Simple Epidemic Models Research Articles

Real-time estimates of the emergence and dynamics of SARS-CoV-2 variants of concern: A modeling approach

The emergence of SARS-CoV-2 variants of concern (VOCs) punctuated the dynamics of the COVID-19 pandemic in multiple occasions. The stages subsequent to their identification have been particularly challenging due to the hurdles associated with a prompt assessment of transmissibility and immune evasion characteristics of the newly emerged VOC. Here, we retrospectively analyze the performance of a modeling strategy developed to evaluate, in real-time, the risks posed by the Alpha and Omicron VOC soon after their emergence. Our approach utilized multi-strain, stochastic, compartmental models enriched with demographic information, age-specific contact patterns, the influence of non-pharmaceutical interventions, and the trajectory of vaccine distribution. The models’ preliminary assessment about Omicron’s transmissibility and immune evasion closely match later findings. Additionally, analyses based on data collected since our initial assessments demonstrate the retrospective accuracy of our real-time projections in capturing the emergence and subsequent dominance of the Alpha VOC in seven European countries and the Omicron VOC in South Africa. This study shows the value of relatively simple epidemic models in assessing the impact of emerging VOCs in real time, the importance of timely and accurate data, and the need for regular evaluation of these methodologies as we prepare for future global health crises.

Bifurcation analysis of a two infection SIR-SIR epidemic model with temporary immunity and disease enhancement

In this paper we study a two infection SIR-SIR compartmental model, considering biological features described in dengue fever epidemiology. Due to a progressive loss of protective antibodies there is waning immunity in the first infection stage and disease enhancement or protection effects by the second infection stage. Bifurcation analysis reveals two codim-2 bifurcations as organizing centers. The unfolding of a cusp bifurcation describes the transition of the disease-free equilibrium into an endemic equilibrium by varying a parameter. These equilibria allow an analytical solution with explicit expressions which allow for a full geometrical interpretation of the occurring bifurcations related to stationary dynamics. A Bogdanov-Takens point is the starting point in the parameter space where oscillatory endemic dynamics occurs including a homoclinic connection. These findings bring additional insights on biological mechanisms able to generate rich and complicated dynamical behavior in simple epidemic models that are, so far, largely unexplored.

Can infectious diseases eradicate host species? The effect of infection-age structure.

It is a fundamental question in mathematical epidemiology whether deadly infectious diseases only lead to a mere decline of their host populations or whether they can cause their complete disappearance. Upper density-dependent incidences do not lead to host extinction in simple, deterministic SI or SIS (susceptible-infectious) epidemic models. Infection-age structure is introduced into SIS models because of the biological accuracy offered by considering arbitrarily distributed infectious periods. In an SIS model with infection-age structure, survival of the susceptible host population is established for incidences that depend on the infection-age density in a general way. This confirms previous host persistence results without infection-age for incidence functions that are not generalizations of frequency-dependent transmission. For certain power incidences, hosts persist if some infected individuals leave the infected class and become susceptible again and the return rate dominates the infection-age dependent infectivity in a sufficient way. The hosts may be driven into extinction by the infectious disease if there is no return into the susceptible class at all.

The advantage of self-protecting interventions in mitigating epidemic circulation at the community level

Protecting interventions of many types (both pharmaceutical and non-pharmaceutical) can be deployed against the spreading of a communicable disease, as the worldwide COVID-19 pandemic has dramatically shown. Here we investigate in detail the effects at the population level of interventions that provide an asymmetric protection between the people involved in a single interaction. Masks of different filtration types, either protecting mainly the wearer or the contacts of the wearer, are a prominent example of these interventions. By means of analytical calculations and extensive simulations of simple epidemic models on networks, we show that interventions protecting more efficiently the adopter (e.g the mask wearer) are more effective than interventions protecting primarily the contacts of the adopter in reducing the prevalence of the disease and the number of concurrently infected individuals (“flattening the curve”). This observation is backed up by the study of a more realistic epidemic model on an empirical network representing the patterns of contacts in the city of Portland. Our results point out that promoting wearer-protecting face masks and other self-protecting interventions, though deemed selfish and inefficient, can actually be a better strategy to efficiently curtail pandemic spreading.

Estimating the effective reproduction number for heterogeneous models using incidence data.

