Epidemic Model Research Articles

An epidemic spread model with nonlinear recovery rates on meta-population networks

An epidemic spread model with nonlinear recovery rates on meta-population networks

Stability and Hopf bifurcation analysis of a networked SIR epidemic model with two delays

Stability and Hopf bifurcation analysis of a networked SIR epidemic model with two delays

Stochastic Optimal Control Analysis for HBV Epidemic Model with Vaccination

In this study, we explore the concept of symmetry as it applies to the dynamics of the Hepatitis B Virus (HBV) epidemic model. By incorporating symmetric principles in the stochastic model, we ensure that the control strategies derived are not only effective but also consistent across varying conditions, and ensure the reliability of our predictions. This paper presents a stochastic optimal control analysis of an HBV epidemic model, incorporating vaccination as a pivotal control measure. We formulate a stochastic model to capture the complex dynamics of HBV transmission and its progression to acute and chronic stages. By leveraging stochastic differential equations, we examine the model’s stationary distribution and asymptotic behavior, elucidating the impact of random perturbations on disease dynamics. Optimal control theory is employed to derive control strategies aimed at minimizing the disease burden and vaccination costs. Through rigorous numerical simulations using the fourth-order Runge–Kutta method, we demonstrate the efficacy of the proposed control measures. Our findings highlight the critical role of vaccination in controlling HBV spread and provide insights into the optimization of vaccination strategies under stochastic conditions. The symmetry within the proposed model equations allows for a balanced approach to analyzing both acute and chronic stages of HBV.

Extinction Dynamics and Equilibrium Patterns in Stochastic Epidemic Model for Norovirus: Role of Temporal Immunity and Generalized Incidence Rates

Norovirus is a leading global cause of viral gastroenteritis, significantly affecting mortality, morbidity, and healthcare costs. This paper develops and analyzes a stochastic SEIQR epidemic model for norovirus dynamics, incorporating temporal immunity and a generalized incidence rate. The model is proven to have a unique positive global solution, with extinction conditions explored. Using Khasminskii’s method, the model’s ergodicity and equilibrium distribution are investigated, demonstrating a unique ergodic stationary distribution when R^s>1. Extinction occurs when R0E<1. Computer simulations confirm that noise level significantly influences epidemic spread.

New metrics for influential spreaders identification in complex networks based on D-spectra of nodes

Identifying important nodes is of key significance in network sciences as it is closely related to information and disease spreading, structural robustness, etc. Till today, no theoretically proved optimal metric for measuring the importance of nodes is known, while a lot of metrics, mostly heuristic, have been studied. In this paper, we propose several heuristic metrics that are constructed from D-spectra of vertices for discussion. The latter is a graph invariant which is induced by D-chain decompositions of graphs, a recently introduced framework for studying network structures by the first author, Bura and Reidys (SIAM J. Appl. Dyn. Syst. 18, 2019, pp. 2181–2201). Statistical analyses based on numerous data from running the SIR model on real-world and random networks show that some of our proposed metrics outperform a number of well-known metrics such as the H-index, the core values, the betweenness centrality and the closeness centrality.

Modeling and analysis of COVID-19 spreading based on complex network theory

Complex networks can effectively describe interactions within real-world complex systems. In researches of epidemic spreading, scientists constructed various physical contact networks between individuals on the microscopic scale and the metapopulation networks on the macroscopic scale. These different types of network structures significantly impact the propagation dynamics of epidemic in human society. For instance, population flows in global airline networks influence the speed and arrival time of epidemics across large-scale space. In this paper we review the epidemic spreading models on various network structures, including fully mixed networks, three types of lower-order networks, three types of higher-order networks, metapopulation networks, and multiple strains competitive epidemic spreading models. We also provide an overview of the application of complex network theory in the COVID-19 pandemic, covering topics of prediction, prevention, and control of the epidemic. Finally, we discuss the strengths and limitations of these models and propose perspectives for future research.

