A practical guide to mathematical methods for estimating infectious disease outbreak risks

Probability of extinction and peak time for multi-type epidemics with application to COVID-19 variants of concern

During the COVID-19 pandemic, the emergence of novel variants of concern (VoCs) prompted different responses from governments across the world aimed at mitigating the impacts of more transmissible or more harmful strains. We model the invasion of a novel VoC into a population with heterogeneous vaccine- and infection-acquired immunity using a multi-type branching process framework with immigration. We define the number of cases needed to be reached to ensure stochastic extinction of this strain is unlikely and, therefore, the strain has become established in the population. To estimate the first-passage time distribution to reach this number of cases we use a mixture of stochastic simulations and analytic results. The first-passage time distribution gives a time window that is useful for policymakers planning interventions aimed at suppressing or delaying the introduction of novel VoC. We apply our method to a model of COVID-19 in the United Kingdom, though our results are applicable to other pathogens and settings.

Heterogeneity in and correlation between host transmissibility and susceptibility can greatly impact epidemic dynamics

While it is well established that host heterogeneity in transmission and host heterogeneity in susceptibility each individually impact disease dynamics in characteristic ways, it is generally unknown how disease dynamics are impacted when both types of heterogeneity are simultaneously present. Here we explore this question. We first conducted a systematic review of published studies from which we determined that the effects of correlations have been drastically understudied. We then filled in the knowledge gaps by developing and analyzing a stochastic, individual-based SIR model that includes both heterogeneity in transmission and susceptibility and flexibly allows for positive or negative correlations between transmissibility and susceptibility. We found that in comparison to the uncorrelated case, positive correlations result in major epidemics that are larger, faster, and more likely, whereas negative correlations result in major epidemics that are smaller and less likely. We additionally found that, counter to the conventional wisdom that heterogeneity in susceptibility always reduces outbreak size, heterogeneity in susceptibility can lead to major epidemics that are larger and more likely than the homogeneous case when correlations between transmissibility and susceptibility are positive, but this effect only arises at small to moderateR 0 . Moreover, positive correlations can frequently lead to major epidemics even with subcriticalR 0 . To illustrate the potential importance of heterogeneity and correlations, we developed an SEIR model to describe mpox disease dynamics in New York City, demonstrating that the dynamics of a 2022 outbreak can be reasonably well explained by the presence of positive correlations between susceptibility and transmissibility. Ultimately, we show that correlations between transmissibility and susceptibility profoundly impact disease dynamics.

Modelling with SPEED: a Stochastic Predictor of Early Epidemic Detection

The frequency of emerging infectious disease outbreaks continues to rise, necessitating predictive frameworks for public health decision-making. This study introduces the Stochastic Predictor of Early Epidemic Detection (SPEED) model, an adaptation of the classic Susceptible-Infected-Recovered model, employing a Gillespie-like algorithm to simulate early-stage stochastic disease transmission. SPEED incorporates individual-level detection probabilities based on the infection time and the lag from GP consultation to lab confirmation. The model dynamically adjusts to public health responses by enhancing testing and reducing detection times once a single case has been identified. SPEED serves two key functionalities. First, as a statistical inference tool refining reproduction number estimates following the detection of a small number of cases. SPEED inference uses specified prior distributions for the reproduction number to provide reliable posterior estimates. Second, to simulate epidemic scenarios under specified values of the reproduction number in order to construct a distribution of the time to subsequent detections. The model is used to evaluate how second case timings can rule out higher values of the reproduction number. Comparisons with simulations under heightened surveillance scenarios demonstrate the model's utility in assessing response efficacy on the initial outbreak spread. Our results demonstrate SPEED applied to a single case of influenza A(H1N2)v, detected through routine flu surveillance on the 23rd November 2023.

