Susceptible-infectious-recovered-susceptible Research Articles

A behaviour and disease transmission model: incorporating the Health Belief Model for human behaviour into a simple transmission model.

The health and economic impacts of infectious diseases such as COVID-19 affect all levels of a community from the individual to the governing bodies. However, the spread of an infectious disease is intricately linked to the behaviour of the people within a community since crowd behaviour affects individual human behaviour, while human behaviour affects infection spread, and infection spread affects human behaviour. Capturing these feedback loops of behaviour and infection is a well-known challenge in infectious disease modelling. Here, we investigate the interface of behavioural science theory and infectious disease modelling to explore behaviour and disease (BaD) transmission models. Specifically, we incorporate a visible protective behaviour into the susceptible-infectious-recovered-susceptible (SIRS) transmission model using the socio-psychological Health Belief Model to motivate behavioural uptake and abandonment. We characterize the mathematical thresholds for BaD emergence in the BaD SIRS model and the feasible steady states. We also explore, under different infectious disease scenarios, the effects of a fully protective behaviour on long-term disease prevalence in a community, and describe how BaD modelling can investigate non-pharmaceutical interventions that target-specific components of the Health Belief Model. This transdisciplinary BaD modelling approach may reduce the health and economic impacts of future epidemics.

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.

A Comparison of Monte Carlo-Based and PINN Parameter Estimation Methods for Malware Identification in IoT Networks

Malware propagation is a growing concern due to its potential impact on the security and integrity of connected devices in Internet of Things (IoT) network environments. This study investigates parameter estimation for Susceptible-Infectious-Recovered (SIR) and Susceptible–Infectious–Recovered–Susceptible (SIRS) models modeling malware propagation in an IoT network. Synthetic data of malware propagation in the IoT network is generated and a comprehensive comparison is made between two approaches: algorithms based on Monte Carlo methods and Physics-Informed Neural Networks (PINNs). The results show that, based on the infection curve measured in the IoT network, both methods are able to provide accurate estimates of the parameters of the malware propagation model. Furthermore, the results show that the choice of the appropriate method depends on the dynamics of the spreading malware and computational constraints. This work highlights the importance of considering both classical and AI-based approaches and provides a basis for future research on parameter estimation in epidemiological models applied to malware propagation in IoT networks.

Extending susceptible-infectious-recovered-susceptible epidemics to allow for gradual waning of immunity.

Susceptible-infectious-recovered-susceptible (SIRS) epidemic models assume that individual immunity wanes in one leap, from complete immunity to complete susceptibility. For many diseases immunity on the contrary wanes gradually, something that has become even more evident during COVID-19 pandemic where also recently infected have a reinfection risk, and booster vaccines are given to increase immunity. Here, a novel mathematical model is presented allowing for the gradual decay of immunity following linear or exponential waning functions. The two new models and the SIRS model are compared assuming all three models have the same cumulative immunity. When no intervention is put in place, we find that the long-term prevalence is higher for the models with gradual waning. If aiming for herd immunity by continuous vaccination, it is shown that larger vaccine quantities are required when immunity wanes gradually compared with results obtained from the SIRS model, and this difference is the biggest for the most realistic assumption of exponentially waning of immunity. For parameter choices fitting to COVID-19, the critical amount of vaccine supply is about 50% higher if immunity wanes linearly, and more than 150% higher when immunity wanes exponentially, when compared with the classic SIRS epidemic model.

A Tutorial on Common Differential Equations and Solutions Useful for Modeling Epidemics Like COVID-19: Linear and Non-Linear Compartmentation Models

Purpose: To review some of the basic models, differential equations and solutions, both analytic and numerical, which produce time courses for the fractions of Susceptible (S), Infectious (I) and Recovered (R) fractions of the population during the epidemic and/or endemic conditions. Methods: Two and three-compartment models with analytic solutions to the proposed linear differential equations as well as models based on the non-linear differential equations first proposed by Kermack and McKendrick (KM) [1] a century ago are considered. The equations reviewed include the ability to slide between so-called Susceptible-Infected-Recovered (SIR), Susceptible-Infectious-Susceptible (SIS), Susceptible-Infectious (SI) and Susceptible-Infectious-Recovered-Susceptible (SIRS) models, effectively moving from epidemic to endemic characterizations of infectious disease. Results: Both the linear and KM model yield typical “curves” of the infected fraction being sought “to flatten” with the effects of social distancing/masking efforts and/or pharmaceutical interventions. Demonstrative applications of the solutions to fit real COVID-19 data, including linear and KM SIR fit data from the first 100 days following “lockdown” in the authors’ locale and to the total number of cases in the USA over the course of 1 year with SI and SIS models are provided. Conclusions: COVID-19 took us all by surprise, all wondering how to help. Spreading a basic understanding of some of the mathematics used by epidemiologists to model infectious diseases seemed like a good place to start and served as the primary purpose for this tutorial.