Complex dynamics of an SIHR epidemic model with variable hospitalization rate depending on unoccupied hospital beds

In this paper, we propose an susceptible-infectious-hospitalized-recovered (SIHR) epidemic model with a nonlinear hospitalization rate depending on the number of unoccupied hospital beds. Note that the number of all hospital beds is used as a measure of all available medical resources. The basic reproduction number is calculated using the next-generation matrix method. We analyze the existence of endemic equilibria and discuss the global stability of the disease-free equilibrium. Existence and stability of endemic equilibria indicate possible occurrences of bifurcations. We confirm the appearance of backward bifurcation, saddle–node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation using normal form theory and central manifold theory. Numerical simulations show that the dynamic behavior of the model undergoes a transition from forward bifurcation to backward bifurcation and saddle–node bifurcation when the number of total hospital beds is reduced. Our findings suggest that when the number of total hospital beds falls below a threshold, backward bifurcation will occur, meaning that the disease cannot be eliminated even if the basic reproduction number is below unity. Therefore, the number of hospital beds should be increased beyond the bed threshold during an outbreak of a disease, which has important implications for disease control.

Minimal wave speed for a two-group epidemic model with nonlocal dispersal and spatial-temporal delay

In this paper, a two-group SIR reaction-diffusion epidemic model with nonlocal dispersal and spatial-temporal delay based on within-group and inter-group transmission mechanisms is investigated. The basic reproduction number R0 is calculated using the method of next-generation matrix. The critical wave speed cm* is determined by analyzing the distribution of roots of the characteristic equation. When R0>1 and wave speed c⩾cm*, the existence of traveling waves connecting disease-free and endemic steady states is obtained by constructing sub- and super-solutions and a Lyapunov functional, and applying Schauder’s fixed-point theorem and a limit argument. When R0>1 and 0<c<cm*, the nonexistence of traveling waves connecting disease-free and endemic steady states is proven by contradiction and two-sided Laplace transform. This indicates that the critical wave speed cm* is exactly the minimum wave speed. Numerical simulations are carried out to illustrate theoretical results. The dependence of the minimal speed cm* on time delay, diffusion rates and contact rates is discussed, showing that the longer the latent period and the lower the diffusion rates of infected individuals and the inter-group transmission rates between groups, the slower the spread of disease.

A clustered-LPSEIRS malware propagation model in complex networks

In this paper, we present a new malware propagation model that integrates epidemic spread, clustering, and link prediction techniques, tailored for complex network networks. Our model is based on the clustered-link prediction-susceptible-exposed-infected-recovered (clustered-LPSEIRS) epidemic model, which simulates malware dissemination within the network. Our findings reveal a significant decrease in the rate of malware spread compared to the traditional SEIR model, with this enhancement in containment attributed to the integration of clustering and link prediction methods. We also compute the basic reproduction ratio (R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{0}$$\\end{document}) for our model, providing insights into the potential ramifications of malware within the network. By examining parameter variations, we enhance our understanding of the model's behavior under diverse scenarios. Additionally, we assess the influence of clustering and link prediction on mitigating malware spread, emphasizing its effectiveness in diminishing the overall impact.

Analyzing steady-state equilibria and bifurcations in a time-delayed SIR epidemic model featuring Crowley-Martin incidence and Holling type II treatment rates

This article presents a time-delayed SIR epidemiological model that has been quantitatively examined. The model incorporates a logistic growth function for the susceptible population, a Crowley-Martin type incidence, and Holling type II treatment rates. We investigated two separate time delays. The first delay refers to the rate at which new infections occur, allowing us to evaluate the impact of the latent period. The second delay relates to the rate of treatment for those who have contracted the infection, which allows us to examine the consequences of postponed access to therapy. The investigation of the steady-state behavior of the model emphasizes two equilibria, namely, the infection-free equilibrium and the endemic equilibrium. The determination of critical values involves the use of the fundamental reproduction number, denoted R0, which serves as a predictive measure to determine the potential elimination of a disease within a specific population. Using the fundamental reproduction number, it can be shown that infection-free equilibrium exhibits local asymptotic stability when the value of R0 is less than 1. In contrast, when R0 exceeds 1, the infection-free equilibrium becomes unstable in the context of the time-delayed system. Furthermore, an analysis of the steady-state dynamics of the endemic equilibrium indicates the appearance of oscillations and periodic solutions with the Hopf bifurcation for all feasible combinations of two-time delays as the bifurcated parameter. In sensitivity analysis, a sensitivity index is utilized to evaluate the relative modification in the fundamental reproduction number caused by each parameter. In summary, numerical simulations are employed to offer empirical evidence for the theoretical findings.

Pattern dynamics of networked epidemic model with higher-order infections.

