Discrete-time Epidemic Model Research Articles

The impact of the Caputo fractional difference operator on the dynamical behavior of a discrete-time SIR model for influenza A virus

Abstract Since there are few studies that deal with the fractional-order discrete-time epidemic models, this paper presents a new fractional-order discrete-time SIR epidemic model that is constructed based on the Caputo fractional difference operator. The effect of the fractional orders on the global dynamics of the SIR model is analyzed. In particular, the existence and stability of equilibrium points of the model are presented. Furthermore, we investigate the qualitative dynamical properties of the SIR model for both commensurate and incommensurate fractional orders using powerful nonlinear tools such as phase attractors, bifurcation diagrams, maximum Lyapunov exponent, chaos diagrams, and 0-1 test. In addition, the complexity of the discrete model is measured via the spectral entropy complexity algorithm. Further, an active controller is designed to stabilize the chaotic dynamics of the fractional-order SIR model. Finally, the suggested model is fitted with real data to show the accuracy of the current stability study. Our goal was achieved by confirming that the proposed SIR model can display a variety of epidiomologically observed states, including stable, periodic, and chaotic behaviors. The findings suggest that any change in parameter values or fractional orders could lead to unpredictable behavior. As a result, there is a need for additional research on this topic.

Infection-induced increases to population size during cycles in a discrete-time epidemic model

One-dimensional discrete-time population models, such as those that involve Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Expanding the system by one dimension to incorporate epidemiological interactions causes an interesting complexity of new behaviors. Here, we examine a discrete-time two-dimensional susceptible-infectious (SI) model with Ricker growth and show that the introduction of infection can not only produce a distinctly different bifurcation structure than that of the underlying disease-free system but also lead to counter-intuitive increases in population size. We use numerical bifurcation analysis to determine the influence of infection on the location and types of bifurcations. In addition, we examine the appearance and extent of a phenomenon known as the ‘hydra effect,’ i.e., increases in total population size when factors, such as mortality, that act negatively on a population, are increased. Previous work, primarily focused on dynamics at fixed points, showed that the introduction of infection that reduces fecundity to the SI model can lead to a so-called ‘infection-induced hydra effect.’ Our work shows that even in such a simple two-dimensional SI model, the introduction of infection that alters fecundity or mortality can produce dynamics can lead to the appearance of a hydra effect, particularly when the disease-free population is at a cycle.

A new discrete-time epidemic model describing information spread and its impact on the agree-disagree model

In this study, we propose a new modified discrete-time epidemic model that characterizes the diffusion of information and its impact on the agreement/disagreement model. So the goal is to increase the amount of information to influence people’s opinions. To do this, we proposed a control strategy based on the increase in the number of posts that influence people to agree with the subject studied (election, vaccine against COVID 19). The Pontryagin maximum principle is used to describe optimal control. Finally, numerical simulations are performed to verify the theoretical analysis using MATLAB.

Spectral Method in Epidemic Time Series: Application to COVID-19 Pandemic.

The age of infection plays an important role in assessing an individual's daily level of contagiousness, quantified by the daily reproduction number. Then, we derive an autoregressive moving average model from a daily discrete-time epidemic model based on a difference equation involving the age of infection. Novelty: The article's main idea is to use a part of the spectrum associated with this difference equation to describe the data and the model. We present some results of the parameters' identification of the model when all the eigenvalues are known. This method was applied to Japan's third epidemic wave of COVID-19 fails to preserve the positivity of daily reproduction. This problem forced us to develop an original truncated spectral method applied to Japanese data. We start by considering ten days and extend our analysis to one month. We can identify the shape for a daily reproduction numbers curve throughout the contagion period using only a few eigenvalues to fit the data.

Modelling optimal vaccination strategies against COVID-19 in a context of Gamma variant predominance in Brazil

