Diffusive Epidemic Model Research Articles

Dynamics for a diffusive epidemic model with a free boundary: spreading-vanishing dichotomy

Dynamics for a diffusive epidemic model with a free boundary: spreading-vanishing dichotomy

Solving fractional differential equations on a quantum computer: A variational approach

We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in terms of time complexity but also has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation costs scale economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering applications, such as the subdiffusion equation, the nonlinear Burgers' equation, and a coupled diffusive epidemic model. We assess quantum hardware performance under realistic noise conditions, further validating the practical utility of our algorithm.

A risk-induced dispersal strategy of the infected population for a disease-free state in the SIS epidemic model

This article proposes a dispersal strategy for infected individuals in a spatial susceptible-infected-susceptible (SIS) epidemic model. The presence of spatial heterogeneity and the movement of individuals play crucial roles in determining the persistence and eradication of infectious diseases. To capture these dynamics, we introduce a moving strategy called risk-induced dispersal (RID) for infected individuals in a continuous-time patch model of the SIS epidemic. First, we establish a continuous-time n-patch model and verify that the RID strategy is an effective approach for attaining a disease-free state. This is substantiated through simulations conducted on 7-patch models and analytical results derived from 2-patch models. Second, we extend our analysis by adapting the patch model into a diffusive epidemic model. This extension allows us to explore further the impact of the RID movement strategy on disease transmission and control. We validate our results through simulations, which provide the effects of the RID dispersal strategy.

Analysis of a diffusive epidemic model with a zero-infection zone

In this paper, we propose and analyze a diffusive epidemic model with a zero-infection zone for the susceptible population. A distinct feature of the model is that the infected people are assumed to not able to cross the boundary of the zero-infection zone while the susceptible people cross the boundary at a prescribed rate. We investigate the global existence and boundedness of the solutions of the model and the existence and uniqueness of the disease-free equilibrium. We define the basic reproduction number and show that it is a threshold parameter for the disease dynamics. Moreover, we study the impact of the population transfer rate at the zero-infection zone boundary on the magnitude of the basic reproduction number. Finally, numerical simulations are performed as a supplement to our theoretical results.

Spreading speeds in an asymptotic autonomous system with application to an epidemic model

This article studies the initial value problem in an asymptotic autonomous reaction–diffusion system. Namely, when the time goes to infinity, these parameters converge to constants. The system may be nonmonotonic. With different decaying initial conditions, the leftward and rightward spreading speeds are estimated by constructing auxiliary functions and systems. As an application, we present the propagation properties in a diffusive epidemic model with different decaying initial values, which does not satisfy the classical comparison principle of mixed quasimonotone systems.

On a diffusive epidemic model with the tendency to move away from the infectious diseases

This paper is purported to investigate an epidemic system with the tendency of the susceptible to move away from the infectious diseases in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without cross-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are considered for this epidemic system. The results show that there exists spatiotemporal pattern formation when the tendency of susceptible individuals is strong to move away from the infectious diseases with 1<R0<1+min⁡{dSd,βσ1Aσ2}, while there does not exist spatiotemporal pattern formation when the self pressure is big enough with a weak tendency of the susceptible individuals to keep away from the infectious individual. Moreover, there exists positive solution bifurcation from the semi-trivial solution when the basic reproduction number is equal to one, which implies that endemic disease exists locally.

Extinction of some strains and asymptotic profiles of coexistence endemic equilibria in a multi-strain epidemic model

This work studies the asymptotic dynamics of classical solutions of a multi-strain diffusive epidemic model in spatially heterogeneous environments. Results on the nonexistence of coexistence endemic equilibria are established. Moreover, explicit conditions on the parameters range which guarantee the extinctions of some strains of the disease are examined. In particular, we establish that the competition-exclusion principle occurs when the population diffuses very fast and one strain uniquely maximizes the ratio of the spatial average of transmission rate over that of the recovery rate. However, several strains of the disease may coexist when either: (i) the population diffuses very slowly, (ii) or a subgroup diffuses very slowly and the other subgroup diffuses very fast. In these two scenarios, the asymptotic profiles of the coexistence endemic equilibria with respect to the diffusion rates of the population are presented. In particular, we observe a spatial segregation of the infected populations when they diffuse very slowly.

