An Introduction to Networks in Epidemic Modeling

Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks

We incorporate the disease state and testing state into the formulation of a COVID-19 epidemic model. For this model, the basic reproduction number is identified and its dependence on model parameters related to the testing process and isolation efficacy is discussed. The relations between the basic reproduction number, the final epidemic and peak sizes, and the model parameters are further explored numerically. We find that fast test reporting does not always benefit the control of the COVID-19 epidemic if good quarantine while awaiting test results is implemented. Moreover, the final epidemic and peak sizes do not always increase along with the basic reproduction number. Under some circumstances, lowering the basic reproduction number increases the final epidemic and peak sizes. Our findings suggest that properly implementing isolation for individuals who are waiting for their testing results would lower the basic reproduction number as well as the final epidemic and peak sizes.

Avian influenza is a global public health threat. Since 2021, the ongoing H5N1 panzootic has brought a major shift in H5Nx epidemiology, including unprecedented spread, wide host range and lack of seasonality. Infections in marine mammals, wildlife and livestock have heightened concern for human-to-human transmission and pandemic potential. Contact tracing and self-isolation are used as public health measures in the UK to manage contacts of confirmed human cases of avian influenza. In this study, we aimed to estimate potential outbreak sizes and evaluate the effectiveness of contact tracing and self-isolation in managing community outbreaks of H5N1 following spillover from birds to people. We characterised contact patterns from an underrepresented agricultural population at high risk of avian influenza exposure through contact with birds (Avian Contact Study). Informed by these realistic social contact data, we modelled outbreak sizes using a stochastic branching process model. Most simulations resulted in small-scale outbreaks, ranging from 0 to 10 cases. When the basic reproduction number was 1.1, contact tracing and self-isolation reduced the average outbreak size from 41 cases (95% Confidence Interval (CI): 37–46 cases) to 7 cases (95% CI: 6–8 cases), preventing, on average, 8 out of every 10 infections. However, controls became less effective in reducing the outbreak size when a higher proportion of cases were asymptomatic. Overall, our findings suggest that contact tracing and self-isolation can be effective at preventing zoonotic infections. Increasing awareness, encouraging self-isolation, and detecting asymptomatic cases through routine surveillance are important components of zoonotic infection containment strategies.

Avian influenza is a global public health threat. Since 2021, the ongoing H5N1 panzootic has brought a major shift in H5Nx epidemiology, including unprecedented spread, wide host range and lack of seasonality. Infections in marine mammals, wildlife and livestock have heightened concern for human-to-human transmission and pandemic potential. Contact tracing and self-isolation are used as public health measures in the UK to manage contacts of confirmed human cases of avian influenza. In this study, we aimed to estimate potential outbreak sizes and evaluate the effectiveness of contact tracing and self-isolation in managing community outbreaks of H5N1 following spillover from birds to people. We characterised contact patterns from an underrepresented agricultural population at high risk of avian influenza exposure through contact with birds (Avian Contact Study). Informed by these realistic social contact data, we modelled outbreak sizes using a stochastic branching process model. Most simulations resulted in small-scale outbreaks, ranging from 0 to 10 cases. When the basic reproduction number was 1.1, contact tracing and self-isolation reduced the average outbreak size from 41 cases (95% Confidence Interval (CI): 37-46 cases) to 7 cases (95% CI: 6-8 cases), preventing, on average, 8 out of every 10 infections. However, controls became less effective in reducing the outbreak size when a higher proportion of cases were asymptomatic. Overall, our findings suggest that contact tracing and self-isolation can be effective at preventing zoonotic infections. Increasing awareness, encouraging self-isolation, and detecting asymptomatic cases through routine surveillance are important components of zoonotic infection containment strategies.

Large epidemics such as the recent Ebola crisis in West Africa occur when local efforts to contain outbreaks fail to overcome the probabilistic onward transmission to new locations. As a result, there may be large differences in total epidemic size from similar initial conditions. This work seeks to determine the extent to which the effects of behavior changes and metapopulation coupling on epidemic size can be characterized. While mathematical models have been developed to study local containment by social distancing, intervention and other behavior changes, their connection to larger-scale transmission is relatively underdeveloped. We make use of the assumption that behavior changes limit local transmission before susceptible depletion to develop a time-varying birth-death process capturing the dynamic decrease of the transmission rate associated with behavior changes. We derive an expression for the mean outbreak size of this model and show that the distribution of outbreak sizes is approximately geometric. This allows a probabilistic extension whereby infected individuals may initiate new outbreaks. From this model we characterize the overall epidemic size as a function of the behavior change rate and the probability that an infected individual starts a new outbreak. We find good agreement between the analytical results and stochastic simulations leading to novel findings including critical learning rates that demarcate large and small epidemic sizes.