IntroductionBrazil experienced moments of collapse in its health system throughout 2021, driven by the emergence of variants of concern (VOC) combined with an inefficient initial vaccination strategy against Covid-19. ObjectivesTo support decision-makers in formulating COVID-19 immunization policy in the context of limited vaccine availability and evolving variants over time, we evaluate optimal strategies for Covid-19 vaccination in Brazil in 2021, when vaccination was rolled out during Gamma variant predominance. MethodsUsing a discrete-time epidemic model we estimate Covid-19 deaths averted, considering the currently Covid-19 vaccine products and doses available in Brazil; vaccine coverage by target population; and vaccine effectiveness estimates. We evaluated a 5-month time horizon, from early August to the end of December 2021. Optimal vaccination strategies compared the outcomes in terms of averted deaths when varying dose intervals from 8 to 12 weeks, and choosing the minimum coverage levels per age group required prior to expanding vaccination to younger target populations. We also estimated dose availability required over time to allow the implementation of optimal strategies. ResultsTo maximize the number of averted deaths, vaccine coverage of at least 80 % should be reached in older age groups before starting vaccination into subsequent younger age groups. When evaluating varying dose intervals for AZD1222, reducing the dose interval from 12 to 8 weeks for the primary schedule would result in fewer COVID-19 deaths, but this can only be implemented if accompanied by an increase in vaccine supply of at least 50 % over the coming six-months in Brazil. ConclusionCovid-19 immunization strategies should be tailored to local vaccine product availability and supply over time, circulating variants of concern, and vaccine coverage in target population groups. Modelling can provide valuable and timely evidence to support the implementation of vaccination strategies considering the local context, yet following international and regional technical evidence-based guidance.

The analysis of discrete-time epidemic model on networks with protective measures on game theory

Protective measures such as wearing mask, quarantine, hand washing and so on can prevent and control the spread of infectious diseases. In this paper, a discrete-time model on networks combines infectious diseases and protective measures is proposed. Protective measures include voluntary protective measures of susceptible individuals and compulsory protective measures of infected individuals. The epidemic threshold is obtained and the time series of optimal control probability of infected individuals are analyzed by optimal control theory. The results of optimal control are compared with other constant control and the influences of protective measures on infectious diseases are analyzed by simulations. The optimal control results indicate that the greater the control proportion of infected individuals, the greater the average social payoff. It is shown that increasing the effectiveness of protective measures could effectively control the spread of infectious diseases.

A discrete-time epidemic model for the analysis of transmission of COVID19 based upon data of epidemiological parameters.

The forecasting of the nature and dynamics of emerging coronavirus (COVID-19) pandemic has gained a great concern for health care organizations and governments. The efforts aim to to suppress the rapid and global spread of its tentacles and also control the infection with the limited available resources. The aim of this work is to employ real data set to propose and analyze a compartmental discrete time COVID-19 pandemic model with non-linear incidence and hence predict and control its outbreak through dynamical research. The Basic Reproduction Number (R_{0}) is calculated analytically to study the disease-free steady state ( R_{0}<1), and also the permanency case (R_{0}>1) of the disease. Numerical results show that the transmission rates alpha left( >0right) and beta left( >0right) are quite effective in reducing the COVID-19 infections in India or any country. The fitting and predictive capability of the proposed discrete-time system are presented for relishing the effect of disease through stability analysis using real data sets.

Global behaviour of a class of discrete epidemiological SI models with constant recruitment of susceptibles

Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:10.1007/s12190-021-01503-2], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number We establish the global asymptotic stability of the disease-free equilibrium (GAS) for We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for . For the transcritical bifurcation appears. For we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number .

The existence of codimension-two bifurcations in a discrete-time SIR epidemic model

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a discrete-time SIR epidemic model. Codimension-two bifurcations associated with 1:2, 1:3 and 1:4 strong resonances are analyzed by using a series of affine transformations and bifurcation theory. Numerical simulations are carried out to verify and illustrate these theoretical results. More precisely, two kinds of high-resolution stability phase diagrams are exhibited to describe how the system's complexity unfolds with control parameters varying.&lt;/p&gt;&lt;/abstract&gt;

Optimal control for a discrete time epidemic model with zones evolution

In this paper, we present a new mathematical model to describe the evolution of an infectious disease in regions and between individuals. For this purpose we considered two systems, the first one for humans SiIiRi, where Si represents the number of susceptible, Ii of infected and Ri of cured. The second system ZSi ZIiZRi represents the different types of regions, where Zsi is the number of susceptible regions, where there are only susceptible people, after visiting an infected person, a susceptible region is likely to be infected, which we will note ZIi, the last compartment ZRi denotes the infected regions, which are restored after the recovery of all infected people. In addition, we considered three control strategies u, v and w to control the spread of the virus within regions and between individuals. Numerical examples are provided to illustrate the effectiveness of our proposed control strategy.