Current research on pattern formations in networked reaction-diffusion (RD) systems predominantly focuses on the impacts of diffusion heterogeneity between nodes, often overlooking the contact heterogeneity between individuals within nodes in the reaction terms. In this paper, we establish a networked RD model incorporating infection through higher-order interaction in simplicial complexes in the reaction terms. Through theoretical and numerical analysis, we find that these higher-order interactions may induce Turing instability in the system. Notably, the relationship between the size of the Turing instability range and the average 2-simplices degree within nodes can be approximated by a quadratic function. Additionally, as the average 2-simplices degree increases, the amplitude of the patterns exhibits three distinct trends: increasing, decreasing, and initially increasing then decreasing, while the average infection density increases consistently. We then provide a possible explanation for these observations. Our findings offer new insights into the effects of contact heterogeneity within nodes on networked pattern formations, thereby informing the development of epidemic prevention and control measures.

Quantifying age-specific household contacts in Aotearoa New Zealand for infectious disease modelling.

Accounting for population age structure and age-specific contact patterns is crucial for accurate modelling of human infectious disease dynamics and impact. A common approach is to use contact matrices, which estimate the number of contacts between individuals of different ages. These contact matrices are frequently based on data collected from populations with very different demographic and socio-economic characteristics from the population of interest. Here we use a comprehensive household composition dataset based on Aotearoa New Zealand census and administrative data to construct a household contact matrix and a synthetic population that can be used for modelling. We investigate the behaviour of a compartment-based and an agent-based epidemic model parametrized using these data, compared with a commonly used contact matrix that was constructed by projecting international data onto New Zealand's population. We find that using the New Zealand household data, either in a compartment-based model or in an agent-based model, leads to lower attack rates in older age groups compared with using the projected contact matrix. This difference becomes larger when household transmission is more dominant relative to non-household transmission. We provide electronic versions of the synthetic population and household contact matrix for other researchers to use in infectious disease models.

Vaccination subsidy allocation under budget constraints considering the human interpersonal contact pattern and vaccine protection effect in epidemics

Vaccination subsidy is one of the most important measures to increase the willingness of people to be vaccinated for epidemic prevention and control. Given the budget constraint of vaccination subsidy and the different roles of various age groups population in the epidemic transmission process, a novel approach is proposed to optimize the utilization of the limited vaccination subsidy budget. First, an age-structured SIR model that integrates the human interpersonal contact pattern and vaccine protection effect is developed to simulate the epidemic trend under a certain vaccination state. Then, a decision model is constructed to determine the optimal vaccination subsidy allocation scheme for various age groups based on the age-structured SIR model. A case study of Wuhan City is used to verify the decision approach, and a series of sensitivity analyses are conducted. The result shows that the peak time of an epidemic is positively related to the total treatment cost for all the infections, and more subsidies should be allocated to the populations with higher interpersonal contact frequencies for epidemic prevention and control.

Computational Thinking Processes in Solving the Corona Epidemic Model: Pre-service Maths Teachers

In the 21st century, pre-service mathematics teachers are expected to have problem-solving skills that are effective, efficient, and solutive and are in line with the mindset of computer experts. In learning mathematics, the concept of computational thinking (CT) is also needed and at this time, many still have difficulty solving mathematical problems in general, especially in solving problems in epidemic mathematical models. The subjects of this study were twenty-seven pre-service mathematics teacher students who took mathematical modeling courses. The researcher used the purposive sampling technique to select two research samples. The research method used was a descriptive qualitative research method in exploring the thinking process of pre-service mathematics teacher students in solving the problem of modeling the epidemic spread of disease. The results showed that the thinking process of the first subject began with identifying the problem and existing information by writing down the data in the form of a graph so as to get a certain pattern, which was then used as the basis for the process of transforming the problem into mathematical language. By adding assumptions related to the existence of environmental limitations in the next epidemic model, the concept of differential equations, in which there are integral properties and natural logarithms, can be used to find the solution to the epidemic model. The second subject was unable to solve the integral at hand. The researcher discovered that pre-service mathematics teacher students who correctly solved the problem in the mathematical model used CT components, namely decomposition, abstraction, pattern recognition, algorithm and mathematical literacy.

MATHEMATICAL ANALYSIS OF SIR MODEL WITH INCUBATION PERIOD

The SIPmodel is a fundamental tool for simulating epidemic dynamics, but it has limitations in accurately representing real-world scenarios. This paper presents a comprehensive review and mathematical analysis of the SIR model when incorporating the incubation period of infectious diseases. We discuss the significance of the incubation period in disease transmission dynamics and explore the modifications made to the SIR model to integrate this parameter. By analyzing the mathematical equations governing the modified SIR model, we demonstrate its enhanced accuracy in predicting disease spread patterns and its implications for public health interventions. Our findings evaluated the importance of incorporating the incubation period into epidemic models.