Reducing transmission in multiple settings is required to eliminate the risk of major Ebola outbreaks: a mathematical modelling study

The Ebola virus (EV) persists in animal populations, with zoonotic transmission to humans occurring every few months or years. When zoonotic transmission arises, it is important to understand which interventions are most effective at preventing a major outbreak driven by human-to-human transmission. Here, we analyse a mathematical model of EV transmission and calculate the probability of a major outbreak starting from a single introduced case. We consider community, funeral and healthcare facility transmission and conduct sensitivity analyses to explore the effects of non-pharmaceutical interventions (NPIs) that influence these transmission routes. We find that, if the index case is treated in the community, then the elimination of transmission at funerals reduces the probability of a major outbreak substantially (from 0.410 to 0.066 under our baseline model parametrization). However, eliminating the risk of major outbreaks entirely requires combinations of measures that limit transmission in different settings, such as community engagement to promote safe burial practices and implementation of barrier nursing in healthcare facilities. In addition to generating insights into the drivers of Ebola outbreaks, this research provides a modelling framework for assessing the effectiveness of interventions at mitigating outbreaks of other infectious diseases with transmission in multiple settings.

Investigating seasonal disease emergence and extinction in stochastic epidemic models

Seasonal disease outbreaks are common in many infectious diseases such as seasonal influenza, Zika, dengue fever, Lyme disease, malaria, and cholera. Seasonal outbreaks are often due to weather patterns affecting pathogens or disease-carrying vectors or by social behavior. We investigate disease emergence and extinction in seasonal stochastic epidemic models. Specifically, we study disease emergence through seasonally varying parameters for transmission, recovery, and vector births and deaths in time-nonhomogeneous Markov chains for SIR, SEIR, and vector-host systems. A branching process approximation of the Markov chain is used to estimate the seasonal probabilities of disease extinction. Several disease outcome measures are used to compare the dynamics in seasonal and constant environments. Numerical investigations illustrate and confirm previous results derived from stochastic epidemic models. Seasonal environments often result in lower probabilities of disease emergence and smaller values of the basic reproduction number than in constant environments, and the time of peak emergence generally precedes the peak time of the seasonal driver. We identify some new results when both transmission and recovery vary seasonally. If the relative amplitude of the recovery exceeds that of transmission or if the periodicity is not synchronized in time, lower average probabilities of disease emergence occur in a constant environment than in a seasonal environment. We also investigate the timing of vector control. This investigation provides new methods and outcome measures to study seasonal infectious disease dynamics and offers new insights into the timing of prevention and control.

Epidemiological Dynamics in Populations Structured by Neighbourhoods and Households

Epidemiological dynamics are affected by the spatial and demographic structure of the host population. Households and neighbourhoods are known to be important groupings but little is known about the epidemiological interplay between them. In order to explore the implications for infectious disease epidemiology of households with similar demographic structures clustered in space we develop a multi-scale epidemic model consisting of neighbourhoods of households. In our analysis we focus on key parameters which control household size, the importance of transmission within households relative to outside of them, and the degree to which the non-household transmission is localised within neighbourhoods. We construct the household reproduction number R ∗ over all neighbourhoods and derive the analytic probability of an outbreak occurring from a single infected individual in a specific neighbourhood. We find that reduced localisation of transmission within neighbourhoods reduces R ∗ when household size differs between neighbourhoods. This effect is amplified by larger differences between household sizes and larger divergence between transmission rates within households and outside of them. However, the impact of neighbourhoods with larger household sizes on an individual's risk of infection is mainly limited to the individuals that reside in those neighbourhoods. We consider various surveillance scenarios and show that household size information from the initial infectious cases is often more important than neighbourhood information while household size and neighbourhood localisation influences the sequence of neighbourhoods in which an outbreak is observed.

The effect of pathogens from environmental breeding and accumulative release by the infected individuals on spread dynamics of a SIRP epidemic model

In this paper, a SIRP epidemic model is proposed, wherein the pathogens derive from two ways, i.e., environmental breeding, and accumulative excretion by the infected individuals. The former is characterized by Logistic growth, while the latter is in the form of infinite integral. First, the positivity and ultimate boundedness of solutions are obtained. Second, the basic reproduction number R 0 is obtained, by which the model is analyzed if either the intrinsic growth rate of environmental pathogens is lower or higher than its clearance rate. For the first case, the disease-free equilibrium is globally asymptotically stable whenR 0 < 1 , while the endemic equilibrium is globally asymptotically stable whenR 0 > 1 . Conversely, if the growth rate exceeds the removal rate, the disease-free equilibrium is always unstable, meanwhile, the uniform persistence of the model indicates that there could exist one or multi-endemic equilibria, and it is globally asymptotically stable if the endemic equilibrium is unique. Finally, the theoretical results are illustrated by numerical simulations. We find that the accumulative release of pathogens by the infected individuals in the form of infinite integral is more realistic and consistent with the disease spread than that of linear form by real data.