Discrete epidemic models with two time scales

The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviors, or others.To include a sufficiently general disease model, we first build up from first principles a discrete-time susceptible–exposed–infectious–recovered–susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number mathcal{R}_{0}.Then, we propose a general full model that includes sequentially the two processes at different time scales and proceed to its analysis through a reduced model. The basic reproduction number overline{mathcal{R}}_{0} of the reduced system gives a good approximation of mathcal{R}_{0} of the full model since it serves at analyzing its asymptotic behavior.As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.

A discrete-time epidemic SISI model

We consider a discrete-time epidemic SISI model in case when the population size is a constant, so the per capita death rate is equal to per capita birth rate. The evolution operator of this model is a non-linear operator which depends on seven parameters. Reducing it to a quadratic stochastic operator, we prove uniqueness of interior fixed point of the operator and study the limit behavior of the trajectory under some conditions to parameters.

Использование стохастической модели для прогнозирования длительных эпидемий

Рассматривается стохастическая дискретная модель эпидемии, в которой уровень инфекционности зависит от продолжительности болезни, с оценкой максимальной вероятности для параметра, отвечающего за скорость распространения инфекции. Для более эффективного использования реальной статистики количества больных введен дополнительный параметр модели – вероятность выявления заболевания. Разработана программа для автоматической оценки параметров модели с использованием прошлой статистики с последующей симуляцией развития эпидемии. Приведены сравнения результатов симуляции с существующими данными количества больных COVID-19 в Киеве с ручным и автоматическим подбором параметров. Рассмотрены возможность и целесообразность разбиения эпидемии на несколько промежутков с разными значениями параметров для описания длительных эпидемий со значительными изменениями динамики. Выполнено сравнение результатов симуляции с разным количеством промежутков для описания долгосрочного развития эпидемии COVID-19 в Киеве и Чехии, имеющих разную динамику развития эпидемии.

The risk of future waves of COVID-19: modeling and data analysis.

After a major outbreak of the coronavirus disease (COVID-19) starting in late December 2019, there were no new cases reported in mainland China for the first time on March 18, 2020, and no new cases reported in Hong Kong Special Administrative Region on April 20, 2020. However, these places had reported new cases and experienced a second wave since June 11, 2020. Here we develop a stochastic discrete-time epidemic model to evaluate the risk of COVID-19 resurgence by analyzing the data from the beginning of the outbreak to the second wave in these three places. In the model, we use an input parameter to represent a few potential risks that may cause a second wave, including asymptomatic infection, imported cases from other places, and virus from the environment such as frozen food packages. The effect of physical distancing restrictions imposed at different stages of the outbreak is also included in the model. Model simulations show that the magnitude of the input and the time between the initial entry and subsequent case confirmation significantly affect the probability of the second wave occurrence. Although the susceptible population size does not change the probability of resurgence, it can influence the severity of the outbreak when a second wave occurs. Therefore, to prevent the occurrence of a future wave, timely screening and detection are needed to identify infected cases in the early stage of infection. When infected cases appear, various measures such as contact tracing and quarantine should be followed to reduce the size of susceptible population in order to mitigate the COVID-19 outbreak.

An automated optimal vaccination control with a multi-region SIR epidemic model

Many mathematical models describing the evolution of infectious diseases underestimate the effect of the Spatio-temporal spread of epidemics. Currently, the COVID-19 epidemic shows the importance of taking into account the spatial dynamic of epidemics and pandemics. In this contribution, we consider a multi-region discretetime epidemic model that describes the spatial spread of an epidemic within different geographical zones assumed to be connected with the movements of their populations. Based on the fact that there are several limitations in medical resources, the authorities and health decision-makers must define a threshold of infections in order to determine if a zone is epidemic or not yet. We propose a new approach of optimal control by defining new importance functions to identify affected zones and then the need for the control intervention there. Numerical results are provided to illustrate our findings by applying this new approach in two adjacent regions of Morocco, the Casablanca-Settat and Rabat-Sale-K´ enitra regions. We investigate different scenarios to show the most effective ´ scenario, based on thresholds’ values.

Automated optimal vaccination and travel-restriction controls with a discrete multi-region SIR epidemic model

Many mathematical models describing the evolution of infectious diseases underestimate the effect of the Spatio-temporal spread of epidemics. Currently, the COVID-19 epidemic shows the importance of taking into account the spatial dynamic of epidemics and pandemics. In this contribution, we consider a multi-region discrete-time epidemic model that describes the spatial spread of an epidemic within different geographical zones assumed to be connected with the movements of their populations. Based on the fact that there are several limitations in medical resources, the authorities and health decision-makers must define a threshold of infections in order to determine if a zone is epidemic or not yet. We propose a new approach of optimal control by defining new importance functions to identify affected zones and then the need for the control intervention. Numerical results are provided to illustrate our findings by applying this new approach in two adjacent regions of Morocco, the Casablanca-Settat and Rabat-Sale-Kenitra regions. We investigate different scenarios to show the most effective scenario, based on thresholds’ values.

