Nonlinear Incidence Function Research Articles

Global stability for age-structured heroin model with general nonlinear incidence rate and time delay

In this article, we consider a mathematical model on the consumption of Heroin, it is an epidemiological model describing the interactions between heroin consumers and the rest of the population. The model is a system of differential equations with delay, integro-differential equations and partial differential equations. The first equation describes the evolution of healthy individuals, who do not consume Heroin. The second equation describes the evolution of consumer individuals, and in the last equation we find the evolution of individuals under treatment, so that they become healthy. In this work, we are interested in the qualitative study of the system considered. We examine the effects of a nonlinear incidence function with lag on the dynamics of an age-of-infection structured Heroin consumption model in the framework of mathematical epidemiology. We prove that incorporating the lag into the model does not change the asymptotic behavior of its equilibrium states. By defining the basic reproduction number R0 as a critical threshold parameter, we show that if R0 < 1, the disease-free equilibrium (DFE) is globally stable regardless of the lag τ. Conversely, if R0 < 1, the model exhibits an endemic equilibrium (EE) that remains locally and globally asymptotically stable for any lag τ. Our results highlight the robustness of the equilibrium stability in response to variations in the lag and provide important insights into the long-term dynamics of Heroin consumption, providing a clearer understanding of the conditions under which epidemic outcomes are stable or unstable.

Global analysis for a modified SEIR model with general non-linear incidence function

Global analysis for a modified SEIR model with general non-linear incidence function

Global stability for SIR models with nonlinear incidence and removal functions

In this work, we provide some sufficient conditions to study the global asymptotic stability of the endemic equilibrium for certain models in mathematical epidemiology with nonlinear incidence and removal functions. We also present numerical examples in order to illustrate our results.

Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages.

Mathematical Modeling of Two Interacting Populations’ Dynamics of Onchocerciasis Disease Spread with Nonlinear Incidence Functions

The transmission dynamics of onchocerciasis in two interacting populations are examined using a deterministic compartmental model with nonlinear incidence functions. The model undergoes qualitative analysis to examine how it behaves near disease-free equilibrium (DFE) and endemic equilibrium. Using the Lyapunov function, it is demonstrated that the DFE is globally stable when the threshold parameter R0≤1 is taken into account. When R0>1, it suffices to show globally how asymptotically stable the endemic equilibrium is and its existence. We conduct the bifurcation analysis by looking at the possibility of the model’s equilibria coexisting at R0<1 but near R0=1 using the Center Manifold Theory. We use the sensitivity analysis method to understand how some parameters influence the R0, hence the transmission and mitigation of the disease dynamics. Furthermore, we simulate the model developed numerically to understand the population dynamics. The outcome presented in this article offers valuable understanding of the transmission dynamics of onchocerciasis, specifically in the context of two populations that interact with each other, considering the presence of nonlinear incidence.

Study of an SIQR model with optimal control techniques: A mathematical approach

In this article, a SIQR deterministic model with a non-linear incidence function for an infectious disease is formulated and analyzed. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The basic reproduction number (R0) of the model is found. The sensitivity analysis of the basic reproduction number is discussed. The local and global stability of different equilibria of the model are discussed in terms of basic reproduction number. We use the center manifold theory to analyze the transcritical bifurcation exhibited by the system at R0=1 and found that the disease does not vanish even if R0<1 due to the occurrence of backward bifurcation. Furthermore, this model is extended to the optimal control model and is analyzed to determine the effect of screening and treatment as controls on the population dynamics through Pontryagin’s Maximum Principle. Moreover, we performed numerical analysis to illustrate our theoretical findings. We aim at determining the best intervention strategy to control the infectious diseases among screening, treatment and quarantine. We conclude that only screening is economically effective as compared to only treatment but the combined effort of screening and treatment is most effective and less expensive than single implementation of any control measure. Our study reveals that both the control interventions not only reduce the severity of disease burden but also minimize the economic burden to control the disease.

