Non-monotonic Incidence Research Articles

Exploring epidemics using non-monotonic incidence function: Effects of sociological factors and hospital bed availability

This study investigates the dynamics of an epidemic employing an (Susceptible-Infected-Hospitalized-Recovered-Susceptible (SIHRS) epidemiological model highlighting the crucial importance of hospital bed availability and a non-monotone incidence function, which incorporates the influences of stringent governmental measures, social behavior dynamics and public responses in both autonomous and non-autonomous scenarios. The analysis investigates the conditions for existence of infection-free and endemic steady states based on the basic reproduction number as it surpasses unity. Sensitivity analysis has been conducted to evaluate the impact of different system parameters on disease transmission. This work also investigates alterations in stability of the system caused by transcritical, Hopf and saddle-node bifurcations. Additionally, two-parameter bifurcation identifies the regions where the stability of both the equilibrium points has been examined. Numerical simulations are shown to illustrate all the theoretically obtained results. Also, the model examines the dynamical behavior of the epidemic when the quantity of available hospital beds varies periodically. This aspect of the study highlights the significant impact of hospital bed availability. Such factors are crucial in preventing disease spread during an epidemic. The results provide valuable insights into how dynamic patterns of disease transmission are influenced by healthcare infrastructure and public health interventions. This comprehensive exploration underscores the importance of integrated approaches combining medical resources and societal measures in managing and mitigating the effects of epidemics.

Stability and Bifurcation Analysis of a Symmetric Fractional-Order Epidemic Mathematical Model with Time Delay and Non-Monotonic Incidence Rates for Two Viral Strains

This study investigates a symmetric fractional-order epidemic model with time delays and non-monotonic incidence rates, considering two viral strains. By confirming the existence, uniqueness, and boundedness of the system’s solutions, the research ensures the model’s well-posedness, guaranteeing its mathematical soundness and practical relevance. The study calculates and evaluates the equilibrium points and the basic reproduction numbers R01 and R02 to understand the dynamic behavior of the model under different parameter settings. Through the application of the Lyapunov method, the research examines the asymptotic global stability of the system, determining whether it will converge to a particular equilibrium state over time. Furthermore, Hopf bifurcation theory is employed to investigate potential periodic solutions and bifurcation scenarios, highlighting how the system might shift from stability to periodic oscillations under certain conditions. By utilizing the Adams-Bashforth-Moulton numerical simulation method, the theoretical results are validated, reinforcing the conclusions and demonstrating the model’s applicability in real-world contexts. It emphasizes the importance of fractional-order models in addressing epidemiological issues related to time delays (τ), individual heterogeneity (m, k), and memory effects (θ), offering greater accuracy compared with traditional integer-order models. In summary, this research provides a theoretical foundation and practical insights, enhancing the understanding and management of epidemic dynamics through fractional-order epidemic models.

Global Dynamics and Optimal Control of a Fractional-Order SIV Epidemic Model with Nonmonotonic Occurrence Rate

This paper performs a detailed analysis and explores optimal control strategies for a fractional-order SIV epidemic model, incorporating a nonmonotonic incidence rate. In this paper, the population of vaccinated individuals is included in the disease dynamics model. After proving the non-negative boundedness of the fractional-order SIV model, we focus on analyzing the equilibrium point characteristics of the model, delving into its existence, uniqueness, and stability analysis. In addition, our research includes formulating optimal control strategies specifically aimed at minimizing the number of infections while keeping costs as low as possible. To validate the theoretical findings and uncover the practical efficacy and prospects of control measures in mitigating epidemic spread, numerical simulations are performed.

Effect of ratio threshold control on epidemic dynamics: analysis of an SIR model with non-monotonic incidence

Effect of ratio threshold control on epidemic dynamics: analysis of an SIR model with non-monotonic incidence

Complex dynamics of an SIRS epidemic model with non-monotone incidence and saturated cure rate

Complex dynamics of an SIRS epidemic model with non-monotone incidence and saturated cure rate

Stochastic analysis of two‐strain epidemic model with non‐monotone incidence rates: Application to COVID‐19 pandemic

This work is devoted to the mathematical analysis of a COVID‐19 two‐strain epidemic model. The COVID‐19 mathematical model described the infection forces of each strain by a nonmonotonic incidence function. First, we establish the well‐posedness of the COVID‐19 stochastic model in terms of existence and uniqueness of the global positive solution. After that, we investigate the results of the stochastic extinction and persistence in mean of the COVID‐19 disease. The findings show that both strains of COVID‐19 pandemic extinct, when the basic reproduction number is less than unity. If the latter is not achieved, then the infection related to the strain with higher stochastic basic reproduction number will persist. Additionally, both strains can persist at the same time, if their related stochastic basic reproduction numbers are both greater than one. Finally, various numerical simulations are carried out in order to validate the theoretical findings concerning the extinction and persistence in mean of the disease. As an application of our work, we have chosen to compare our deterministic and stochastic results with COVID‐19 clinical data.

Exploring the complex dynamics of a diffusive epidemic model: Stability and bifurcation analysis.

The recent pandemic has highlighted the need to understand how we resist infections and their causes, which may differ from the ways we often think about treating epidemic diseases. The current study presents an improved version of the susceptible-infected-recovered (SIR) epidemic model, to better comprehend the community's overall dynamics of diseases, involving numerous infectious agents. The model deals with a non-monotone incidence rate that exhibits psychological or inhibitory influence and a saturation treatment rate. It has been identified that depending on the measure of medical resources and the effectiveness of their supply, the model exposes both forward and backward bifurcations where two endemic equilibria coexist with infection-free equilibrium. The model also experiences local and global bifurcations of codimension two, including saddle-node, Hopf, and Bogdanov-Takens bifurcations. Additionally, the stability of equilibrium points is investigated. For a spatially extended SIR model system, we have shown that cross-diffusion allows S and I populations to coexist in a habitat. Also, the Turing instability requirements and Turing bifurcation regime are derived. The relationship between distinct role-playing model parameters and various pattern formations like spot and stripe patterns is validated by carrying out in-depth numerical simulations. The findings in the vicinity of the endemic equilibrium solution demonstrate the significance of positive and negative valued cross-diffusion coefficients in regulating the genesis of spatial patterns in susceptible as well as diseased individuals. The discussion of the findings of epidemiological ramifications concludes the manuscript.

Stochastic two-strain epidemic model with bilinear and non-monotonic incidence rates

Stochastic two-strain epidemic model with bilinear and non-monotonic incidence rates

Global Dynamics and Optimal Control of a Two-Strain Epidemic Model with Non-monotone Incidence and Saturated Treatment

Global Dynamics and Optimal Control of a Two-Strain Epidemic Model with Non-monotone Incidence and Saturated Treatment

Three dimensional epidemic model with non-monotonic incidence and saturated treatment: A case study of SARS infection of Hong Kong 2003 scenario

In this paper, we have proposed a mathematical compartmental model with non-monotonic incidence and saturated treatment and we have validated the model with SARS infection in Hong Kong, 2003. We have analysed the stability of disease free and endemic equilibria as well as different bifurcations. We have shown that the epidemic disappears if the cure rate of treatment crosses a threshold value. We have obtained a necessary and sufficient condition for backward bifurcation, which shows the basic reproduction number less than unity is not sufficient to eradicate the disease completely. Saddle–node and Hopf bifurcation with respect to awareness factor have been investigated, which shows that the awareness factor is effective to change the disease dynamics. The model has been fitted to SARS cases in Hong Kong. The most effective parameters for controlling infections have been identified through sensitivity analysis. Moreover, we have investigated how the number of infected cases reduces if there was some vaccination polices in SARS infection. Finally, the model has been also used as an optimal control problem as vaccination and treatment controls are time dependent functions.

Dynamics analysis of a spatiotemporal SI model

In this article, we investigate a susceptible-infected (SI) model with the saturated treatment, the non-monotonic incidence rate, the logistic growth, and the homogeneous Neumann boundary conditions. The global existence and the uniform boundedness of the parabolic system are performed. After that, we investigate the global stability of the disease free equilibrium (DFE) and the endemic equilibrium (EE), respectively. In the end, we give a priori estimates, some propositions of the nonconstant steady states to the elliptic system. Meanwhile, we find that the diffusion rates of the susceptible and the infected population can affect the nonexistence of the nonconstant steady states. An interesting finding is DFE and basic reproduction number do not exist when the intrinsic growth rate of the susceptible class less than the rate of susceptible individuals being vaccinated.

Dynamics and data fitting of a time-delayed SIRS hepatitis B model with psychological inhibition factor and limited medical resources

Hepatitis B is an infectious disease worthy of attention. Considering the incubation period, psychological inhibition factor, vaccine, limited medical resources and horizontal transmission, an SIRS model is proposed to describe hepatitis B transmission dynamics. In order to describe the behavior changes caused by people’s psychological changes, the non-monotonic incidence rate is adopted in the model. We use the saturated treatment rate to describe the limited medical resources. Mathematical analysis shows the existence conditions of the equilibria, forward or backward bifurcation, Hopf bifurcation and the Bogdanov–Takens bifurcation. During the observation of the case data of hepatitis B in China, it is found that there are mainly three features, periodic outbreaks, aperiodic outbreaks, and periodic outbreaks turns to aperiodic outbreaks. According to the above features, we select three different representative regions, Jiangxi, Zhejiang province and Beijing, and then use our model to fit the actual monthly hepatitis B case data. The basic reproduction numbers that we estimated are 1.7712, 1.4805 and 1.4132, respectively. The results of data fitting are consistent with those of theoretical analysis. According to the sensitivity analysis of [Formula: see text], we conclude that reducing contact, increasing treatment rate, strengthening vaccination and revaccinating can effectively prevent and control the prevalence of hepatitis B.

On a two-strain epidemic mathematical model with vaccination

In this paper, we study mathematically a two strains epidemic model taking into account non-monotonic incidence rates and vaccination strategy. The model contains seven ordinary differential equations that illustrate the interaction between the susceptible, the vaccinated, the exposed, the infected and the removed individuals. The model has four equilibrium points, namely, disease free equilibrium, endemic equilibrium with respect to the first strain, endemic equilibrium with respect to the second strain and the endemic equilibrium with respect to both strains. The global stability of the equilibria has been demonstrated using some suitable Lyapunov functions. The basic reproduction number is found depending on the first strain reproduction number R 0 1 and the second reproduction number R 0 2 . We have shown that the disease dies out when the basic reproduction number is less than unity. It was remarked that the global stability of the endemic equilibria depends, on the strain basic reproduction number and on the strain inhibitory effect reproduction number. We have also observed that the strain with high basic reproduction number will dominate the other strain. Finally, the numerical simulations are presented in the last part of this work to support our theoretical results. We notice that our suggested model has some limitations and does not predicting the long-term dynamics for some reproduction numbers cases.

Asymptotic behavior of an SIQR epidemic model driven by Lévy jumps on scale-free networks

To investigate the effect of information transmission, Lévy jumps and contact heterogeneity of individuals on the asymptotic behavior of epidemic, a stochastic SIQR epidemic model with non-monotone incidence rate and Lévy jumps on scale-free networks is constructed. At first, the global dynamics of the deterministic model is studied by constructing appropriate Lyapunov functions. Then the stochastic model is made in accordance with the ecological significance, the existence and uniqueness of the global positive solution of the stochastic SIQR model is manifested. Next, by constructing suitable stochastic Lyapunov functions and applying Itô formula with jump, the asymptotic behavior of solutions of stochastic model around equilibrium of the corresponding deterministic model is checked. At last, the correctness of the analytical results is verified by numerical simulations.

Dynamics of a stochastic epidemic model with quarantine and non-monotone incidence

<abstract><p>In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts.</p></abstract>

Qualitative analysis of a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate.

In this paper, a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate is analyzed. The studied problem is formulated using susceptible, infectious and recovered compartmental model. A Caputo fractional operator is incorporated in each compartment to describe the memory effect related to an epidemic evolution. First, the global existence, positivity and boundedness of solutions of the proposed model are proved. The basic reproduction numbers associated with studied problem are calculated. Four steady states are given, namely the disease-free equilibrium, the strain 1 endemic equilibrium, the strain 2 endemic equilibrium, and the endemic equilibrium associated with both strains. By considering appropriate Lyapunov functions, the global stability of the equilibrium points is proven according to the model parameters. Our modeling approach using a generalized non-monotonic incidence functions encloses a variety of fractional-order epidemic models existing in the literature. Finally, the theoretical findings are illustrated using numerical simulations.

An SIRS model with nonmonotone incidence and saturated treatment in a changing environment.

Nonmonotone incidence and saturated treatment are incorporated into an SIRS model under constant and changing environments. The nonmonotone incidence rate describes the psychological or inhibitory effect: when the number of the infected individuals exceeds a certain level, the infection function decreases. The saturated treatment function describes the effect of infected individuals being delayed for treatment due to the limitation of medical resources. In a constant environment, the model undergoes a sequence of bifurcations including backward bifurcation, degenerate Bogdanov-Takens bifurcation of codimension 3, degenerate Hopf bifurcation as the parameters vary, and the model exhibits rich dynamics such as bistability, tristability, multiple periodic orbits, and homoclinic orbits. Moreover, we provide some sufficient conditions to guarantee the global asymptotical stability of the disease-free equilibrium or the unique positive equilibrium. Our results indicate that there exist three critical values r_1, r_2 and r_3 for the treatment rate r: (i) when rge max {r_1, r_2}, the disease will disappear; (ii) when r<min {r_1, r_3}, the disease will persist. In a changing environment, the infective population starts along the stable disease-free state (or an endemic state) and surprisingly continues tracking the unstable disease-free state (or a limit cycle) when the system crosses a bifurcation point, and eventually tends to the stable endemic state (or the stable disease-free state). This transient tracking of the unstable disease-free state when {mathscr {R}}_0>1 predicts regime shifts that cause the delayed disease outbreak in a changing environment. Furthermore, the disease can disappear in advance (or belatedly) if the rate of environmental change is negative and large (or small). The transient dynamics of an infectious disease heavily depend on the initial infection number and rate or the speed of environmental change. Supplementary InformationThe online version contains supplementary material available at 10.1007/s00285-022-01787-3.

A Fractional-Order Epidemic Model with Quarantine Class and Nonmonotonic Incidence: Modeling and Simulations.

In any outbreak of infectious disease, the timely quarantine of infected individuals along with preventive measures strategy are the crucial methods to control new infections in the population. Therefore, this study aims to provide a novel fractional Caputo derivative-based susceptible-infected-quarantined-recovered-susceptible epidemic mathematical model along with a nonmonotonic incidence rate of infection. A new quarantined individual compartment is incorporated into the susceptible-infected-recovered-susceptible compartmental model by dividing the total population into four subpopulations. The nonmonotonic incidence rate of infection is considered as Monod–Haldane functional type to understand the psychological effects in the population. Qualitative analysis of the study shows that the model solutions are well-posed i.e., they are nonnegative and bounded in an attractive region. It is revealed that the model has two equilibria, namely, disease-free (DFE) and endemic (EE). The stability analysis of equilibria is investigated for local as well as global behaviors. Mathematical analysis of the model reveals that DFE is locally asymptotically stable when the basic reproduction number ({R}_{0}) is lower than one. The basic reproduction number {R}_{0} is computed using the next-generation matrix method. The existence of EE is shown and it is investigated that EE is locally asymptotically stable when {R}_{0}>1 under some appropriate conditions. Moreover, the global stability behaviors of DFE and EE are analyzed under some conditions using {R}_{0}. Finally, some numerical simulations are performed to interpret the theoretical findings.

Delay effect of an e-epidemic SEIRS malware propagation model with a generalized non-monotone incidence rate

Malware such as virus, worm, and Trojan horse, is a software designed to pose harmful and undesirable effects on communication networks. It can cause numerous sever damage like data loss , repeated occupation of the memory space and corrupting databases etc. To overcome such and other related drastic threats, we need to fight against and control spread of malware at large scale. For this goal, a delayed e-epidemic SEIRS malware propagation model with a generalized non-monotone incidence rate is formulated in our paper. The basic reproduction number is derived. Then local stability and Hopf bifurcation analysis of the viral equilibrium is performed. Further, properties of the Hopf bifurcation are carried out. Finally, numerical simulations are implemented to justify the theoretical findings.

Complex dynamics and control analysis of an epidemic model with non-monotone incidence and saturated treatment.

In this manuscript, we consider an epidemic model having constant recruitment of susceptible individuals with non-monotone disease transmission rate and saturated-type treatment rate. Two types of disease control strategies are taken here, namely vaccination for susceptible individuals and treatment for infected individuals to minimize the impact of the disease. We study local as well as global stability analysis of the disease-free equilibrium point and also endemic equilibrium point based on the values of basic reproduction number R_0. Therefore, disease eradicates from the population if basic reproduction number less than unity and disease persists in the population if basic reproduction number greater than unity. We use center manifold theorem to study the dynamical behavior of the disease-free equilibrium point for R_0 = 1. We investigate different bifurcations such as transcritical bifurcation, backward bifurcation, saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of co-dimension 2. The biological significance of all types of bifurcations are described. Some numerical simulations are performed to check the reliability of our theoretical approach. Sensitivity analysis is performed to identify the influential model parameters which have most impact on the basic reproduction number of the proposed model. To control or eradicate the influence of the emerging disease, we need to control the most sensitive model parameters using necessary preventive measures. We study optimal control problem using Pontryagin’s maximum principle. Finally using efficiency analysis, we determine most effective control strategy among applied controls.