The effective reproduction number, , plays a key role in the study of infectious diseases, indicating the current average number of new infections caused by an infected individual in an epidemic process. Estimation methods for the time evolution of , using incidence data, rely on the generation interval distribution, g(τ), which is usually obtained from empirical data or theoretical studies using simple epidemic models. However, for systems that present heterogeneity, either on the host population or in the expression of the disease, there is a lack of data and of a suitable general methodology to obtain g(τ). In this work, we use mathematical models to bridge this gap. We present a general methodology for obtaining explicit expressions of the reproduction numbers and the generation interval distributions, within and between model sub-compartments provided by an arbitrary compartmental model. Additionally, we present the appropriate expressions to evaluate those reproduction numbers using incidence data. To highlight the relevance of such methodology, we apply it to the spread of COVID-19 in municipalities of the state of Rio de Janeiro, Brazil. Using two meta-population models, we estimate the reproduction numbers and the contributions of each municipality in the generation of cases in all others.

Simple epidemic models with segmentation can be better than complex ones.

Given a sequence of epidemic events, can a single epidemic model capture its dynamics during the entire period? How should we divide the sequence into segments to better capture the dynamics? Throughout human history, infectious diseases (e.g., the Black Death and COVID-19) have been serious threats. Consequently, understanding and forecasting the evolving patterns of epidemic events are critical for prevention and decision making. To this end, epidemic models based on ordinary differential equations (ODEs), which effectively describe dynamic systems in many fields, have been employed. However, a single epidemic model is not enough to capture long-term dynamics of epidemic events especially when the dynamics heavily depend on external factors (e.g., lockdown and the capability to perform tests). In this work, we demonstrate that properly dividing the event sequence regarding COVID-19 (specifically, the numbers of active cases, recoveries, and deaths) into multiple segments and fitting a simple epidemic model to each segment leads to a better fit with fewer parameters than fitting a complex model to the entire sequence. Moreover, we propose a methodology for balancing the number of segments and the complexity of epidemic models, based on the Minimum Description Length principle. Our methodology is (a) Automatic: not requiring any user-defined parameters, (b) Model-agnostic: applicable to any ODE-based epidemic models, and (c) Effective: effectively describing and forecasting the spread of COVID-19 in 70 countries.

Early warning signals of infectious disease transitions: a review.

Early warning signals (EWSs) are a group of statistical time-series signals which could be used to anticipate a critical transition before it is reached. EWSs are model-independent methods that have grown in popularity to support evidence of disease emergence and disease elimination. Theoretical work has demonstrated their capability of detecting disease transitions in simple epidemic models, where elimination is reached through vaccination, to more complex vector transmission, age-structured and metapopulation models. However, the exact time evolution of EWSs depends on the transition; here we review the literature to provide guidance on what trends to expect and when. Recent advances include methods which detect when an EWS becomes significant; the earlier an upcoming disease transition is detected, the more valuable an EWS will be in practice. We suggest that future work should firstly validate detection methods with synthetic and historical datasets, before addressing their performance with real-time data which is accruing. A major challenge to overcome for the use of EWSs with disease transitions is to maintain the accuracy of EWSs in data-poor settings. We demonstrate how EWSs behave on reported cases for pertussis in the USA, to highlight some limitations when detecting disease transitions with real-world data.

COVID-19 in Chile: The usefulness of simple epidemic models in practice.

The purpose of this article is to describe and develop the predictive value of three models during the COVID-19 epidemic in Chile, providing knowledge for decision-making in health. We developed three models during the epidemic: a discrete model to predict the maximum burden on the health system in a short time framea basic SEIR (susceptible-exposed-infected-removed) model with discrete equations; a stochastic SEIR model with the Monte Carlo method; and a Gompertz-type model for metropolitan city of Santiago. The maximum potential burden model has been useful throughout the monitoring of the epidemic, providing an upper bound for the number of cases, intensive care unit occupancy, and deaths. Deterministic and stochastic SEIR models were very useful in predicting the rise of cases and the peak and onset of case decline; however, they lost utility in the current situation due to the asynchronous recruitment of cases in the regions and the persistence of a strong endemic. The Gompertz model had a better fit in the decline since it best captures the epidemic curves asymmetry in Santiago. The models have shown great utility in monitoring the epidemic in Chile, with different objectives in different epidemic stages. They have complemented empirical indicators such as reported cases, fatality, deaths, and others, making it possible to predict situations of interest and visualization of the short and long-term local behavior of this pandemic.

COVID-19 intervention models: An initial aggressive treatment strategy for controlling the infection.

The novel coronavirus (COVID-19) outbreak emerged in December 2019. The disease has caused loss of many lives and has become an unprecedented threat to public health worldwide. We develop simple COVID-19 epidemic models to study treatment strategies to control the pandemic. The results show that eradication of the disease is possible if the efficacy of treatment is perfect. We also investigate the existence of a dual-rate effect. Conditions under which the effect occurs are derived. When the effect is present, a tactic to control the infection might be to initially treat infected individuals aggressively at a relatively high rate to drive the prevalence to a lower region that can be maintained in the long run at relatively moderate rate and cost.