Bifurcations of an SIRS epidemic model with a general saturated incidence rate.

This paper is concerned with the bifurcations of a susceptible-infectious-recovered-susceptible (SIRS) epidemic model with a general saturated incidence rate $ k I^p/(1+\alpha I^p) $. For general $ p > 1 $, it is shown that the model can undergo saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, and degenerate Hopf bifurcation of codimension two with the change of parameters. Combining with the results in [1] for $ 0 < p\leq 1 $, this type of SIRS model has Hopf cyclicity $ 2 $ for any $ p > 0 $. These results also improve some previous ones in [2] and [3], which are dealt with the special case of $ p = 2 $.

Quantifying the Impact of COVID-19 Nonpharmaceutical Interventions on Influenza Transmission in the United States.

BackgroundNonpharmaceutical interventions (NPIs) have been implemented to suppress the transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Evidence indicates that NPIs against COVID-19 may also have effects on the transmission of seasonal influenza.MethodsIn this study, we use an absolute humidity-driven susceptible-infectious-recovered-susceptible (SIRS) model to quantify the reduction of influenza incidence and transmission in the US and HHS (the US Department of Health and Human Services) regions after implementation of NPIs in 2020. We investigate the long-term effect of NPIs on influenza incidence by projecting influenza transmission at the national scale over the next five years, using the SIRS model.ResultsWe estimate that incidence of influenza A/H1 and B, which circulated in early 2020, was reduced by more than 60% in the US during the first ten weeks following implementation of NPIs. The reduction of influenza transmission exhibits clear geographical variation. After the control measures are relaxed, potential accumulation of susceptibility to influenza infection may lead to a large outbreak, the scale of which may be affected by length of the intervention period and duration of immunity to influenza.DiscussionHealthcare systems need to prepare for potential influenza patient surges and advocate vaccination and continued precautions.

Improving wireless sensor networks performance through epidemic model

ABSTRACTWireless sensor networks (WSNs) encounter a critical challenge of ‘Network Security’ due to extreme operational constraints. The origin of the challenge begins with the entry of worms in the wireless network. Just one infected node is enough to spread the worms across the entire network. The infected node rapidly infects the neighbouring nodes in an unstoppable manner. In this paper, a mathematical model is proposed based on epidemic theory. It is an improvement of the Susceptible-Infectious-Recovered-Susceptible (SIRS) and Susceptible-Exposed-Infectious-Susceptible (SEIS) model. We propose Susceptible-Exposed-Infectious-Recovered-Susceptible (SEIRS) model that overcomes the drawbacks of existing models. The proposed ameliorated model includes a finite communication radius and the associated node density. We obtain basic reproduction number which determines the local and global propagation dynamics of worm in the WSNs. Also, we deduce expression for threshold for node density and communication radius. We investigated the control mechanism against worm propagation. We compare the proposed model with various existing models and evaluate its performance on the basis of various performance metrics. The study confirms melioration in the vital aspects (security, network reliability, transmission efficiency, energy efficiency) for WSNs. The proposed SEIRS model provides an improved technique to restraint worms’ transmission in comparison to the existing models.

First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model

The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions.

Permanence of a regime-switching malaria transmission model

Malaria transmission is a complex system, and lots of malaria transmission models have been showed by determined differential equations. Considering that variability of the environment may have an important impact on its transmission, we introduce the regime-switching to a malaria transmission model with susceptible–infectious–recovered–susceptible (SIRS) structure for human population and susceptible–infectious (SI) structure for mosquito population. First, we show the existence of positive solutions for the model. Then, the dynamics of the model is carried out. By constructing a suitable Lyapunov function, we show that there exists a stationary distribution, and so the threshold for the disease to prevail is given. At last an example with two states is given to illustrate the results.

Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching

Abstract We discuss the effect of introducing telegraph noise, which is an example of an environmental noise, into the susceptible–infectious–recovered–susceptible (SIRS) model by examining the model using a finite-state Markov Chain (MC). First we start with a two-state MC and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence. We then explain how the results can be generalised to a finite-state MC. The results for the SIR (Susceptible–Infectious–Removed) model with Markovian Switching (MS) are a special case. Numerical simulations are produced to confirm our theoretical results.

