Nonlinear Incidence Research Articles

Intelligent Adaptive Networks for Chaos Analysis in Stochastic SIS Differential Epidemic Model with the Impact of Unpredictable Environmental Factors

In this study, an intelligent computing framework is presented to explore the influence of unpredictable environmental factors on the spread of infections. A stochastic non-linear SIS epidemic model (SISEM) is examined that incorporates a non-linear incidence rate, and impact of random fluctuations on the disease transmission. The knacks of artificial neural networks Levenberg–Marquardt (ANNLMA) technique are exploited to illustrate the behavioral complexities of stochastic non-linear SISEM, which explores the dynamics of disease transmission in a stochastic environment and provides insights into the interactions between susceptible and infectious populations. The Milstein approach of order 1 generates a dataset for the proposed ANNLMA technique by varying integrated parameters, such as the rates of recovery, death, disease transmission, noise standard deviation and the entire population, for various SISEM scenarios. The referenced dataset is randomly divided into test, validation and train sets for the weight learning of the proposed ANNLMA. The viability of the ANNLMA is measured using intensive simulation-based investigations through analysis of mean squared errors, histograms, regression and autocorrelation studies. The investigations highlighted that the ANNLMA results closely match the reference results, illustrating convergence within the 10[Formula: see text] to 10[Formula: see text] range, indicating the validity of the proposed methodology.

A Novel Nonlinear SAZIQHR Epidemic Transmission Model: Mathematical Modeling, Simulation, and Optimal Control

Abstract This study presents a seven-compartmental nonlinear epidemic model to explore the dynamics and control of the epidemic. The model extends the traditional susceptible-infected-recovered (SIR) framework by including additional compartments for aware, infodemic, hospitalized, and quarantined populations. It also incorporates three Holling type II saturated nonlinear incidence rates to better represent the disease transmission process. The "infodemic population" refers to those spreading misinformation about the disease, its spread, control, and treatment. The study examines disease dynamics in the absence of a vaccine and identifies two key equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). It is found that the DFE is locally asymptotically stable when the basic reproduction number (R_0) is below one and becomes unstable when R_0 exceeds one. The model's behavior at R_0=1 is analyzed using center manifold theory, revealing a forward bifurcation. Additionally, the existence of a positive endemic equilibrium is confirmed for R_0>1, with no backward bifurcation observed when R_0≤1. The EE is also shown to be locally asymptotically stable when R_0 is greater than one. Furthermore, the study formulates and mathematically analyzes an optimal control problem. Finally, numerical simulations are presented to support the analytical results.

Dynamics analysis of a delayed stochastic SIRS epidemic model with a nonlinear incidence rate

. This article proposes a stochastic delayed SIRS epidemic model with nonlinear incidence to describe disease transmission in a stochastic environment. First, it proves that the model has a unique global positive solution and derives sufficient conditions for the extinction and persistence of the epidemic in the population under appropriate conditions. Then, by constructing suitable Lyapunov functions, the asymptotic behavior of the model around the disease-free equilibrium and endemic equilibrium points of the deterministic model is analyzed, and it is concluded that under certain conditions, the solution of the stochastic system randomly oscillates around these two equilibrium points. Finally, several numerical examples are provided to validate the theoretical results, illustrating the significant impact of stochastic perturbations and time delays on disease control.

Chaos and Sub-Harmonic Bifurcations in a Simple SIR Model with Periodically and Parametrically Nonlinear Incidence Rate

Chaos and sub-harmonic bifurcations of a simple SIR model with periodically and parametrically nonlinear incidence rate are investigated analytically with the Melnikov method and sub-harmonic Melnikov method, respectively. The critical curves of chaos for the model with the periodically and parametrically nonlinear incidence rate are studied analytically for the first time. Some new interesting dynamic phenomena are presented and proved rigorously, including the nonchaotic band, controllable frequencies and even controllable frequency interval, which may exist for some system parameters in this model. The sub-harmonic periodic orbit family is derived. It is presented that sub-harmonic bifurcations with even order and integer order may exist in the system. The sub-harmonic bifurcation conditions are obtained rigorously for two different starting points. Numerical simulations including phase portraits, Poincaré sections, basins of attraction and Lyapunov exponents are given to verify the chaos threshold obtained by the analytical results. It is also shown that the system can undergo chaos through even-order sub-harmonic bifurcations.

Stochastic delayed analysis of coronavirus model through efficient computational method

Stochastic delayed modeling has a significant non-pharmaceutical intervention to control transmission dynamics of infectious diseases and its results are close to the reality of nature. The covid-19 has been controlled globally but there is still a threat and appears in different variants like omicron and SARS-CoV-2 etc. globally. This article, considered pattern a mathematical model based on Susceptible, Infected, and recovered populations with highly nonlinear incidence rates. we studied the dynamics of the coronavirus model; a newly proposed version is a stochastic delayed model that is based on nonlinear stochastic delayed differential equations (SDDEs). Transition probabilities and parametric perturbation methods were used for the construction of the stochastic delayed model. The fundamental properties like positivity, boundedness, existence and uniqueness, and stability results of equilibria of the model with certain conditions of reproduction number are studied regularly. Also, the extinction and persistence of disease are studied with the help of well-known theorems. The numerical methods used to find a visualization of results due to the complexity of stochastic delayed differential equations. Furthermore, for computational analysis, we implemented existing methods in the literature and compared their results with the proposed method like nonstandard finite difference for stochastic delayed model. The proposed method restores all dynamical properties of the model with a free choice of time steps.

