness of the solutions and the basic reproduction number mathfrak{R}_{0}. Moreover, we show that the disease-free steady state is globally asymptotically stable when mathfrak{R}_{0}<1, whereas the disease will be persistent when mathfrak{R}_{0}>1. Furthermore, using the method of Lyapunov functional, we prove the global stability of the positive steady state for the spatial homogeneous model.

Global dynamic behavior of a vaccination–age SVIR model with treatment and general nonlinear incidence rate

In this paper, we propose a vaccination–age SVIR model with general nonlinear incidence rate and general treatment function. Firstly, we establish the existence, uniqueness and positivity of solutions for the considered SVIR model in the case where the linear part is nondensely defined and satisfies the Hille–Yosida condition by the use of the integrated semigroup theory. Secondly, based on the basic reproduction number R0, we investigate the existence and the local stability of the equilibrium points. Moreover, we discuss the global stability of the equilibrium points under the light of the Lyapunov–LaSalle approach. Finally, an application with numerical simulations is done to check the validity of our theoretical results and to exemplify the impact of the treatment and the vaccination on the disease propagation.

A nonlinear HCV model in deterministic and randomly fluctuating environments

Two novel deterministic and stochastic nonlinear hepatitis C virus (HCV) models with nonlinear incidence rates as , and , containing acute and chronic infections, are developed, and an endeavor to understand their dynamics has been accomplished. For the former model, both the existence of unique equilibria and the sharp sufficient conditions of globally stable equilibria under different cases are established. Whereafter, the efforts taken to detect the asymptotic behaviors around equilibria, the existence of ergodic stationary distribution, and stochastic extinction of the latter model have been supplemented with meticulous theoretical proofs. At last, several numerical examples and discussions are illustrated to sustain and visualize our theoretical results. Compared and analyzed the numerical and theoretical results, we can find that the noise can influence the dynamics of disease and turn the disease from persistence to extinction. Moreover, several control measures are suggested based on our study.

Relaxation Oscillation and Canard Explosion for a SIRS Model with Nonlinear Incidence Rate

Relaxation Oscillation and Canard Explosion for a SIRS Model with Nonlinear Incidence Rate

Impact of nonlinear infection rate on HIV/AIDS considering prevalence‐dependent awareness

An HIV/AIDS transmission model is constructed and analyzed that assimilates spontaneous change in human behavior depending on the disease prevalence incorporating self‐shielding mechanism and treatment control. It is assumed that human population become aware of HIV disease from exposed environment and are conscious enough to make their knowledge fruitful by adopting self‐protective measures to reduce disease transmission. We introduce Hill type of behavioral change function and an awareness‐induced transmission function associated with nonlinear infection rate. We emphasize the effectiveness of human awareness in controlling HIV epidemic by investigating the impact of nonlinear incidence rate. Further, an optimal control model is formulated and solved to find the strategy of applying treatment considering spontaneous response of people towards HIV disease. In the numerical simulation segment, impacts of significant parameters have been assessed using sophisticated techniques. We calibrate the proposed model with the real data of HIV cases in India and estimate some crucial model parameters. Moreover, different amalgamations of preventive and treatment controls are introduced. The optimal treatment control has massive impact in curbing HIV epidemic. Also, combined strategy of prevention and treatment can curtail the disease with economic gain. It reveals that effective response of individuals can suppress the disease in such an unexpected situation when treatment rate is low. Increasing the efficacy of human consciousness is much beneficial in curbing HIV outbreak. Thus, our findings highlight a path to control HIV epidemic with uninstructed modification of human behavior by boosting up the productivity of their existing knowledge.

The Stability Analysis and Transmission Dynamics of the SIR Model with Nonlinear Recovery and Incidence Rates

In the present paper, the SIR model with nonlinear recovery and Monod type equation as incidence rates is proposed and analyzed. The expression for basic reproduction number is obtained which plays a main role in the stability of disease-free and endemic equilibria. The nonstandard finite difference (NSFD) scheme is constructed for the model and the denominator function is chosen such that the suggested scheme ensures solutions boundedness. It is shown that the NSFD scheme does not depend on the step size and gives better results in all respects. To prove the local stability of disease-free equilibrium point, the Jacobean method is used; however, Schur–Cohn conditions are applied to discuss the local stability of the endemic equilibrium point for the discrete NSFD scheme. The Enatsu criterion and Lyapunov function are employed to prove the global stability of disease-free and endemic equilibria. Numerical simulations are also presented to discuss the advantages of NSFD scheme as well as to strengthen the theoretical results. Numerical simulations specify that the NSFD scheme preserves the important properties of the continuous model. Consequently, they can produce estimates which are entirely according to the solutions of the model.

Optimal control for a multi-group reaction–diffusion SIR model with heterogeneous incidence rates

This paper is devoted to the study of an optimal control problem for a generalized multi-group reaction–diffusion SIR epidemic model, with heterogeneous nonlinear incidence rates. The proposed model incorporates a wide range of spatiotemporal epidemic models. Primarily, the ones used to describe the propagation of zoonotic and sexually transmitted diseases, as well as epidemics that propagate disparately within populations. For the aforementioned diseases, dividing the susceptible, infected and recovered populations into several subpopulations is necessary in order to capture all the possible ways of the disease transmission. This makes the problem of finding the possible optimal control strategies and the division of the available control resources complicated. To address this problem mathematically, for each subpopulation, we introduce two types of control variables, namely vaccination for the susceptible and treatment for the infected. The existence and uniqueness of a biologically feasible solution to the proposed model, for fixed controls, is derived by means of a truncation technique and a semigroup approach. Moreover, first-order necessary optimality conditions for the introduced optimal control problem are obtained using the adjoint state method. Finally, numerical simulations are performed for a two-group epidemic model with particular incidence rates and by considering three cases in the maximal control resources allowed for each subpopulation.

Asymptotic analysis of a nonlinear stochastic eco-epidemiological system with feedback control

This paper proposes a new stochastic eco-epidemiological model with nonlinear incidence rate and feedback controls. First, we prove that the stochastic system has a unique global positive solution. Second, by constructing a series of appropriate stochastic Lyapunov functions, the asymptotic behaviors around the equilibria of deterministic model are obtained, and we demonstrate that the stochastic system exists a stationary Markov process. Third, the conditions for persistence in the mean and extinction of the stochastic system are established. Finally, we carry out some numerical simulations with respect to different stochastic parameters to verify our analytical results. The obtained results indicate that the stochastic perturbations and feedback controls have crucial effects on the survivability of system.

MATHEMATICAL ANALYSIS OF A DELAYED MALWARE PROPAGATION MODEL ON MOBILE WIRELESS SENSOR NETWORK

The security of mobile wireless sensor networks has captivated extensive attention of researchers because of their wide range of applications and vulnerability to attacks caused by malware. In this paper, we investigate a delayed malware propagation model on mobile wireless sensor network incorporating nonlinear incidence rate, logistic growth rate and recovery rate. Local asymptotic stability of the endemic equilibrium and existence of Hopf bifurcation at crucial value of the time delay are analyzed. Then, properties of Hopf bifurcation are explored. Specifically, global exponential stability is investigated via linear matrix inequality. An example is presented finally to underline the effectiveness of findings in our paper numerically and graphically.

To study the transmission dynamic of SARS-CoV-2 using nonlinear saturated incidence rate

In this work, we construct a new SARS-CoV-2 mathematical model of SQIR type. The considered model has four compartments as susceptible S, quarantine Q, infected I and recovered R. Here saturated nonlinear incidence rate is used for the transmission of the disease. We formulate our model first and then the disease-free and endemic equilibrium (EE) are calculated. Further, the basic reproduction number is computed via the next generation matrix method. Also on using the idea of Dulac function, the global stability for the proposed model is discussed. By using the Routh–Hurwitz criteria, local stability is investigated. Through nonstandard finite difference (NSFD) scheme, numerical simulations are performed. Keeping in mind the significant importance of fractional calculus in recent time, the considered model is also investigated under fractional order derivative in Caputo sense. Finally, numerical interpretation of the model by using various fractional order derivatives are provided. For fractional order model, we utilize fractional order NSFD method. Comparison with some real data is also given.

Traveling waves of an epidemic model with general nonlinear incidence rate and infection-age structure

Traveling waves of an epidemic model with general nonlinear incidence rate and infection-age structure

A nonlinear SAIR epidemic model: Effect of awareness class, nonlinear incidences, saturated treatment and time delay

Awareness plays a vital role in informing and educating people about infection risk during an outbreak and hence helps to reduce the epidemic’s health burden by lowering the peak incidence. Therefore, this paper studies a susceptible-aware-infected-recovered (SAIR) epidemic model with the novel combinations of Michaelis-Menten functional type nonlinear incidence rates for unaware and aware susceptible with the inclusion of time delay as a latent period and a saturated treatment rate for infected people. The model is analyzed mathematically to describe disease transmission dynamics in two obtained equilibria: disease-free and endemic. We derive the basic reproduction number R_0 and investigate the local and global stability behavior of obtained equilibria for the time delay . A bifurcation analysis is performed using center manifold theory when there is no time delay, revealing the forward bifurcation when R_0 varies from unity. Moreover, the presence of Hopf bifurcation around EE is shown depending on the bifurcation parameter time delay. Lastly, the numerical simulations validate the analytical findings.

Dynamical analysis of a multi-group SIR epidemic model with nonlocal diffusion and nonlinear incidence rate

In this paper, a multi-group SIR epidemic model with nonlocal diffusion and nonlinear incidence rate in spatially heterogeneous environment is proposed. The well-posedness, including the existence, positivity and boundedness of solutions, is achieved. The basic reproduction number R0 is defined, and the existence of principal eigenvalue is studied. Further, the threshold criteria on the global dynamics of the model are established in terms of R0. That is, when R0<1, the disease-free steady state is globally asymptotically stable, when R0>1, the disease is uniformly persistent. Particularly, a single-group degenerated SIR epidemic model is also studied. We give some properties on the principle eigenvalue and prove that there exists a compact global attractor for the solution semiflow. Furthermore, when the basic reproduction number R̄0>1, we obtain that the endemic steady state is unique to exist and globally asymptotically stable.

Theoretical and numerical analysis of hepatitis B virus model with non-singular kernels

Hepatitis B virus (HBV) has a strong influence on global public health and economics. The transmission of hepatitis B infection has different phases, i.e. acute and chronic carrier stages play an important role. Here we assessed the transmissibility associated with different infection stages of HBV and generated an epidemic model with a nonlinear incidence rate. Furthermore, we fictionalized the proposed model by using the Atangana Baleanu operator with vaccination effects. First, we find the basic reproduction number and equilibria of the proposed model. And then we introduce the dynamical behavior of the proposed model in Atangana Baleanu derivative fractional sense. The uniqueness and existence of the numerical solution of the proposed model are proved by using the fixed point theory. A numerical scheme using the Adams–Bashforth method for solving the proposed fractional model involving the AB derivative is presented. Finally, numerical simulation is carried out to illustrate the theoretical results and compared fractional order ζ for different values.

THE IMPACT OF DELAY STRATEGIES ON THE DYNAMICS OF CORONAVIRUS PANDEMIC MODEL WITH NONLINEAR INCIDENCE RATE

Currently, the world is facing a devastating pandemic of a novel coronavirus, which started as an outbreak of pneumonia of unknown cause in Wuhan city of China in December of 2019. According to the recent report of the World Health Organization (WHO), 210 countries convicted badly 1.8 million infections and almost 200,000 causalities. Due to the non-availability of the vaccination, delay strategies such as community distancing, travel restrictions, extension in breaks, use of face-mask, and self-quarantine are the effective treatments to control the pandemic of coronavirus. So, we proposed the delayed susceptible-exposed-infected-recovered model with a nonlinear incidence rate to study the effective role of control strategies. For this analysis, we discussed three types of equilibria of the model such as trivial, coronavirus free, and coronavirus existence with delay terms. The local and global stabilities are investigated by using well-posed notations like the Lasalle invariance principle, Routh–Hurwitz criterion, and Lyapunov function. In the end, some useful replications are presented.

Qualitative analysis on a reaction-diffusion host-pathogen model with incubation period and nonlinear incidence rate

In this paper, a degenerate reaction-diffusion host-pathogen model with an incubation period and a nonlinear incidence rate in a spatially heterogeneous environment is proposed. We analyze the dynamics of this model on a bounded domain. Firstly, we establish the well-posedness, including the global existence of solutions and the existence of a global attractor by defining a noncompact measure. Then, the basic reproduction number is given and a threshold dynamics is established. Finally, when there is a positive steady state, we investigate the asymptotic profiles of the positive steady state when host individuals disperse at small or large rate. Our results show that the incubation period can significantly enhance the persistence of the disease if the dispersal rate of susceptible hosts or exposed hosts is small or large.