Mathematical Modelling of Onchocerciasis Transmission Potentials of Female Blackflies in Hyper-Endemic Communities after a Decade Annual Ivermectin Treatment

Hyper-Endemic communities are the major focus for the elimination of Onchocerciasis disease. This has been one of the major public health problems in tropical countries. In this paper, an SI (Susceptible - Infectious) and SIRS (Susceptible - Infectious - Recovered - Susceptible) compartmental model that estimates the Onchocerciasis transmission potentials of female blackflies in hyper-endemic communities after a decade long annual ivermectin treatment are proposed and analyzed. The data collected from the identified hyper-endemic communities in Imo and Abia State Nigeria, that shows the status of Onchocerciasis in some pre-community directed treatment with ivermectin (1994) hyper-endemic communities located in the Imo River Basin transmission foci of Onchocerciasis were analyzed. We show that the threshold that determines the stability of the equilibrium is the ivermectin administration reproduction number, which incorporates the female blackflies and humans living in hyper-endemic communities. The numerical results are presented and are found to be in agreement to the real life situation data.

Stochastic parameter search for events.

BackgroundWith recent increase in affordability and accessibility of high-performance computing (HPC), the use of large stochastic models has become increasingly popular for its ability to accurately mimic the behavior of the represented biochemical system. One important application of such models is to predict parameter configurations that yield an event of scientific significance. Due to the high computational requirements of Monte Carlo simulations and dimensionality of parameter space, brute force search is computationally infeasible for most large models.ResultsWe have developed a novel parameter estimation algorithm—Stochastic Parameter Search for Events (SParSE)—that automatically computes parameter configurations for propagating the system to produce an event of interest at a user-specified success rate and error tolerance. Our method is highly automated and parallelizable. In addition, computational complexity does not scale linearly with the number of unknown parameters; all reaction rate parameters are updated concurrently at the end of each iteration in SParSE. We apply SParSE to three systems of increasing complexity: birth-death, reversible isomerization, and Susceptible-Infectious-Recovered-Susceptible (SIRS) disease transmission. Our results demonstrate that SParSE substantially accelerates computation of the parametric solution hyperplane compared to uniform random search. We also show that the novel heuristic for handling over-perturbing parameter sets enables SParSE to compute biasing parameters for a class of rare events that is not amenable to current algorithms that are based on importance sampling.ConclusionsSParSE provides a novel, efficient, event-oriented parameter estimation method for computing parametric configurations that can be readily applied to any stochastic systems obeying chemical master equation (CME). Its usability and utility do not diminish with large systems as the algorithmic complexity for a given system is independent of the number of unknown reaction rate parameters.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-014-0126-y) contains supplementary material, which is available to authorized users.

Invasion and transmission of Salmonella Kentucky in an adult dairy herd using approximate Bayesian computation

BackgroundAn outbreak of Salmonella Kentucky followed by a high level of sustained endemic prevalence was recently observed in a US adult dairy herd enrolled in a longitudinal study involving intensive fecal sampling. To understand the invasion ability and transmission dynamics of Salmonella Kentucky in dairy cattle, accurate estimation of the key epidemiological parameters from longitudinal field data is necessary. The approximate Bayesian computation technique was applied for estimating the transmission rate (β), the recovery rate (γ) and shape (n) parameters of the gamma distribution for the infectious (shedding) period, and the basic reproduction ratio (R0), given a susceptible-infectious-recovered-susceptible (SIRS) compartment model with a gamma distribution for the infectious period.ResultsThe results report that the mean transmission rate (β) is 0.417 month-1 (median: 0.417, 95% credible interval [0.406, 0.429]), the average infectious period (γ-1) is 7.95 months (median: 7.95, 95% credible interval [7.70, 8.22]), the mean shape parameter (n) of the gamma distribution for the infectious period is 242 (median: 182, 95% credible interval [16, 482]), and the mean basic reproduction ratio (R0) is 2.91 (median: 2.91, 95% credible interval [2.83, 3.00]).ConclusionsThis study shows that Salmonella Kentucky in this herd was of mild infectiousness and had a long infectious period, which together provide an explanation for the observed prevalence pattern after invasion. The transmission rate and the recovery rate parameters are inferred with better accuracy than the shape parameter, therefore these two parameters are more sensitive to the model and the observed data. The estimated shape parameter (n) has large variability with a minimal value greater than one, indicating that the infectious period of Salmonella Kentucky in dairy cattle does not follow the conventionally assumed exponential distribution.