Are Models Useful? Reflections on Simple Epidemic Projection Models and the Covid-19 Pandemic.

Are Models Useful? Reflections on Simple Epidemic Projection Models and the Covid-19 Pandemic.

Coinfections by noninteracting pathogens are not independent and require new tests of interaction.

If pathogen species, strains, or clones do not interact, intuition suggests the proportion of coinfected hosts should be the product of the individual prevalences. Independence consequently underpins the wide range of methods for detecting pathogen interactions from cross-sectional survey data. However, the very simplest of epidemiological models challenge the underlying assumption of statistical independence. Even if pathogens do not interact, death of coinfected hosts causes net prevalences of individual pathogens to decrease simultaneously. The induced positive correlation between prevalences means the proportion of coinfected hosts is expected to be higher than multiplication would suggest. By modelling the dynamics of multiple noninteracting pathogens causing chronic infections, we develop a pair of novel tests of interaction that properly account for nonindependence between pathogens causing lifelong infection. Our tests allow us to reinterpret data from previous studies including pathogens of humans, plants, and animals. Our work demonstrates how methods to identify interactions between pathogens can be updated using simple epidemic models.

Social and Networking Effects on the Online Diffusion of Opinion on a UK Government Petition - A Phase Transition?

The UK online Government petition ‘Revoke Article 50’ was both the largest and fastest growing petition ever registered, peak growth achieved 60,000 signatures per hour with the total number reaching 6,087,845. The pattern of signatures followed a sinusoidal curve, consistent with the logistic process and simple epidemic diffusion models such as S:I (susceptible, infected). This suggests that the growth rate of signatures is proportional to the frequency of contacts between actors as opposed to a direct transmission from a single source. I argue that this is also consistent with the “two step flow” (Lazarsfeld & Katz) theory of message diffusion. The assumption behind the logistics process and the S:I model is that of a well mixed homogenous population. Networks like Twitter and Facebook are: homophilic, small-world, long-tailed, exhibit one or more “large components”, and so on. As an alternative I consider threshold models. In particular the work of Watts. The two step flow theory is controversial. Watts,questions whether or not influentials are really influential. He develops simulations to sustain this. These however are centred on random (E_R) networks, and the arguments do not work so well in the context of scale free networks (A_R), where hyper-influentials do exist as the root of the largest trees in s-f distributions. Empirically hyper-influentials did play a role in disseminating Revoke Art 50 retweets. There is psychological evidence of thresholds (personality) in both voting and in following particular websites and that these traits occur in geographical clusters Clusters of vote Remain voters and Revoke Art 50 endorsers are compared. A 1000 vertex scale free network was simulated and diffusion processes with high and low density, central and marginal original tweeters (i.e., either inside or outside a giant connected component, high and low threshold barriers. The objective being to simulate whether Watt’s transition point (“cascade conditions”) would in fact trigger a global cascade as did indeed happen empirically. This did, offer support for Watt’s conjecture but in the context of scale free rather than random networks. A threshold process of this kind could be operating on social networks and that the phase change and subsequent global cascade were the result of the endorsement of “hyper-influentials” who in effect engineered a transition to a high density, low threshold giant hub. I also argue for a qualified revival of the “two step” flow model with hyper-influentials.

Learning the seasonality of disease incidences from empirical data

Investigating the seasonality of disease incidences is very important in disease surveillance in regions with periodical climatic patterns. In lieu of the paradigm about disease incidences varying seasonally in line with meteorology, this work seeks to determine how well simple epidemic models can capture such seasonality for better forecasts and optimal futuristic interventions. Once incidence data are assimilated by a periodic model, asymptotic analysis in relation to the long-term behavior of the disease occurrences can be performed using the classical Floquet theory, which explains the stability of the existing periodic solutions. For an illustrative case, we employed infected-recovered models with an infection rate of a single period and that of commensurate periods to assimilate weekly dengue incidence data from the city of Jakarta, Indonesia, which we present in their raw and moving-average-filtered version. To estimate the infection rate of a single period, eight optimization schemes were assigned returning magnitudes of the rate that vary insignificantly across schemes. Three schemes were assigned to estimate the infection rate of commensurate periods based on three different sets of periods used. Each scheme involving commensurate periods gives better fitting than that involving only a single period. The computation results combined with the analytical results indicate that if the disease surveillance in the city does not improve, then the incidence will raise to a certain positive orbit and remain cyclical.

