Nonlinear Incidence Research Articles

Spatiotemporal Dynamics of a Delayed Viral Infection Model with Two Modes of Transmission and Immune Impairment

In this paper, the dynamical behaviors for a diffusive virus infection model with two general nonlinear incidence functions, CTL immune impairment and discrete delays, are investigated. Cell‐to‐cell transmission plays a more important role in virus spreading in addition to virus‐to‐cell infection. Nonnegativity and boundcedness properties of the solutions are proven to show the well‐posedness of the model. Five reproduction numbers are calculated and verified to be thresholds determining the global properties of the virus model. A significant impact of cell‐to‐cell transmission is that they increase the basic reproduction number. If one neglects either the cell‐to‐cell transmission or virus‐to‐cell infection, the basic reproduction number of the model is underevaluated. Numerical simulations are carried out to illustrate the dynamical behavior of the model.

Dynamics of stochastic chemostat models with mixed nonlinear incidence

In this paper, we proposes a stochastic chemostat model with mixed nonlinear incidence. Firstly, the existence and uniqueness of the global positive solutions are proved. Secondly, we demonstrate that the chemostat model is persistence in mean and the solution of this stochastic chemostat model is bounded for any initial condition by constructing the Lyapunov function. Then we obtain the sufficient condition for the existence of an ergodic stationary distribution in this system. Finally, the numerical simulation results of the model are given. The simulation results show that a particular random perturbation can change the fate of microorganisms.

Mathematical analysis of a fractional-order epidemic model with nonlinear incidence function

<abstract><p>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.</p></abstract>

On the Dynamics of a Fractional-Order Ebola Epidemic Model with Nonlinear Incidence Rates

We propose a new fractional-order model to investigate the transmission and spread of Ebola virus disease. The proposed model incorporates relevant biological factors that characterize Ebola transmission during an outbreak. In particular, we have assumed that susceptible individuals are capable of contracting the infection from a deceased Ebola patient due to traditional beliefs and customs practiced in many African countries where frequent outbreaks of the disease are recorded. We conducted both epidemic and endemic analysis, with a focus on the threshold dynamics characterized by the basic reproduction number. Model parameters were estimated based on the 2014-2015 Ebola outbreak in Sierra Leone. In addition, numerical simulation results are presented to demonstrate the analytical findings.

Estabilidad de un modelo matemático del VIH/SIDA considerando el tratamiento y el impacto de la cobertura mediática

A mathematical model is proposed to study the dynamics of the spread of HIV/AIDS with treatment considering media coverage. The subpopulations involved in the study are; susceptible individuals, slow-latency infected individuals, fast-latency infected individuals, symptomatic individuals undergoing treatment, and finally individuals with AIDS. We consider a system of ordinary differential equations that let us to understand the dynamics of the spread of HIV/AIDS, taking into account two non-linear incidence rates that show the influence of media coverage as a disease control. We establish conditions for the stability of this model. The result enables us to evaluate the media coverage impact on the dynamics of the disease. Finally, we show a numerical simulation analysis of the model and a sensitivity analysis corresponding to the parameters.

Analysis and simulation of a two-strain disease model with nonlinear incidence

We analysed and simulated a two-strain Susceptible-Infected-Recovered (SIR) disease model with varying population size and nonlinear incidence. We found that in addition to the infection-free and two single-strain endemic equilibria, the non-linear incidence term induces a fourth equilibrium point where the two strains co-exist. We determined the conditions for the existence and stability of each of the four equilibrium points, and showed that these depend on both the traditional basic reproduction numbers (R0) of each strain and a set of generalized threshold parameters (ψand γ). In particular, we used appropriate Lyapunov functions to show that these generalized quantities define strict boundaries that separate regions of parameter space in which each of the equilibrium points are unstable or globally asymptotically stable. Further, we used sensitivity analysis and the Partial Rank Correlation Coefficient (PRCC) method to identify the most influential model parameters on transmission and disease prevalence, finding that the contact rate of both strains had the largest influence on both. Finally, numerical simulations were carried out to support the analytic results.

Coronavirus epidemic model with isolation and nonlinear incidence rate

Coronavirus epidemic model with isolation and nonlinear incidence rate

Dynamics of a reaction-diffusion SIRS model with general incidence rate in a heterogeneous environment.

In this paper, we study a diffusive SIRS-type epidemic model with transfer from the infectious to the susceptible class. Our model includes a general nonlinear incidence rate and spatially heterogeneous diffusion coefficients. We compute the basic reproduction number mathcal {R}_0 of our model and establish the global stability of the disease-free steady state when mathcal {R}_0<1. Furthermore, we study the uniform persistence when mathcal {R}_0>1 and perform a bifurcation analysis for a special case of our model. Some numerical simulations are presented to illustrate the dynamics of the solutions as the model parameters are varied.