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.

Dynamics of a Stochastic Vector-Borne Model with Plant Virus Disease Resistance and Nonlinear Incidence

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 R0s<1, then the disease will be extinct. By using the reproduction number R0, 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.

Exploring the Landscape of Fractional-Order Models in Epidemiology: A Comparative Simulation Study

Mathematical models play a crucial role in evaluating real-life processes qualitatively and quantitatively. They have been extensively employed to study the spread of diseases such as hepatitis B, COVID-19, influenza, and other epidemics. Many researchers have discussed various types of epidemiological models, including deterministic, stochastic, and fractional order models, for this purpose. This article presents a comprehensive review and comparative study of the transmission dynamics of fractional order in epidemiological modeling. A significant portion of the paper is dedicated to the graphical simulation of these models, providing a visual representation of their behavior and characteristics. The article further embarks on a comparative analysis of fractional-order models with their integer-order counterparts. This comparison sheds light on the nuances and subtleties that differentiate these models, thereby offering valuable insights into their respective strengths and limitations. The paper also explores time delay models, non-linear incidence rate models, and stochastic models, explaining their use and significance in epidemiology. It includes studies and models that focus on the transmission dynamics of diseases using fractional order models, as well as comparisons with integer-order models. The findings from this study contribute to the broader understanding of epidemiological modeling, paving the way for more accurate and effective strategies in disease control and prevention.

Upper bounds on overshoot in SIR models with nonlinear incidence

We expand the calculation of the upper bound on epidemic overshoot in SIR models to account for nonlinear incidence. We lay out the general procedure and restrictions to perform the calculation analytically for nonlinear functions in the number of susceptibles. We demonstrate the procedure by working through several examples and also numerically study what happens to the upper bound on overshoot when nonlinear incidence manifests in the form of epidemic dynamics over a contact network. We find that both steeper incidence terms and larger contact heterogeneity can increase the range of communicable diseases at which the overshoot remains a relatively large public health hazard.

Global dynamical behavior of a delayed cytokine-enhanced viral infection model with nonlinear incidence

In this paper, a delayed cytokine-enhanced viral infection model incorporating nonlinear incidence and immune response is proposed and studied. The global stabilities of the equilibria of the model are characterized by constructing suitable Lyapunov functionals. The existence and properties of local Hopf bifurcation are discussed by regarding the immunity delay as the bifurcation parameter. Moreover, the global existence of Hopf bifurcation has been proved by regarding the immune delay as the bifurcation parameter. Numerical simulations are carried out to validate the obtained results. The results show that ignoring the cytokine-enhanced effect makes the infection risk underevaluated. The simulations show that increasing the immune delay destabilizes the model and generates a Hopf bifurcation and stability switches occurs. Moreover, it shows that immune delay may dominate the intracellular delays in such a viral infection model which means that the immune system of the host itself is complicated during virus infection.

Analytical analysis and bifurcation of pine wilt dynamical transmission with host vector and nonlinear incidence using sustainable fractional approach

To study the dynamical system, it is necessary to formulate a mathematical model to comprehend the dynamics of the diseases that are prevalent around the world by using fractional calculus. A mathematical model is developed with the hypothesis created by adding control and asymptomatic variables to observe the rate of change of pine wilt and the ABC operator is used to turn the model into a fractional ordered model for continuous monitoring. The Boundedness and uniqueness of the developed model are investigated for bounded findings by using Banach space, which are the key properties of such an epidemic model. A newly developed system is examined both qualitatively and quantitatively to determine its stable position, and the verification of flip bifurcation has been made for developed systems. Derived reproductive numbers using the next-generation technique as well as the sensitivity of each involved parameter are verified. The Atangana–Toufik scheme is employed to find the solution for the developed system using different fractional values, which are advanced tools for reliable bounded solutions. Simulations have been made to see the real behavior and effects of pine wilt disease with control and asymptomatic battels in the community. Also, identify the real situation of the spread as well as the control of pine wilt after employing control and asymptomatic battels due to treatment. Such a type of investigation will be useful in investigating the spread of disease as well as helpful in developing control strategies based on our justified outcomes.

Global stability and optimal control of an age-structured SVEIR epidemic model with waning immunity and relapses.

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established.

A simple approach for studying stability properties of an SEIRS epidemic model

Abstract In this work, we study stability properties of a well-known integer-order SEIRS model with nonlinear incidence and vertical transmission. Firstly, we introduce a simple approach to the analysis of global asymptotic stability (GAS) of the integer-order model. This approach is based on general quadratic Lyapunov functions and characteristic of quadratic forms associated with real matrices. The result is that the GAS of disease-free and disease-endemic equilibrium points is completely established. This provides an important improvement for results constructed in two previous works. Secondly, we generalize the integer-order SEIRS model by considering it in the context of the Caputo fractional-order derivative. After that, the present approach is utilized to investigate the GAS of the proposed fractional-order model. As an important consequence, not only the GAS but also the uniform stability of the fractional-order model are determined fully. Therefore, the applicability of the approach is shown. Finally, a series of numerical experiments is conducted to illustrate and support the theoretical findings.