On the stability analysis of a fractional order epidemic model including the most general forms of nonlinear incidence and treatment function

In this paper, we propose to study a SEIR model of fractional order with an incidence and a treatment function. The incidence and treatment functions included in the model are general nonlinear functions that satisfy some meaningful biological hypotheses. Under these hypotheses, it is shown that the disease free equilibrium point of the proposed model is locally and globally asymptotically stable when the reproduction number $R_{0} $ is smaller than 1. When $ R_{0}&amp;gt;1 $, it is established that the endemic equilibrium of the studied system is uniformly asymptotically stable. Finally, some numerical simulations are provided to illustrate the theory.

Local Stability Analysis of Onchocerciasis Transmission Dynamics With Nonlinear Incidence Functions in Two Interacting Populations

A deterministic compartmental model for the transmission dynamics of onchocerciasis with nonlinear incidence functions in two interacting populations is studied. The model is qualitatively analyzed to investigate its local asymptotic behavior with respect to disease-free and endemic equilibria. It is shown, using Routh-Hurwitz criteria, that the disease-free equilibrium is locally asymptotically stable when the associated basic reproduction number is less than the unity. When the basic reproduction number is greater than the unity, we prove the existence of a locally asymptotically stable endemic equilibrium.

Mathematical model of a SCIR epidemic system with migration and nonlinear incidence function

Mathematical model of a SCIR epidemic system with migration and nonlinear incidence function

Analysis of an optimal control problem for a spatio‐temporal SIR model with nonlinear density dependent diffusion terms

SummaryThis paper focuses on the study of an optimal control problem for a new spatio‐temporal SIR epidemic model with nonlinear density dependent diffusion terms and a class of nonlinear incidence functions. We consider two types of control variables, vaccination for the susceptible and treatment for the infected. For fixed controls, by means of Schauder fixed point theorem, we prove that the proposed model admits a weak biologically feasible solution, the uniqueness of the latter is also investigated. Furthermore, using the state and adjoint problems, first order necessary optimal conditions are obtained. Finally, numerical simulations are carried out for particular diffusion terms incorporating the heard mentality of individuals, when it comes to the spatial movement, and for particular incidence functions, as well as by varying the parameters of the objective functional, to illustrate the possible optimal control strategies and their effect on the studied population.

A mathematical model of Zika virus transmission with saturated incidence and optimal control: A case study of 2016 zika outbreak in Puerto Rico

ABSTRACT Zika virus outbreak is creating a global public health hazard due to its increasing prevalence rate and neurological complication in affected individuals. In this work we have developed and studied an ODE compartmental Zika model with non-linear saturated incidence function and three control measures including prevention control from vector bite, treatment control of infected host and vector killing control. Our theoretical analysis shows that the disease free equilibrium is locally as well as globally asymptotically stable when the basic reproduction number is less than unity. The virus is uniformly persistent when the basic reproduction number is greater than unity. An endemic equilibrium point exists and locally stable when the basic reproduction number is greater than unity. The model is validated by fitting the model to real reported data of 2016 Zika outbreak in Puerto Rico and estimated the model parameters. We also have studied the basic reproduction number and effective reproduction number of the outbreak. Sensitivity analysis has been carried out to find most sensitive parameters. The optimal control and efficiency analysis has been done to find the policy of control variables. Lastly, the manuscript is ended by some conclusions.

Asymptotic analysis of SIR epidemic model with nonlocal diffusion and generalized nonlinear incidence functional

We aim in this research to determine the global stability of equilibrium states for an SIR epidemic model with nonlocal diffusion and nonlinear incidence function in a heterogeneous environment. For achieving this result, we consider that the model parameters are Lipschitz continuous functions. At the infection free‐equilibrium, the eigenvalue problem has a principal simple eigenvalue corresponding to strictly positive eigenfunction which is expressed as . By a Lyapunov function approach, we show the global stability of drug‐free equilibrium for . For , and using persistence theory for dynamical systems, we show that the epidemic will persist and the unique endemic equilibrium state is globally stable by constructing a proper Lyapunov function.