Global behavior of Heroin epidemic model with time distributed delay and nonlinear incidence function

In this research, we investigate the global properties of the heroin epidemic model with time distributed delay and nonlinear incidence function. We show that the system has threshold dynamics in terms of R0, and we prove, a Lyapunov function, that for R0<1 the drug-free equilibrium is globally asymptotically stable. For R0>1, we give the persistence result of the heroin consumption. We also show the global stability of the endemic equilibrium for R0>1 using a suitable Lyapunov function. The mathematical results are illustrated by numerically simulations.

Analysis of a reaction–diffusion HCV model with general cell-to-cell incidence function incorporating B cell activation and cure rate

A spatio-temporal model for HCV dynamics incorporating the spatial mobility of viruses and immune B cells, with general nonlinear incidence functions for both modes of infection transmission, namely, virus-to-cell as well as cell-to-cell transmission, is proposed and analyzed with the homogeneous Neumann boundary conditions. In addition, the model includes the role of B cell or antibody response and considers the non-cytolytic cure of productively infected hepatocytes. The existential conditions and global stability for the system equilibrium points are examined. The analytical results shows that the three types of equilibria for the model with general incidence functions are globally asymptotically stable under certain conditions along with criteria on the basic reproductive ratio. The numerical illustration of the analytical findings are presented for the case of bilinear as well as Holling type-II incidence function in a one-dimensional spatial domain. For both the incidence functions, the model populations converge to the corresponding levels of the infected or infection-free equilibrium, contingent upon the case of the basic reproductive ratio being greater than unity or not, respectively. Furthermore, it is observed that the diffusive coefficients do not influence the asymptotic behavior of the system. The infection will be eliminated when basic reproductive ratio is less than unity.

Wave propagation in a diffusive SEIR epidemic model with nonlocal transmission and a general nonlinear incidence rate

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}=partial _{I}F(S_{0},0)/gamma 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.

Ergodic property, extinction, and density function of an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbations

In this paper, we investigate an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbation. First, we obtain a stochastic threshold related to the basic reproduction number . A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if . Next, by solving the corresponding Fokker‐Planck equation, the exact expression of probability density function of the stochastic model is obtained. Moreover, we establish the sufficient condition for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical results.

Stability and optimal control strategy analysis for a class of SEIQR model with time delay on scale-free networks

Numerous experiences in fighting epidemic diseases have revealed that isolation before treatment is an effective way to prevent the further spread of the epidemic, which scholars in current researches mostly ignore. Also, medical research shows that most infectious diseases have an incubation period, and the length of the incubation period will affect the final therapeutic effect. Therefore, in this paper, to deeply analyze the epidemic transmission with latency and quarantine states, we construct a class of health state-latent state - infected state-quarantined state-recovered state (SEIQR), epidemic model, with a power-law distribution of nodes based on considering the non-linear incidence formed by the psychological suppressor. Furthermore, the system’s basic reproduction number and the equilibrium points’ stability are discussed. The results show that depends on birth rate, death rate, recovery rate, vaccination rate, isolation rate, disease transmission rate, and network topology. Interestingly, the latency does not influence And if system’s disease-free equilibrium is global asymptotically stable, so is endemic equilibrium if Also, we demonstrate that both the latency and psychological inhibitory factor can influence the evolutionary trend of infected nodes in the system. Similarly, this paper verifies the quarantine parameter’s effectiveness, and considers that the parameter variation within (0,0.3) is the most effective. It is worth noting that this paper focuses not only on the dynamics of the SEIQR system, but also on the control of infectious diseases’ spread. Because of this, We use Pontryagin’s maximum principle to solve optimal control solutions of multiple control system under the constraints of minimizing the objective function. Furthermore, the simulation results show that applying multiple controls can effectively inhibit the epidemic’s spread, especially when

Global Stability for Novel Complicated SIR Epidemic Models with the Nonlinear Recovery Rate and Transfer from Being Infectious to Being Susceptible to Analyze the Transmission of COVID-19

Epidemiological models play pivotal roles in predicting, anticipating, understanding, and controlling present and future epidemics. The dynamics of infectious diseases is complex, and therefore, researchers need to consider more complicated mathematical models. In this paper, we first describe the dynamics of a complex SIR epidemic model with nonstandard nonlinear incidence and recovery rates. In this model, we consider the rate at which individuals lose immunity. Rigorous mathematical results have been established from the point of view of stability and bifurcation. The basic reproduction number ( R 0 ) is determined. We then apply LaSalle’s invariance principle and Lyapunov’s direct method to prove that the disease-free equilibrium is globally asymptotically stable when R 0 &lt; 1 . The model has a unique endemic equilibrium when R 0 &gt; 1 . A nonlinear Lyapunov function is used together with LaSalle’s invariance principle to show that the endemic equilibrium is globally asymptotically stable under some conditions. Further, for the case when R 0 = 1 , we analyze the model and show a backward bifurcation under certain conditions. In the second part of this paper, we analyze a modified SIR model with a vaccination term, which must be a function of time. We show that the modified model agrees well with COVID-19 data in Saudi Arabia. We then investigate different future scenarios. Simulation results suggest that a two-pronged strategy is crucial to control the COVID-19 pandemic in Saudi Arabia.