The epidemiological footprint of contact structures in models with two levels of mixing.

Models with several levels of mixing (households, workplaces), as well as various corresponding formulations for , have been proposed in the literature. However, little attention has been paid to the impact of the distribution of the population size within social structures, effect that can help plan effective interventions. We focus on the influence on the model outcomes of teleworking strategies, consisting in reshaping the distribution of workplace sizes. We consider a stochastic SIR model with two levels of mixing, accounting for a uniformly mixing general population, each individual belonging also to a household and a workplace. The variance of the workplace size distribution appears to be a good proxy for the impact of this distribution on key outcomes of the epidemic, such as epidemic size and peak. In particular, our findings suggest that strategies where the proportion of individuals teleworking depends sublinearly on the size of the workplace outperform the strategy with linear dependence. Besides, one drawback of the model with multiple levels of mixing is its complexity, raising interest in a reduced model. We propose a homogeneously mixing SIR ODE-based model, whose infection rate is chosen as to observe the growth rate of the initial model. This reduced model yields a generally satisfying approximation of the epidemic. These results, robust to various changes in model structure, are very promising from the perspective of implementing effective strategies based on social distancing of specific contacts. Furthermore, they contribute to the effort of building relevant approximations of individual based models at intermediate scales.

Stability analysis of a delayed fractional-order $$\mathcal {SIR}$$ epidemic model with Crowley–Martin type incidence rate and Holling type II treatment rate

Stability analysis of a delayed fractional-order $$\mathcal {SIR}$$ epidemic model with Crowley–Martin type incidence rate and Holling type II treatment rate

Positivity and Boundedness Preserving Numerical Scheme for a Stochastic Multigroup Susceptible-Infected-Recovering Epidemic Model with Age Structure.

Since the stochastic age-structured multigroup susceptible-infected-recovering (SIR) epidemic model is nonlinear, the solution of this model is hard to be explicitly represented. It is necessary to construct effective numerical methods so as to predict the number of infections. In addition, the stochastic age-structured multigroup SIR model has features of positivity and boundedness of the solution. Therefore, in this article, in order to ensure that the numerical and analytical solutions must have the same properties, by modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving EM (PBPEM) method on temporal space for stochastic age-structured multigroup SIR model, which is proved to have a strong convergence to the true solution over finite time intervals. Moreover, by combining the standard finite element method and the PBPEM method, we propose a full-discrete scheme to show the numerical solutions, as well as analyze the error estimations. Finally, the full-discrete scheme is applied to a general stochastic two-group SIR model and the Chlamydia epidemic model, which shows the superiority of the numerical method.

Stationary distribution of a stochastic generalized SIRI epidemic model with reinfection and relapse

In this paper, we propose and investigate the stochastic SIRI epidemic model with two generalized incidence rate functions. We firstly study the existence and uniqueness of the globally positive solution to the stochastic SIRI model with positive initial value. Then we obtain sufficient conditions for the extinction of the disease in the stochastic epidemic model, and find that the large noise can make the disease die out exponentially. Meanwhile, we obtain that the solution to the stochastic model has a unique stationary distribution when R0s˜ is greater than one. Our results show that the intensity of white noise can affect the dynamical behaviors of the model. Finally, we use numerical simulation to illustrate theoretical results, and apply both the stochastic and deterministic models to analyze the outbreak of COVID-19 epidemic in Serbia.

Analysis and Optimal Control of a Two-Strain SEIR Epidemic Model with Saturated Treatment Rate

In this paper, we conducted a study on the optimal control problem of an epidemic model which consists of two strain with different types of incidence rates: bilinear and non-monotonic. We also considered use of the saturation treatment function. Two basic regeneration numbers are calculated from the epidemic model, which are denoted as R1 and R2. The global stability of the disease-free equilibrium point was studied by the Lyapunov method, and it was proved that the disease-free equilibrium point is globally asymptotically stable when R1 and R2 are less than one. Finally, we formulated a time-dependent optimal control problem by Pontryagin’s maximum principle. Numerical simulations were performed to establish the effects of model parameters for disease transmission as well as the effects of control.