Dynamics of a Fokker–Planck type diffusion epidemic model with general incidence and relapse

In this paper, a spatially heterogeneous SIRI epidemic model with general morbidity is studied, which conforms to the Fokker–Planck type diffusion law. The basic reproduction number R0 of the model is introduced, and the threshold dynamics based on R0 is discussed. In particular, with the help of elliptic eigenvalue theory, we determine the asymptotic profiles of the endemic equilibrium when the diffusion rate of susceptible individuals approaches zero or infinity.

Global asymptotic stability of endemic equilibria and stability of traveling waves for a diffusive SIR epidemic model with logistic growth

AbstractThis paper deals with the asymptotic behavior of solutions for the diffusive epidemic model with logistic growth. In the first part, we consider the initial boundary value problem on the bounded domain and derive the stabilization of the solutions of the reaction–diffusion system to a constant equilibrium. In the second part, we consider the initial value problem on $${\mathbb {R}},$$ R , and derive the stability of forced waves under certain perturbations of a class of initial data.

Precision in disease dynamics: Finite difference solutions for stochastic epidemics with treatment cure and partial immunity

The complex and ever-changing characteristics of epidemic modelling, particularly when considering random elements, provide a substantial obstacle in creating precise and practical numerical methods for solving differential equations. This study contributes to this effort by introducing an innovative finite difference method for linear and non-linear stochastic and deterministic differential equations. This scheme expands explicitly upon the Euler Maruyama method, improving its precision for the deterministic aspect while ensuring coherence in dealing with stochastic terms. This contribution provides a numerical scheme that can be used to find solutions to linear and non-linear stochastic and deterministic differential equations. The scheme can be considered as the extension of the Euler Maruyama scheme for solving stochastic differential equations. The Euler Maruyama scheme offers a first-order accuracy of the deterministic model. Still, this scheme provides second-order accuracy for the deterministic part, whereas the integration of stochastic terms is the same in both schemes. The scheme is employed in a stochastic diffusive epidemic model with the effect of treatment, cure, and partial immunity. The comparison of the proposed scheme with the existing nonstandard finite difference method is made, and it is shown that the proposed scheme performs better than the nonstandard finite difference method in accuracy for the deterministic differential equations. It is also demonstrated that susceptible people rise whereas infected and recovered people decline by enhancing treatment cure rate. How does the cure rate of the treatment influence the number of the three populations, i.e., S(t), I(t), and R(t)? The results from the numerical simulation have provided useful insights into the dynamics of the epidemic model under various settings. This is particularly useful for influencing any public health plan and intervention. Thus, this work contributes numerical approaches and is an essential tool for epidemiological studies.

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.

Effectiveness of imperfect quarantine in controlling infectious diseases: A mathematical analysis of a general diffusive epidemic model

Effectiveness of imperfect quarantine in controlling infectious diseases: A mathematical analysis of a general diffusive epidemic model

Dynamics of a reaction–diffusion epidemic model with general incidence and protection awareness for multi-transmission pathways

Dynamics of a reaction–diffusion epidemic model with general incidence and protection awareness for multi-transmission pathways

Modelling Infectious Disease Dynamics: A Robust Computational Approach for Stochastic SIRS with Partial Immunity and an Incidence Rate

For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and an incidence rate. Considering the randomness of individual interactions and the spread of illnesses via space, this model is a powerful instrument for studying the spread and evolution of infectious diseases in populations with different immunity levels. A stochastic explicit finite difference scheme is proposed for solving stochastic partial differential equations. The scheme is comprised of predictor–corrector stages. The stability and consistency in the mean square sense are also provided. The scheme is applied to diffusive epidemic models with incidence rates and partial immunity. The proposed scheme with space’s second-order central difference formula solves deterministic and stochastic models. The effect of transmission rate and coefficient of partial immunity on susceptible, infected, and recovered people are also deliberated. The deterministic model is also solved by the existing Euler and non-standard finite difference methods, and it is found that the proposed scheme forms better than the existing non-standard finite difference method. Providing insights into disease dynamics, control tactics, and the influence of immunity, the computational framework for the stochastic SIRS reaction–diffusion model with partial immunity and an incidence rate has broad applications in epidemiology. Public health and disease control ultimately benefit from its application to the study and management of infectious illnesses in various settings.