Stability Analysis of SEIRS Epidemic Model with Nonlinear Incidence Rate Function

This paper addresses the global stability analysis of the SEIRS epidemic model with a nonlinear incidence rate function according to the Lyapunov functions and Volterra-Lyapunov matrices. By creating special conditions and using the properties of Volterra-Lyapunov matrices, it is possible to recognize the stability of the endemic equilibrium (E1) for the SEIRS model. Numerical results are used to verify the presented analysis.

The Dynamics of a Stochastic SIR Epidemic Model with Nonlinear Incidence and Vertical Transmission

In this study, we build a stochastic SIR epidemic model with vertical infection and nonlinear incidence. The influence of the fluctuation of disease transmission parameters and state variables on the dynamic behaviors of the system is the focus of our study. Through the theoretical analysis, we obtain that there exists a unique global positive solution for any positive initial value. A threshold R 0 s is given. When R 0 s &lt; 1 , the diseases can be extincted with probability one. When R 0 s &gt; 1 , we construct a stochastic Lyapunov function to prove that the system exists an ergodic stationary distribution, which means that the disease will persist. Then, we obtain the conditions that the solution of the stochastic model fluctuates widely near the equilibria of the corresponding deterministic model. Finally, the correctness of the results is verified by numerical simulation. It is further found that the fluctuation of disease transmission parameters and infected individuals with the environment can reduce the threshold of disease outbreak, while the fluctuation of susceptible and recovered individuals has a little effect on the dynamic behavior of the system. Therefore, we can make the disease extinct by adjusting the appropriate random disturbance.

A delayed reaction–diffusion viral infection model with nonlinear incidences and cell-to-cell transmission

In this paper, we propose a reaction–diffusion viral infection model with nonlinear incidences, cell-to-cell transmission, and a time delay. We impose the homogeneous Neumann boundary condition. For the case where the domain is bounded, we first study the well-posedness. Then we analyze the local stability of homogeneous steady states. We establish a threshold dynamics which is completely characterized by the basic reproduction number. For the case where the domain is the whole Euclidean space, we consider the existence of traveling wave solutions by using the cross-iteration method and Schauder’s fixed point theorem. Finally, we study how the speed of spread in space affects the spread of cells and viruses. We obtain the existence of the wave speed, which is dependent on the diffusion coefficient.

Mathematical modeling of COVID-19 in India and Nepal with optimal control and sensitivity analysis.

The pandemic started in the late 2019 and is still waving in claiming millions of lives with virus being mutated to deadlier form. This pandemic has caught attention toward interventions like improved detection of the infected, better quarantine facilities and adequate medical facilities in terms of hospital beds and other medical aid. In this study, we developed a 7-compartment epidemiological model, with inclusion of identified and unidentified infected population along with media factor associated with the aware identified infected population. This is included by using Holling function in the nonlinear incidence, that is responsible for reduction in infection rate via identified infected. The model is fitted to the observed active COVID-19 cases data, collected for a period of 11 months between July 2020 to May 2021 of Nepal and India, and the infection rate as well as the basic reproduction number is obtained for the first wave and second wave of the pandemic in both countries. A comparative analysis on the effect of different parameters on the disease prevalence for both the countries is presented in this work. Sensitivity analysis, time series behavior and optimal control analysis with control parameters equating with reduced infection rate, enhanced detection rate, improved quarantine and hospitalization rate are presented in detail. By means of PRCC, sensitivity analysis is performed and the key parameters influencing the disease prevalence are identified. A detailed study on impact of several parameters in the COVID-19 prevalence, thereby suggesting the interventions to be implemented is discussed in the work. Predictions till June 30, 2021, are obtained using the second wave data for both the countries, and a declining trend is observed for both the countries for the next 30 to 40 days. The estimated values of the infection rates and the hospitalization rates obtained are higher for India compared to Nepal. An optimal control analysis for both the countries is described in detail providing the difference in infectives and recoveries with and without any controls or interventions. The study suggests that improved treatment facilities, testing drives and other non-pharmaceutical interventions would bring down the infected cases to a major extent.

Global Dynamics of an SIRSI Epidemic Model with Discrete Delay and General Incidence Rate

In this paper, we propose the global dynamics of an SIRSI epidemic model with discrete latent period and general nonlinear incidence function. By analyzing the corresponding characteristic equations, the local stability of the endemic equilibrium is established. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of the disease-free equilibrium and the endemic equilibrium are established for the SIRSI epidemic model with discrete latent period.

Bifurcation of Periodic Solutions of a Delayed SEIR Epidemic Model with Nonlinear Incidence Rate

In this paper, we propose the SEIR epidemic model with delay and nonlinear incidence rate. The resulting model has two possible equilibria: if $R_{0} \leq 1,$ then the SEIR epidemic model has a disease-free equilibrium and if $R_{0} > 1,$ then the SEIR epidemic model admits a unique endemic equilibrium. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of a disease-free equilibrium is established. Our main contribution affirms the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. Finally, some numerical simulations are presented to illustrate our theoretical results.