Disease X epidemic control using a stochastic model and a deterministic approximation: Performance comparison with and without parameter uncertainties

BACKGROUND

The spread of infectious diseases can be modeled using deterministic or stochastic models. A deterministic approximation of a stochastic model can be appropriate under some conditions, but is unable to capture the discrete nature of populations. We look into the choice of a model from the perspective of decision making.

METHOD

We consider an emerging disease (Disease X) in a closed population modeled by a stochastic SIR model or its deterministic approximation. The objective of the decision maker is to minimize the cumulative number of symptomatic infected-days over the course of the epidemic by picking a vaccination policy. We consider four decision making scenarios: based on the stochastic model or the deterministic model, and with or without parameter uncertainty. We also consider different sample sizes for uncertain parameter draws and stochastic model runs. We estimate the average performance of decision making in each scenario and for each sample size.

RESULTS

The model used for decision making has an influence on the picked policies. The best achievable performance is obtained with the stochastic model, knowing parameter values, and for a large sample size. For small sample sizes, the deterministic model can outperform the stochastic model due to stochastic effects. Resolving uncertainties may bring more benefit than switching to the stochastic model in our example.

CONCLUSION

This article illustrates the interplay between the choice of a type of model, parameter uncertainties, and sample sizes. It points to issues to be considered when optimizing a stochastic model.

Is SARS-CoV-2 elimination or mitigation best? Regional and disease characteristics determine the recommended strategy

Public health responses to the COVID-19 pandemic varied across the world. Some countries (e.g. mainland China, New Zealand and Taiwan) implemented elimination strategies involving strict travel measures and periods of rigorous non-pharmaceutical interventions (NPIs) in the community, aiming to achieve periods with no disease spread; while others (e.g. many European countries and the USA) implemented mitigation strategies involving less strict NPIs for prolonged periods, aiming to limit community spread. Travel measures and community NPIs have high economic and social costs, and there is a need for guidelines that evaluate the appropriateness of an elimination or mitigation strategy in regional contexts. To guide decisions, we identify key criteria and provide indicators and visualizations to help answer each question. Considerations include determining whether disease elimination is: (1) necessary to ensure healthcare provision; (2) feasible from an epidemiological point of view and (3) cost-effective when considering, in particular, the economic costs of travel measures and treating infections. We discuss our recommendations by considering the regional and economic variability of Canadian provinces and territories, and the epidemiological characteristics of different SARS-CoV-2 variants. While elimination may be a preferable strategy for regions with limited healthcare capacity, low travel volumes, and few ports of entry, mitigation may be more feasible in large urban areas with dense infrastructure, strong economies, and with high connectivity to other regions.

Analysis of the risk and pre-emptive control of viral outbreaks accounting for within-host dynamics: SARS-CoV-2 as a case study

In the era of living with COVID-19, the risk of localised SARS-CoV-2 outbreaks remains. Here, we develop a multiscale modelling framework for estimating the local outbreak risk for a viral disease (the probability that a major outbreak results from a single case introduced into the population), accounting for within-host viral dynamics. Compared to population-level models previously used to estimate outbreak risks, our approach enables more detailed analysis of how the risk can be mitigated through pre-emptive interventions such as antigen testing. Considering SARS-CoV-2 as a case study, we quantify the within-host dynamics using data from individuals with omicron variant infections. We demonstrate that regular antigen testing reduces, but may not eliminate, the outbreak risk, depending on characteristics of local transmission. In our baseline analysis, daily antigen testing reduces the outbreak risk by 45% compared to a scenario without antigen testing. Additionally, we show that accounting for heterogeneity in within-host dynamics between individuals affects outbreak risk estimates and assessments of the impact of antigen testing. Our results therefore highlight important factors to consider when using multiscale models to design pre-emptive interventions against SARS-CoV-2 and other viruses.