Modeling the transmission dynamics of a time-delayed epidemic model with saturated treatment rate

In this paper, an attempt has been made to explore a new delayed epidemiological model assuming that the disease is transmitted among the susceptible population and possessing nonlinear incidence function along with a saturated treatment rate. Due attention is paid to the positivity and boundedness of the solutions and the bifurcation of the dynamical system as well. Basic reproduction number is being calculated, and considering the latent period as a bifurcation parameter, it has been examined that a Hopf-bifurcation occurs near the endemic equilibrium point while the parameter attains critical values. We have also discussed the stability and direction of Hopf-bifurcation near the endemic equilibrium point, the global stability analysis and the optimal control theory. We conclude that the system reveals chaotic dynamics through a specific time-delay value. Numerical simulations are being performed in order to explain the accuracy and effectiveness of the acquired theoretical results.

Theoretical analysis of a measles model with nonlinear incidence functions

Measles is a highly contagious respiratory disease of global public health concern. A deterministic mathematical model for the transmission dynamics of measles in a population with Crowley–Martin incidence function to account for the inhibitory effect due to susceptible and infected individuals and vaccination is formulated and analyzed using standard dynamical systems methods. The basic reproduction number is computed. By constructing a suitable Lyapunov function, the disease-free equilibrium is shown to be globally asymptotically stable. Using the Center Manifold theory, the model exhibits a forward bifurcation, which implies that the endemic equilibrium is also globally asymptotically stable. To determine the optimal choice of intervention measures to mitigate the spread of the disease, an optimal control problem is formulated (by introducing a set of three time-dependent control variables representing the first and second vaccine doses, and the palliative treatment) and analyzed using Pontryagin’s Maximum Principle. To account for the scarcity of measles vaccines during a major outbreak or other causes such as the COVID-19 pandemic, a Holling type-II incidence function is introduced at the model simulation stage. The control strategies have a positive population level impact on the evolution of the disease dynamics. Graphical results reveal that when the mass-action incidence function is used, the number of individuals who received first and second vaccine dose is smaller compared to the numbers when the Crowley–Martin incidence-type function is used. Inhibitory effect of susceptibles tends to have the same effect on the population level as the Crowley–Martin incidence function, while the control profiles when inhibitory effect of the infectives is considered have similar effect as when the mass-action incidence is used, or when there is limitation in the availability of measles vaccines. Missing out the second measles vaccine dose has a negative impact on the initial disease prevalence.

Global stability of a HCV dynamics model with cellular proliferation and delay

In this work, we propose and investigate a delay cell population model of hepatitis C virus (HCV) infection with cellular proliferation, absorption effect and a nonlinear incidence function. First of all, after having shown the existence of the local solutions of our model, we show the existence of the global solutions and positivity. Moreover, we determine the uninfected equilibrium point and the basic reproduction rate R0, which is a threshold number in mathematical epidemiology. After showing the existence and uniqueness of the infected equilibrium point, we proceed to the study of the local and global stability of this equilibrium. We show that if R0 &lt; 1, the uninfected equilibrium point is globally asymptotically stable, which means that the disease will disappear and if R0 &gt; 1, we have a unique infected equilibrium that is globally asymptotically stable under some conditions. Finally, we perform some numerical simulations to illustrate the obtained theoretical results.

Global threshold analysis on a diffusive host–pathogen model with hyperinfectivity and nonlinear incidence functions

In this paper, we are concerned with the mathematical analysis of a host–pathogen model with diffusion, hyperinfectivity and nonlinear incidence. We define the basic reproduction number ℜ0 by the spectral radius of the next generation operator, and study the relation between ℜ0 and the principal eigenvalue of the problem linearized at the disease-free steady state (DFSS). Under some assumptions, we show the threshold property of ℜ0: if ℜ0<1, then the DFSS is globally asymptotically stable (GAS), whereas if ℜ0>1, then the system is uniformly persistent and a positive steady state (PSS) exists. Moreover, for the special case where all parameters are constants, we show that the PSS is GAS for ℜ0>1. Numerical simulation suggests that the spatial heterogeneity could enhance the intensity of epidemic, whereas the diffusion effect could reduce it.

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.

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.

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.

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.