A mathematical model for a disease outbreak considering waning-immunity class with nonlinear incidence and recovery rates

In the spread of infectious diseases, intervention levels play a crucial role in shaping interactions between healthy and infected individuals, leading to a nonlinear transmission process. Additionally, the availability of medical resources limits the recovery rate of infected patients, adding further nonlinear dynamics to the healing process. Our research introduces novelty by combining nonlinear incidence and recovery rates alongside waning immunity in an epidemic model. We present a modified SIRW-type model, examining the epidemic problem with these factors. Through analysis, we explore conditions for non-endemic and co-existing cases based on the basic reproduction ratio. The local stability of equilibria is verified using the Routh-Hurwitz criteria, while global stability is assessed using Lyapunov functions for each equilibrium. Furthermore, we investigate bifurcations around both non-endemic and co-existing equilibria. Numerically, we give some simulations to support our analytical findings.

Spatiotemporal Patterns in a Space–Time Discrete SIRS Epidemic Model with Self- and Cross-Diffusion

In this work, a two-dimensional Coupled Map Lattice (CML) method is applied to a continuous Reaction–Diffusion (RD) SIRS epidemic model. The model involves a complex, nonlinear incidence rate and the spatially heterogeneous self- and cross-diffusion coefficients. The CML method applied here results in a space–time discrete counterpart of the model. The basic reproduction number is derived, and the local stability of the fixed points of the discrete model is established. The two typical bifurcations of codimension 1, such as the flip and the Neimark–Sacker (NS) bifurcations, are investigated by utilizing the center manifold theory and the normal form theorem. Furthermore, the codimension-2 fold–flip bifurcation about the endemic fixed point is discussed in detail. Turing instability criterion of the cross-diffusion epidemic model is established by utilizing the CML method under periodic boundary conditions. Finally, the numerical simulations clarify both the effectiveness of all theoretical findings and the effects of time step size and cross-diffusion on the spatiotemporal patterns. The main contribution of this work is the discovery of some interesting Turing and non-Turing patterns which may be induced by flip–Turing instability, NS–Turing instability or fold–flip–Turing instability. Concretely speaking, spiral patterns are observed near the flip bifurcation threshold value and ultimately evolve into an irregular defect pattern. Stripe pattern, labyrinth pattern and circular pattern are observed near the NS bifurcation threshold value. The disorderly interwoven ribbon pattern, mosaic pattern and some irregular oscillatory patterns are observed near the fold–flip bifurcation point. Our results greatly help in predicting the long-term behavior of the space–time discrete SIRS epidemic model under study.

Dynamics and probability density function of a stochastic COVID-19 epidemic model with nonlinear incidence

In this paper, a stochastic SEIQ model with nonlinear incidence rate is proposed. The existence and uniqueness of global positive solution of the stochastic model are proved. Then, by constructing some Lyapunov functions, we derive a sufficient condition for the ergodic stationary distribution when [Formula: see text] is greater than one. By solving a four-dimensional Fokker–Planck equation, we get the exact expression of log-normal probability density function of stationary distribution for the stochastic model. Sufficient condition for the extinction of the exposed and infected population is also provided. Finally, numerical simulations are presented to verify the above theoretical results.

Based on epidemiological parameter data, probe into a stochastically perturbed dominant variant of the COVID-19 pandemic model

During the COVID-19 pandemic, the SARS-CoV-2 gene mutation has been rapidly emerging and spreading all over the world. Experts worldwide regularly monitor genetic mutations and variants through genome-sequence-based surveillance, laboratory testing, outbreak investigation, and epidemiological probing. Clinical pathologists and medical laboratory scientists prefer developing or endorsing COVID-19 vaccines with a broader immune response involving various antibodies and cells to protect against mutations or new variants. Randomness plays an enormous role in pathology and epidemiology. Hence, based on epidemiological parameter data, we construct and probe a stochastically perturbed dominant variant of the coronavirus epidemic model with three nonlinear saturated incidence rates. We reveal the existence of a unique global positive solution to the constructed stochastic COVID-19 model. The Lyapunov function method is used to determine the presence of a stationary distribution of positive solutions. We derive sufficient conditions for the coronavirus to be eradicated. Eventually, numerical simulations validate the effectiveness of our theoretical outcomes.

Dynamic analysis and application of a stochastic rumor spreading model with education and punishment age under generalized nonlinear incidence on homogeneous networks

Dynamic analysis and application of a stochastic rumor spreading model with education and punishment age under generalized nonlinear incidence on homogeneous networks

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

Dynamic analysis of a SIS epidemic model with nonlinear incidence and ratio dependent pulse control

Dynamic analysis of a SIS epidemic model with nonlinear incidence and ratio dependent pulse control