Structural and practical identifiability analysis of outbreak models

Estimating the reproduction number of an emerging infectious disease from an epidemiological data is becoming more essential in evaluating the current status of an outbreak. However, these studies are lacking the fundamental prerequisite in parameter estimation problem, namely the structural identifiability of the epidemic model, which determines the possibility of uniquely determining the model parameters from the epidemic data. In this paper, we perform both structural and practical identifiability analysis to classical epidemic models such as SIR (Susceptible-Infected-Recovered), SEIR (Susceptible-Exposed-Infected-Recovered) and an epidemic model with the treatment class (SITR). We performed structural identifiability analysis on these epidemic models using a differential algebra approach to investigate the well-posedness of the parameter estimation problem. Parameters of these models are estimated from different data types, namely prevalence, cumulative incidences and treated individuals. Furthermore, we carried out practical identifiability analysis on these models using Monte Carlo simulations and Fisher’s Information Matrix. Our study shows that the SIR model is both structurally and practically identifiable from the prevalence data. It is also structurally identifiable to cumulative incidence observations, but due to high correlations of the parameters, it is practically unidentifiable from the cumulative incidence data. Furthermore, we found that none of these simple epidemic models are practically identifiable from the cumulative incidence data which is the standard type of epidemiological data provided by CDC or WHO. Our analysis with simple SIR model suggest that the health agencies, if possible, should report prevalence rather than incidence data.

A primer on stable parameter estimation and forecasting in epidemiology by a problem-oriented regularized least squares algorithm.

Public health officials are increasingly recognizing the need to develop disease-forecasting systems to respond to epidemic and pandemic outbreaks. For instance, simple epidemic models relying on a small number of parameters can play an important role in characterizing epidemic growth and generating short-term epidemic forecasts. In the absence of reliable information about transmission mechanisms of emerging infectious diseases, phenomenological models are useful to characterize epidemic growth patterns without the need to explicitly model transmission mechanisms and the natural history of the disease. In this article, our goal is to discuss and illustrate the role of regularization methods for estimating parameters and generating disease forecasts using the generalized Richards model in the context of the 2014–15 Ebola epidemic in West Africa.

Rich Bifurcation Structure in a Two-Patch Vaccination Model

We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible--infected--vaccinated--susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifur...

Reduction of nonautonomous population dynamics models with two time scales.

The purpose of this work is reviewing some reduction results to deal with systems of nonautonomous ordinary differential equations with two time scales. They could be included among the so-called approximate aggregation methods. The existence of different time scales in a system, together with some long-term features, are used to build up a simpler system governed by a lesser number of state variables. The asymptotic behavior of the latter system is then used to describe the asymptotic behaviour of the former one. The reduction results are stated in two particular but important cases: periodic systems and asymptotically autonomous systems. The reduction results are illustrated with the help of simple spatial SIS epidemic models including either periodic or asymptotically autonomous terms.

Spatial heterogeneity in simple deterministic SIR models assessed ecologically

Patchy or divided populations can be important to infectious disease transmission. We first show that Lloyd’s mean crowding index, an index of patchiness from ecology, appears as a term in simple deterministic epidemic models of the SIR type. Using these models, we demonstrate that the rate of movement between patches is crucial for epidemic dynamics. In particular, there is a relationship between epidemic final size and epidemic duration in patchy habitats: controlling inter-patch movement will reduce epidemic duration, but also final size. This suggests that a strategy of quarantining infected areas during the initial phases of a virulent epidemic might reduce epidemic duration, but leave the population vulnerable to future epidemics by inhibiting the development of herd immunity. doi:10.1017/S1446181113000035

SPATIAL HETEROGENEITY IN SIMPLE DETERMINISTIC SIR MODELS ASSESSED ECOLOGICALLY

AbstractPatchy or divided populations can be important to infectious disease transmission. We first show that Lloyd’s mean crowding index, an index of patchiness from ecology, appears as a term in simple deterministic epidemic models of the SIR type. Using these models, we demonstrate that the rate of movement between patches is crucial for epidemic dynamics. In particular, there is a relationship between epidemic final size and epidemic duration in patchy habitats: controlling inter-patch movement will reduce epidemic duration, but also final size. This suggests that a strategy of quarantining infected areas during the initial phases of a virulent epidemic might reduce epidemic duration, but leave the population vulnerable to future epidemics by inhibiting the development of herd immunity.

Analytic Comparison of Some Epidemic Models with Vaccination

In this paper, we discuss the elementary properties of some simple SI, SR, SIR and SEIR epidemic models whose parameterizing functions (such as per-capita death rate, disease transmission, removal rate etc.) might be eventually time-varying but nonnecessarily time-integrable. Vaccination rules based of feedback, measuring the numbers of some of the partial populations defining the disease progress, are also discussed.