Large time behavior in a reaction diffusion epidemic model with logistic source

This paper concerns a reaction diffusion epidemic model under logistic source. First, we explore uniform boundedness of the system. Moreover, by constructing two Lyapunov functions, we show that the disease-free equilibrium of system is globally asymptotically stable and the large time dynamics of constant endemic equilibrium is also considered. Especially, we further investigate the convergence rate of the nonnegative globally bounded classical solution. Finally, two numerical simulations are displayed to confirm the results.

On Stability of a Fractional Discrete Reaction–Diffusion Epidemic Model

This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global asymptotic stability analysis has been undertaken. We conducted a global stability analysis using a specialized Lyapunov function that captures the system’s historical data, distinguishing it from the integer-order version. This approach significantly advanced our comprehension of the complex stability properties within discrete fractional reaction–diffusion epidemic models. To substantiate the theoretical underpinnings, this paper is accompanied by numerical examples. These examples serve a dual purpose: not only do they validate the theoretical findings, but they also provide illustrations of the practical implications of the proposed discrete fractional system.

Intuitionistic fuzzy multi-objective transportation model during pandemic COVID-19

The given paper discusses an intuitionistic fuzzy discrete time-space optimization model while taking into account the COVID-19 pandemic. The novelty of this study is characterized by the proposal of a distinctive multi-objective intuitionistic fuzzy transportation model in which the demand function varies with time as the number of infected, exposed and susceptible persons increase. Forecasting of demand is executed using the epidemic diffusion model. Also, a new approach to defuzzify triangular intuitionistic fuzzy numbers, depending upon the concept of the golden ratio, is presented. Intuitionistic fuzzy programming approach is used to obtain the pareto-optimal solution. A real-life numerical illustration is explained using the proposed methodology to examine its practical suitability. Outperforming the existing traditional method of solving a transportation problem verifies its proficiency.

Spatial propagation phenomena for a diffusive epidemic model with vaccination

Spatial propagation phenomena for a diffusive epidemic model with vaccination

The Rionero’s special type of Lyapunov function and its application to a diffusive epidemic model with information

We consider a SIR-like reaction-diffusion epidemic model which embeds opinion-driven human behavioural changes. We assume that the contagion rate is theoretically saturated with respect to the density of the disease prevalence. The model extends the general reaction-diffusion epidemic model proposed in 1993 by Capasso and Di Liddo. We study the nonlinear attractivity of the endemic steady state solution by employing a special Lyapunov function introduced in 2006 by S. Rionero. Sufficient conditions for the conditional nonlinear stability of the endemic equilibrium are derived.

Spreading dynamics of a diffusive epidemic model with free boundaries and two delays

AbstractA delayed reaction-diffusion system with free boundaries is investigated in this paper to understand how the bacteria spread spatially to larger area from the initial infected habitat. Under the assumptions that the nonlinearities are of monostable type and the initial values satisfy some compatible condition, we show that the free boundary problem is well-posed and discuss the long-time behaviour of solution (including spreading and vanishing) in terms of the spatial-temporal risk index. Furthermore, to determine the spreading speed of free boundaries when spreading occurs, we first study the distribution of roots of a transcendental equation containing a polynomial of degree four and then establish the existence and uniqueness of monotone solution to a delay-induced nonlocal semi-wave problem by employing the approximation method, lower-upper solutions technique and Schauder fixed point theorem. It is shown that time delays slow down the spreading of bacteria.