Nonlinear Incidence Research Articles

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.

A delayed diffusive HBV model with nonlinear incidence and CTL immune response

In this paper, we develop a diffusive and delayed hepatitis B virus (HBV) infection model, which incorporates both virus‐to‐cell and cell‐to‐cell transmissions along with cytotoxic T lymphocyte (CTL) immune response. Two threshold parameters (basic reproduction number) and (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Employing the approach of Lyapunov functionals, we show that the infection‐free steady state is globally asymptotically stable when ; the immune‐free infected steady state is globally asymptotically stable when ; and the immune‐present steady state is globally asymptotically stable when , and . We perform the sensitivity and uncertainly analysis on with respect to parameters and find that the most important parameter that drives the HBV infection is the mortality of HBV DNA capsids. Moreover, the theoretical results are supported with numerical simulations.

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.

On the Dynamics of Sexually Transmitted Diseases Under Awareness and Treatment

In this paper, we develop and extend the work of Jia and Qin on sexually transmitted disease models with a novel class of non-linear incidence. Awareness plays a central role both in the susceptible and the infectious classes. The Existence, uniqueness, boundedness, and positivity of solutions are systematically established. Concavity arguments and the occurrence of a vertical asymptote are essential in the proof of the existence of a unique endemic equilibrium. Conditions for the stability of all steady states are investigated. In particular, numerical simulations are performed in order to capture the asymptotic behavior of solutions.AMS Classification: 92D30, 34D23.

Modelling the Role of Human Behaviour in Ebola Virus Disease (EVD) Transmission Dynamics.

The role of human behaviour in the dynamics of infectious diseases cannot be underestimated. A clear understanding of how human behaviour influences the spread of infectious diseases is critical in establishing and designing control measures. To study the role that human behaviour plays in Ebola disease dynamics, in this paper, we design an Ebola virus disease model with disease transmission dynamics based on a new exponential nonlinear incidence function. This new incidence function that captures the reduction in disease transmission due to human behaviour innovatively considers the efficacy and the speed of behaviour change. The model's steady states are determined and suitable Lyapunov functions are built. The proofs of the global stability of equilibrium points are presented. To demonstrate the utility of the model, we fit the model to Ebola virus disease data from Liberia and Sierra Leone. The results which are comparable to existing findings from the outbreak of 2014 − 2016 show a better fit when the efficacy and the speed of behaviour change are higher. A rapid and efficacious behaviour change as a control measure to rapidly control an Ebola virus disease epidemic is advocated. Consequently, this model has implications for the management and control of future Ebola virus disease outbreaks.

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.

Analysis of SEIR epidemic patch model with nonlinear incidence rate, vaccination and quarantine strategies

In this paper, a susceptible–exposed–infectious–recovered (SEIR) epidemic patch model with nonlinear incidence rate, vaccination and quarantine strategies is presented. According to the properties of nonnegative matrices, the range of basic reproduction number is obtained. Moreover, the stability of disease-free equilibrium is studied. Some numerical simulations are given to validate the theoretical results. The effects of vaccination, quarantine strategies and population migration on disease transmission are also investigated by numerical simulations.

Bifurcations analysis of a discrete time [formula omitted] epidemic model with nonlinear incidence function

In this paper, we present a discrete-time SIR epidemic model and investigate the stability of its fixed points, as well as the bifurcations of the one and two parameters. The numerical normal form is used to analyze bifurcations. This model exhibits Neimark–Sacker transcritical, flip, and strong resonance bifurcations. Using the critical coefficients, a scenario is identified for each bifurcation. We verify analytical results using the MATLAB package MatContM, which employs the numerical continuation method.

Global dynamics and bifurcation analysis of an insect-borne plant disease model with two transmission routes

Huanglongbing (HLB) is a plant disease mainly spread by the insect-borne citrus psyllid. It is the most destructive citrus pathosystem worldwide. To understand the impact of sexual transmission on HLB dynamics, we propose a host–vector–HLB compartment model incorporating two transmission routes. The basic reproduction number [Formula: see text] is derived. Various interventions of the disease are assessed. We also investigate the effect of different incidence functions to simulate sexual transmission. For the case of sublinear incidence functions, the disease-free equilibrium is globally asymptotically stable (GAS) provided [Formula: see text]. For mass action incidence of sexual transmission, the endemic equilibrium is GAS provided [Formula: see text]. However, under nonlinear incidence, it is proved that the model may exhibit backward bifurcation. Theoretical and numerical studies reveal that (i) different forces of infection between heterosexual psyllids in the model may have a distinct impact on disease dynamics; (ii) sensitivity analysis shows that for [Formula: see text], the transmission rate between host and vector is more sensitive parameter than that between heterosexual psyllids; (iii) if the sexual transmission is ignored, the disease burden is likely to be underestimated in comparison with realistic scenarios; (iv) in the absence of chemical insecticides, the combined use of yellow sticky traps and injection of nutrient solutions can be more effective in suppressing the spread of HLB. These findings provide valuable insights for public policymakers to determine the long-term viability of implemented HLB management strategies and highlight the urgency of finding sustainable HLB solutions.

Spatial dynamics for an SIRE epidemic model with diffusion and prevention in contaminated environments

In this paper, a reaction–diffusion SIRE epidemic model in contaminated environments is proposed, in which the effect of protection for susceptible individuals is included by the nonlinear incidence functions b ( S ) E $b(S)E$ and g ( S ) I $g(S)I$ . When the space is heterogeneous, the basic reproduction number R 0 $\mathcal {R}_{0}$ is derived, by which we find that if R 0 ≤ 1 $\mathcal {R}_{0}\le 1$ , the disease-free steady state is globally asymptotically stable, while R 0 > 1 $\mathcal {R}_{0}>1$ , the disease is uniform persistent. Furthermore, when R 0 > 1 $\mathcal {R}_{0}>1$ and additional conditions hold, the global asymptotic stability of special endemic steady state is obtained in homogeneous space. Finally, the theoretical results are validated by numerical simulations, some open questions are illustrated.

Global stability and control strategies of a SIQRS epidemic model with time delay

The existing research models of SIQRS are often defined by linear incidence, ignoring the impact of lagging public awareness during outbreaks of infectious diseases. Their research results often deviate from the actual situation. For decision‐makers, the delay of quarantine measures according to its formulation is liable to bring losses. Given this, we establish a new class of SIQRS models based on the nonlinear incidence and focus on the effect of time delay on the propagation characteristics of the system. Firstly, we calculate the basic reproduction number of the transmission dynamics model to obtain the expression of the transmission threshold. Secondly, by constructing suitable Lyapunov functions, we prove the stability of the disease‐free equilibrium point and endemic equilibrium point of the system. Thirdly, we add model immune parameters to study three different immunization strategies, namely, uniform immunization, targeted immunization, and acquaintance immunization. Studies show that compared with uniform immunization, targeted and acquaintance immunization have better control effects. When the immunization ratio increases, the number of infected people decreases, and immune factors can control the spread of the disease. Meanwhile, we calculate the optimal quarantine level value to solve the problem of financial costs in quarantine measures to control infectious diseases. Finally, we verify conclusions of the theoretical research by numerical simulation and further discover that when an infection is in the high‐density outbreak, the time delay will lead to a slower convergence rate of the disease. And with the increase of time delay, the control cost will also increase gradually.

An avian influenza model with nonlinear incidence and recovery rates in deterministic and stochastic environments

An avian influenza model with nonlinear incidence and recovery rates in deterministic and stochastic environments

Global stability of a cytokine-enhanced viral infection model with nonlinear incidence rate and time delays

To discuss the influence of cytokine-enhanced rate on viral infection, we propose a delay differential equation model with nonlinear incidence rate. By constructing suitable Lyapunov functionals and using LaSalle invariance principle, the global asymptotic stability of the equilibria of the proposed model is investigated.

Analysis of fractional-order dynamics of dengue infection with non-linear incidence functions

The infection of dengue is a devastating mosquito-borne infection around the globe that affects human health, social and economic sectors in low-income areas. Therefore, policymakers and health experts are trying to point out better policies to reduce these losses and provide better information for the development of vaccination and medication. Here, we formulated a compartmental model for the transmission phenomena of dengue fever with nonlinear forces of infection through fractional derivative. We established several results related to the solution of our dengue model by using the basic properties of fractional calculus. We determined the basic reproduction number of our fractional-order system, symbolized by [Formula: see text]. We established the local asymptotic stability of the infection-free equilibrium of our dengue system for [Formula: see text], and proved that the infection-free equilibrium is globally asymptotically stable without vaccination. The threshold dynamics [Formula: see text] is tested through partial rank correlation coefficient method to notice the importance of parameters in the transmission of dengue infection. In addition, we have shown the impact of memory on the basic reproduction number numerically with the variation of different parameters. We conclude that the biting rate, recruitment rate of mosquitoes and index of memory are the most sensitive factors, which can effectively lower the level of dengue fever. The dynamical behavior of the proposed fractional system is presented through a numerical scheme to explore the overall transmission process. We predict that the fractional-order model can explore more accurately and preciously the intricate dengue disease transmission model rather than the integer-order derivative.

Dynamical Behavior of SEIR-SVS Epidemic Models with Nonlinear Incidence and Vaccination.

For some infectious diseases such as mumps, HBV, there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time. In this paper, we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model. For the age-structured model, we consider the positivity, boundedness, and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals. Moreover, for the reduced delay differential equation model, we study the existence of the endemic equilibrium and prove the global stability of equilibria. Finally, some numerical simulations are provided to support our theoretical results and a brief discussion is given.

Stability of a stochastic discrete SIS epidemic model with general nonlinear incidence rate

The purpose of this paper is to study the stability of a stochastic discrete SIS epidemic model with a general nonlinear incidence rate at the equilibrium. Based on a continuous SIS model with stochastic white noise directly proportional to the deviation of the state from the equilibrium, the model is discretized by the Euler–Marryama method, and then the stochastic discrete SIS epidemic model is obtained. A sufficient condition for the stability in probability of the nonlinear difference equations with stochastic perturbations at the equilibriums is proposed. The sufficient conditions for the stochastic stability of the model at positive equilibrium and boundary equilibrium are obtained. Numerical simulations show the influence of noise intensity on the stability of stochastic discrete model.

Qualitative analysis of an epidemic model with nonlinear incidence rate in the time of COVID-19

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.

Modelling the transmission dynamics of Lassa fever with nonlinear incidence rate and vertical transmission

In this paper, a seven-dimensional nonlinear mathematical model featuring the vertical transmission in humans, saturated incidence functions, and human population with good and poor community hygiene is formulated to describe the dynamics of Lassa fever (LF) disease transmission between the interacting human and rodent populations. The model is analysed to investigate the dynamical behaviour of its solutions using some theories of dynamical system of ordinary differential equations. The basic reproduction number, R0, of the model is established. A suitable Lyapunov function is constructed to establish the global asymptotic behaviour of the model about the Lassa fever-free equilibrium. Sensitivity analysis is carried out to gain insightful information about how R0 is influenced by the variation in the model parameters. Numerical simulations are conducted to examine the effects of the most sensitive model parameters on the population dynamics of LF. The results obtained provide epidemiological insights into the impact of key model parameters on the transmission dynamics of LF and suggest some measures to fight or guide against the disease spread in a population.