Inference of Type-Specific HPV Transmissibility, Progression and Clearance Rates: A Mathematical Modelling Approach

Quantifying rates governing the clearance of Human Papillomavirus (HPV) and its progression to clinical disease, together with viral transmissibility and the duration of naturally-acquired immunity, is essential in estimating the impact of vaccination programmes and screening or testing regimes. However, the complex natural history of HPV makes this difficult. We infer the viral transmissibility, rate of waning natural immunity and rates of progression and clearance of infection of 13 high-risk and 2 non-oncogenic HPV types, making use of a number of rich datasets from Sweden. Estimates of viral transmissibility, clearance of initial infection and waning immunity were derived in a Bayesian framework by fitting a susceptible-infectious-recovered-susceptible (SIRS) transmission model to age- and type-specific HPV prevalence data from both a cross-sectional study and a randomised controlled trial (RCT) of primary HPV screening. The models fitted well, but over-estimated the prevalence of four high-risk types with respect to the data. Three of these types (HPV-33, -35 and -58) are among the most closely related phylogenetically to the most prevalent HPV-16. The fourth (HPV-45) is the most closely related to HPV-18; the second most prevalent type. We suggest that this may be an indicator of cross-immunity. Rates of progression and clearance of clinical lesions were additionally estimated from longitudinal data gathered as part of the same RCT. Our estimates of progression and clearance rates are consistent with the findings of survival analysis studies and we extend the literature by estimating progression and clearance rates for non-16 and non-18 high-risk types. We anticipate that such type-specific estimates will be useful in the parameterisation of further models and in developing our understanding of HPV natural history.

THE INFLUENCE OF INCREASING LIFE EXPECTANCY ON THE DYNAMICS OF SIRS SYSTEMS WITH IMMUNE BOOSTING

AbstractEndemic infectious diseases constantly circulate in human populations, with prevalence fluctuating about a (theoretical and unobserved) time-independent equilibrium. For diseases for which acquired immunity is not lifelong, the classic susceptible–infectious–recovered–susceptible (SIRS) model provides a framework within which to consider temporal trends in the observed epidemiology. However, in some cases (notably pertussis), sustained multiannual fluctuations are observed, whereas the SIRS model is characterized by damped oscillatory dynamics for all biologically meaningful choices of model parameters. We show that a model that allows for “boosting” of immunity may naturally give rise to undamped oscillatory behaviour for biologically realistic parameter choices. The life expectancy of the population is critical in determining the characteristic dynamics of the system. For life expectancies up to approximately$50$years, we find that, even with boosting, damped oscillatory dynamics persist. For increasing life expectancy, the system may sustain oscillatory dynamics, or even exhibit bistable behaviour, in which both stable point attractor and limit cycle dynamics may coexist. Our results suggest that rising life expectancy may induce changes in the characteristic dynamics of infections for which immunity is not lifelong, with potential implications for disease control strategies.

Natural immune boosting in pertussis dynamics and the potential for long-term vaccine failure

Incidence of whooping cough, unlike many other childhood diseases for which there is an efficacious vaccine, has been increasing over the past twenty years despite high levels of vaccine coverage. Its reemergence has been particularly noticeable among teenagers and adults. Many hypotheses have been put forward to explain these two patterns, but parsimonious reconciliation of clinical data on the limited duration of immunity with both pre- and postvaccine era age-specific incidence remains a challenge. We consider the immunologically relevant, yet epidemiologically largely neglected, possibility that a primed immune system can respond to a lower dose of antigen than a naive one. We hypothesize that during the prevaccine era teenagers' and adults' primed immunity was frequently boosted by reexposure, so maintaining herd immunity in the face of potentially eroding individual immunity. In contrast, low pathogen circulation in the current era, except during epidemic outbreaks, allows immunity to be lost before reexposure occurs. We develop and analyze an age-structured model that encapsulates this hypothesis. We find that immune boosting must be more easily triggered than primary infection to account for age-incidence data. We make age-specific and dynamical predictions through bifurcation analysis and simulation. The boosting model proposed here parsimoniously captures four key features of pertussis data from highly vaccinated countries: (i) the shift in age-specific incidence, (ii) reemergence with high vaccine coverage, (iii) the possibility for cyclic dynamics in the pre- and postvaccine eras, and (iv) the apparent shift from susceptible-infectious-recovered (SIR)-like to susceptible-infectious-recovered-susceptible (SIRS)-like phenomenology of infection and immunity to Bordetella pertussis.

Fuzzy epidemic model for the transmission of worms in computer network

An e-epidemic SIRS (susceptible–infectious–recovered–susceptible) model for the fuzzy transmission of worms in computer network is formulated. We have analyzed the comparison between classical basic reproduction number and fuzzy basic reproduction number, that is, when both coincide and when both differ. The three cases of epidemic control strategies of worms in the computer network–low, medium, and, high–are analyzed, which may help us to understand the attacking behavior and also may lead to control of worms. Numerical methods are employed to solve and simulate the system of equations developed.

Calculation of Disease Dynamics in a Population of Households

Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-susceptible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks.