SIS Epidemiological Model Research Articles

Production optimization in the time of pandemic: an SIS-based optimal control model with protection effort and cost minimization.

The COVID-19 pandemic wreaks havoc in supply chains by reducing the production capacity of some essential suppliers, closure of production facilities or the absence of infected workers. In this paper, we present three decision support models for a plant manager to help in deciding on (a) the level of protection of the workforce against the spread of the virus in the absence of regional protection measures, (b) on the duration of the protection, and (c) the level of protection of the workforce with regional protection measures enforced by health authorities. These decision models are based on a SIS epidemiological model which takes into account the possibility that a worker can infect others but also that even when recovered can be infected again. The first and third models prescribe how, in time, the protection effort in terms of prophylactic measures must be deployed. The second model extends the first one as it also determines the length the protection effort must be deployed. The proposed models have been applied to the case of a meat processing plant that must satisfy the demand of a large-scale retailer. Clearly, to achieve production targets and satisfy customers' demand, plants in this labor-intensive industry rely on the number of healthy workers and the service level of suppliers. Our results indicate that these models provide managers with the tools to understand and measure the impact of an infection on production and the corresponding cost. Along the way, this work illustrates the ripple effect as suppliers affected by the pandemic are unable to fulfill the processing plant requirements and so the retailer's orders. Our findings provide normative guidance for supply chain decision support systems under risk of pandemic induced disruptions using a quantitative model-based approach.

Stochastic dynamics of an SIS epidemiological model with media coverage

In this paper, we establish a stochastic SIS epidemic model with general transmission function and media coverage, and prove that two thresholds R1s and R2s (R2s<R1s) can be used to govern the stochastic dynamics of the stochastic SIS model. If R1s<1, the disease will die out with probability one and the distribution of the susceptible converges weakly to a boundary distribution; while if R2s>1, the disease is persistent almost surely and there exists a unique stationary distribution. Furthermore, we study the disease dynamics when R2s<1<R1s numerically, and find that the disease dynamics in this range is rich and complex, i.e., the disease may persist or extinct. Epidemiologically, we find that the smaller the intensity of random disturbance, the smaller the oscillation amplitude of the solution, and as the intensities of random disturbance increase, the mean of the infectious decreases and the distribution of them becomes more and more right-skewed, which provide a theoretical basis for disease control.

New results on the asymptotic behavior of an SIS epidemiological model with quarantine strategy, stochastic transmission, and Lévy disturbance

The spread of infectious diseases is a major challenge in our contemporary world, especially after the recent outbreak of Coronavirus disease 2019 (COVID‐19). The quarantine strategy is one of the important intervention measures to control the spread of an epidemic by greatly minimizing the likelihood of contact between infected and susceptible individuals. In this study, we analyze the impact of various stochastic disturbances on the epidemic dynamics during the quarantine period. For this purpose, we present an SIQS epidemic model that incorporates the stochastic transmission and the Lévy noise in order to simulate both small and massive perturbations. Under appropriate conditions, some interesting asymptotic properties are proved, namely, ergodicity, persistence in the mean, and extinction of the disease. The theoretical results show that the dynamics of the perturbed model are determined by parameters that are closely related to the stochastic noises. Our work improves many existing studies in the field of mathematical epidemiology and provides new techniques to predict and analyze the dynamic behavior of epidemics.

The ratio of single to co-colonization is key to complexity in interacting systems with multiple strains.

The high number and diversity of microbial strains circulating in host populations have motivated extensive research on the mechanisms that maintain biodiversity. However, much of this work focuses on strain‐specific and cross‐immunity interactions. Another less explored mode of pairwise interaction is via altered susceptibilities to co‐colonization in hosts already colonized by one strain. Diversity in such interaction coefficients enables strains to create dynamically their niches for growth and persistence, and “engineer” their common environment. How such a network of interactions with others mediates collective coexistence remains puzzling analytically and computationally difficult to simulate. Furthermore, the gradients modulating stability‐complexity regimes in such multi‐player endemic systems remain poorly understood. In a recent study (Madec & Gjini, Bulletin of Mathematical Biology, 82), we obtained an analytic representation for N‐type coexistence in an SIS epidemiological model with co‐colonization. We mapped multi‐strain dynamics to a replicator equation using timescale separation. Here, we examine what drives coexistence regimes in such co‐colonization system. We find the ratio of single to co‐colonization, µ, critically determines the type of equilibrium and number of coexisting strains, and encodes a trade‐off between overall transmission intensity R 0 and mean interaction coefficient in strain space, k. Preserving a given coexistence regime, under fixed trait variation, requires balancing between higher mean competition in favorable environments, and higher cooperation in harsher environments, and is consistent with the stress gradient hypothesis. Multi‐strain coexistence tends to steady‐state attractors for small µ, whereas as µ increases, dynamics tend to more complex attractors. Following strain frequencies, evolutionary dynamics in the system also display contrasting patterns with µ, interpolating between multi‐stable and fluctuating selection for cooperation and mean invasion fitness, in the two extremes. This co‐colonization framework could be applied more generally, to study invariant principles in collective coexistence, and to quantify how critical shifts in community dynamics get potentiated by mean‐field and environmental gradients.

Using lattice SIS epidemiological model with clustered treatment to investigate epidemic control

In epidemic control, how the size of treatment blocks influences the disease is a very interesting problem. Here, we construct an SIS model with spatially clustered treatment blocks using cellular automata and pair approximation to investigate the disease control and optimal arrangement of treatment blocks. On a same treatment intensity, the treatment blocks are built randomly and the individuals in the treatment blocks are cured in the next interval. The migration and neighboring structures are also considered in this model. We find that the size of treatment blocks will influence the control of disease. The bigger size of treatment blocks is more effective in epidemic control. The results obtained by lattice simulations and pair approximation respectively are consistent. On the same treatment arrange, higher migration rate has completely different influence on the epidemic depending on different neighboring structures. Besides, in different neighboring structures, both infected and susceptible populations will get clustered under the treatment blocks. Meanwhile, the neighboring structures can influence the degree of aggregation and the bigger treatment blocks can promote the aggregation compared to the smaller ones. The results here are significant for the control of epidemic diseases, especially when available medical resources are limited.

Optimal control and the value of information for a stochastic epidemiological SIS-model

We analyze a continuous time stochastic SIS epidemiological model, describing the transmission dynamics of infectious diseases where infected individuals can become susceptible again after recovery. The infectivity period of infected individuals can be influenced dynamically by a decision maker, with the aim to minimize the expected aggregated economic costs due to illness and drug treatment. This stochastic control problem is investigated under two alternative assumptions about the information available. If a complete and exact measurement of the size of the infected population is available at any time, then the optimal control is sought in a state-feedback form and the related Hamilton-Jacobi-Bellman (HJB) equation can be used. If no state measurement is available, then the optimal control is sought in an open-loop form and the problem can be reformulated as an optimal control problem for the Kolmogorov forward equation. In both cases, deriving optimality conditions requires non-standard arguments due to the degeneracy of the involved HJB and Kolmogorov equations. Based on the obtained theoretical results, the role of the information pattern is numerically investigated.

Global asymptotic stability of an SIS epidemic model with variable population size and a delay

An SIS epidemiological model with an exponential demographic structure and a delay corresponding to the infectious period is studied. We derive sufficient conditions for the global asymptotic stability of the infected steady state. The study is mainly based on very recent results of the monotone dynamical systems theory rarely used in mathematical epidemiology. We propose an improved and practical version of an important result in the theory. The obtained results are a partial - but important - resolution of the open problem proposed by Hethcote and van den Driessche in 1995. Numerical simulations are conducted to demonstrate our theoretical results.

Stability analysis of a delayed SIS epidemiological model

In this paper, we investigate global stability of the endemic steady state of the SIS epidemic model studied recently in terms of fluctuations. The epidemiological model has an exponential demographic structure, disease-related deaths and a delay corresponding to the infectious period. The disease spread is governed by a scalar delay differential equation. Our study is mainly based on the monotone dynamical systems theory. We begin by simplifying the classical framework of stability analysis for non-quasi-monotone scalar autonomous delay differential equations by using recent results on essentially strongly order-preserving semiflows. Under certain conditions, it is proved that the endemic equilibrium is globally asymptotically stable on a closed subset of the phase space C. To our knowledge, this is the first study proving the endemic steady state's global stability for this model. Numerical simulations which illustrate the results are carried out.

A nonstandard Volterra difference equation for the SIS epidemiological model

By considering the contact rate as a function of infective individuals and by using a general distribution of the infective period, the SIS-model extends to a Volterra integral equation that exhibits complex behaviour such as the backward bifurcation phenomenon. We design a nonstandard finite difference (NSFD) scheme, which is reliable in replicating this complex dynamics. It is shown that the NSFD scheme has no spurious fixed-points compared to the equilibria of the continuous model. Furthermore, there exist two threshold parameters \(\mathcal {R}_0^c\) and \(\mathcal {R}_0^m,\; ~\mathcal {R}_0^c\le 1\le \mathcal {R}_0^m\), such that the disease-free fixed-point is globally asymptotically stable (GAS) for \(\mathcal {R}_0\), the basic reproduction number, less than \(\mathcal {R}_0^c\) and unstable for \(\mathcal {R}_0>1\), while it is locally asymptotically stable (LAS) and coexists with a LAS endemic fixed-point for \(\mathcal {R}_0^c<\mathcal {R}_0<1\). A unique GAS endemic fixed-point exists when \(\mathcal {R}_0 >\mathcal {R}_0^m\) and \( \mathcal {R}_0^m <\infty \). Numerical experiments that support the theory are provided.

Variability in a Community-Structured SIS Epidemiological Model

We study an SIS epidemiological model of a population partitioned into groups referred to as communities, households, or patches. The system is studied using stochastic spatial simulations, as well as a system of ordinary differential equations describing moments of the distribution of infectious individuals. The ODE model explicitly includes the population size, as well as the variability in infection levels among communities and the variability among stochastic realizations of the process. Results are compared with an earlier moment-based model which assumed infinite population size and no variance among realizations of the process. We find that although the amount of localized (as opposed to global) contact in the model has little effect on the equilibrium infection level, it does affect both the timing and magnitude of both types of variability in infection level.

Backward bifurcation analysis of epidemiological model with partial immunity

This paper presents a two stage SIS epidemiological model in animal population with bovine tuberculosis (BTB) in African buffalo as a guiding example. The proposed model is rigorously analyzed. The analysis reveals that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) coexists with a stable endemic equilibrium (EE) when the associated reproduction number (Rv) is less than unity. It is shown under two special cases of the presented model, that this phenomenon of backward bifurcation does not arise depending on vaccination coverage and efficacy of vaccine. Numerical simulations of the model show that, the use of an imperfect vaccine can lead to effective control of the disease if the vaccination coverage and the efficacy of vaccine are high enough.

Turing Instability in a SIS Epidemiological Model in Discrete Space with Self and Cross Migration

In this paper an epidemic model is proposed to describe the dynamics of disease spread among patches due to population migration. We formulate a Susceptible–Infective–Susceptible type of epidemiological disease transmission model in a habitat of two identical patches linked by migration and we study the effect of the self and cross diffusion on the stability of the endemic equilibrium with disease induced mortality and nonlinear incidence rate. We show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation giving rise to non-constant stationary solutions. Numerical examples are also included.

Fast Response to Infection Spread and Cyber Attacks on Large-Scale Networks

We present a strategy for designing fast methods of response to cyber attacks and infection spread on complex weighted networks. In these networks, nodes can be interpreted as primitive elements of the system, and weighted edges reflect the strength of interaction among these elements. The proposed strategy belongs to the family of multiscale methods whose goal is to approximate the system at multiple scales of coarseness and to obtain a solution of microscopic scale by combining the information from coarse scales. In recent years these methods have demonstrated their potential for solving optimization and analysis problems on large-scale networks. We consider an optimization problem that is based on the SIS epidemiological model. The objective is to detect the network nodes that have to be immunized in order to keep a low level of infection in the system.

GLOBAL DYNAMICS OF A DIFFERENTIAL SUSCEPTIBILITY MODEL

An SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals has been analyzed. We prove that all the solutions tend to the equilibria of the system. Then we use the Poincaré Index theorem to determine the number of the rest points and their stability properties. It has been shown that bistability occurs for suitable values of the involved parameters. We use the perturbations of the pitchfork bifurcation points to give examples of all possible dynamics of the system. Some numerical examples of bistability and hysteresis behavior of the system has been also provided.

Diversify or focus? Spending to combat infectious diseases when budgets are tight

We consider a health authority seeking to allocate annual budgets optimally over time to minimize the discounted social cost of infection(s) evolving in a finite set of groups. This optimization problem is challenging since the standard SIS epidemiological model describing the spread of the disease contains a nonconvexity. Neither optimal control nor standard discrete-time dynamic programming can be used to identify the optimal policy. We modify the standard dynamic programming algorithm and show how familiar, elementary arguments can be used to reach conclusions about the optimal policy. We show that under certain conditions it is optimal to focus the entire annual budget on one group at a time rather than divide it among several groups, as is often done in practice. We also show that under certain conditions it remains optimal to focus on one group when faced with a wealth constraint instead of an annual budget.

Global analysis of a model of differential susceptibility induced by genetics

This paper is concerned with global analysis of an SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals. We prove that this system has no periodic solutions and use the Poincaré index theorem to determine the number of rest points and their stability properties. It has been shown that multiple equilibria (bistability) occurs for suitable values of parameters. We also give some numerical examples of all possible bifurcations of this system.

Epidemics and chaotic synchronization in recombining monogamous populations

We analyze the critical transitions (a) to endemic states in an SIS epidemiological model, and (b) to full synchronization in an ensemble of coupled chaotic maps, on networks where, at any given time, each node is connected to just one neighbour. In these “monogamous” populations, the lack of connectivity in the instantaneous interaction pattern—that would prevent both the propagation of an infection and the collective entrainment into synchronization—is compensated by occasional random reconnections which recombine interacting couples by exchanging their partners. The transitions to endemic states and to synchronization are recovered if the recombination rate is sufficiently large, thus giving rise to a bifurcation as this rate varies. We study this new critical phenomenon both analytically and numerically.

Fluctuations in a SIS epidemic model with variable size population

In an epidemiological model, time spent in one compartment is often modeled by a delay in the model. In general the presence of delay in differential equations can change the stability of an equilibrium to instability and causes the appearance of oscillatory solutions. In this paper we consider a SIS epidemiological model with demographic effects: birth, mortality and mortality caused by infection. The delay is the period of infection. We define the concept of oscillation in the sense that solutions of the model studied fluctuate around a steady state. Our goal is to show that in this model, there are oscillating solutions for certain parameters values. We determine a large set of initial data for which solutions of this model are slowly oscillating.

Sexually transmitted infections and the marriage problem

We study an SIS epidemiological model for a sexually transmitted infection in a monogamous population where the formation and breaking of couples is governed by individual preferences. The mechanism of couple recombination is based on the so-called bar dynamics for the marriage problem. We compare the results with those of random recombination – where no individual preferences exist – for which we calculate analytically the infection incidence and the endemic threshold. We find that individual preferences give rise to a large dispersion in the average duration of different couples, causing substantial changes in the incidence of the infection and in the endemic threshold. Our analysis yields also new results on the bar dynamics, that may be of interest beyond the field of epidemiological models.

Infection Spreading in a Population with Evolving Contacts

We study the spreading of an infection within an SIS epidemiological model on a network. Susceptible agents are given the opportunity of breaking their links with infected agents. Broken links are either permanently removed or reconnected with the rest of the population. Thus, the network coevolves with the population as the infection progresses. We show that a moderate reconnection frequency is enough to completely suppress the infection. A partial, rather weak isolation of infected agents suffices to eliminate the endemic state.