Multitask learning and nonlinear optimal control of the COVID-19 outbreak: A geometric programming approach

We propose a multitask learning approach to learn the parameters of a compartmental discrete-time epidemic model from various data sources and use it to design optimal control strategies of human-mobility restrictions that both curb the epidemic and minimize the economic costs associated with implementing non-pharmaceutical interventions. We develop an extension of the SEIR epidemic model that captures the effects of changes in human mobility on the spread of the disease. The parameters of the model are learned using a multitask learning approach that leverages both data on the number of deaths across a set of regions, and cellphone data on individuals’ mobility patterns specific to each region. Using this model, we propose a nonlinear optimal control problem aiming to find the optimal mobility-based intervention strategy that curbs the spread of the epidemic while obeying a budget on the economic cost incurred. We also show that the solution to this nonlinear optimal control problem can be efficiently found, in polynomial time, using tools from geometric programming. Furthermore, in the absence of a straightforward mapping from human mobility data to economic costs, we propose a practical method by which a budget on economic losses incurred may be chosen to eliminate excess deaths due to over-utilization of hospital resources. Our results are demonstrated with numerical simulations using real data from the COVID-19 pandemic in the Philadelphia metropolitan area.

Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

BackgroundThe spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued.ResultsA discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number mathcal {R}_0. Then the stability of the equilibria is given in terms of mathcal {R}_0, and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed.ConclusionIt is proven that the disease-free equilibrium and endemic equilibrium are stable if mathcal {R}_0<1 and mathcal {R}_0>1, respectively. Also, the disease-free equilibrium is globally stable when mathcal {R}_0le 1. The system has a transcritical bifurcation when mathcal {R}_0=1 and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

Separate seasons of infection and reproduction can lead to multi-year population cycles

Many host–pathogen systems are characterized by a temporal order of disease transmission and host reproduction. For example, this can be due to pathogens infecting certain life cycle stages of insect hosts; transmission occurring during the aggregation of migratory birds; or plant diseases spreading between planting seasons. We develop a simple discrete-time epidemic model with density-dependent transmission and disease affecting host fecundity and survival. The model shows sustained multi-annual cycles in host population abundance and disease prevalence, both in the presence and absence of density dependence in host reproduction, for large horizontal transmissibility, imperfect vertical transmission, high virulence, and high reproductive capability. The multi-annual cycles emerge as invariant curves in a Neimark–Sacker bifurcation. They are caused by a carry-over effect, because the reproductive fitness of an individual can be reduced by virulent effects due to infection in an earlier season. As the infection process is density-dependent but shows an effect only in a later season, this produces delayed density dependence typical for second-order oscillations. The temporal separation between the infection and reproduction season is crucial in driving the cycles; if these processes occur simultaneously as in differential equation models, there are no sustained oscillations. Our model highlights the destabilizing effects of inter-seasonal feedbacks and is one of the simplest epidemic models that can generate population cycles.

New automated optimal vaccination control with a multi-region SIRS epidemic mode

Many mathematical models describing the evolution of infectious diseases underestimate the effect of the Spatio-temporal spread of epidemics. Currently, the COVID-19 epidemic shows the importance of taking into account the spatial dynamic of epidemics and pandemics. In this paper, we consider a multi-region discrete-time SIRS epidemic model that describes the spatial spread of an epidemic within different geographical zones assumed to be connected with the movements of their populations (cities, towns, neighbors...). Judging by the fact that there are several restrictions in medical resources and some delay in decision-making, the authorities and health decision-makers must define a threshold of infections in order to determine if a zone is epidemic or not yet. We propose a new approach of optimal control by defining new importance functions to identify affected zones and then the need for the control intervention. This optimal control strategy allows to reduce the infectious individuals and increase the number of recovered ones in the targeted domain and this with an optimal cost. Numerical results are provided to illustrate our findings by applying this new approach in the Casablanca-Settat region of Morocco. We investigate different scenarios to show the most effective scenario, based on thresholds’ values.