Epidemics spreading in interconnected complex networks

BackgroundCholera epidemics continue to challenge disease control, particularly in fragile and conflict-affected states. Rapid detection and response to small cholera clusters is key for efficient control before an epidemic propagates. To understand the capacity for early response in fragile states, we investigated delays in outbreak detection, investigation, response, and laboratory confirmation, and we estimated epidemic sizes. We assessed predictors of delays, and annual changes in response time.MethodsWe compiled a list of cholera outbreaks in fragile and conflict-affected states from 2008 to 2019. We searched for peer-reviewed articles and epidemiological reports. We evaluated delays from the dates of symptom onset of the primary case, and the earliest dates of outbreak detection, investigation, response, and confirmation. Information on how the outbreak was alerted was summarized. A branching process model was used to estimate epidemic size at each delay. Regression models were used to investigate the association between predictors and delays to response.ResultsSeventy-six outbreaks from 34 countries were included. Median delays spanned 1–2 weeks: from symptom onset of the primary case to presentation at the health facility (5 days, IQR 5–5), detection (5 days, IQR 5–6), investigation (7 days, IQR 5.8–13.3), response (10 days, IQR 7–18), and confirmation (11 days, IQR 7–16). In the model simulation, the median delay to response (10 days) with 3 seed cases led to a median epidemic size of 12 cases (upper range, 47) and 8% of outbreaks ≥ 20 cases (increasing to 32% with a 30-day delay to response). Increased outbreak size at detection (10 seed cases) and a 10-day median delay to response resulted in an epidemic size of 34 cases (upper range 67 cases) and < 1% of outbreaks < 20 cases. We estimated an annual global decrease in delay to response of 5.2% (95% CI 0.5–9.6, p = 0.03). Outbreaks signaled by immediate alerts were associated with a reduction in delay to response of 39.3% (95% CI 5.7–61.0, p = 0.03).ConclusionsFrom 2008 to 2019, median delays from symptom onset of the primary case to case presentation and to response were 5 days and 10 days, respectively. Our model simulations suggest that depending on the outbreak size (3 versus 10 seed cases), in 8 to 99% of scenarios, a 10-day delay to response would result in large clusters that would be difficult to contain. Improving the delay to response involves rethinking the integration at local levels of event-based detection, rapid diagnostic testing for cluster validation, and integrated alert, investigation, and response.

Background: The World Health Organization declared the ongoing Zika virus (ZIKV) epidemic in the Americas a Public Health Emergency of International Concern on February 1, 2016. ZIKV disease in humans is characterized by a “dengue-like” syndrome including febrile illness and rash. However, ZIKV infection in early pregnancy has been associated with severe birth defects, including microcephaly and other developmental issues. Mechanistic models of disease transmission can be used to forecast trajectories and likely disease burden but are currently hampered by substantial uncertainty on the epidemiology of the disease (e.g., the role of asymptomatic transmission, generation interval, incubation period, and key drivers). When insight is limited, phenomenological models provide a starting point for estimation of key transmission parameters, such as the reproduction number, and forecasts of epidemic impact.Methods: We obtained daily counts of suspected Zika cases by date of symptoms onset from the Secretary of Health of Antioquia, Colombia during January-April 2016. We calibrated the generalized Richards model, a phenomenological model that accommodates a variety of early exponential and sub-exponential growth kinetics, against the early epidemic trajectory and generated predictions of epidemic size. The reproduction number was estimated by applying the renewal equation to incident cases simulated from the fitted generalized-growth model and assuming gamma or exponentially-distributed generation intervals derived from the literature. We estimated the reproduction number for an increasing duration of the epidemic growth phase.Results: The reproduction number rapidly declined from 10.3 (95% CI: 8.3, 12.4) in the first disease generation to 2.2 (95% CI: 1.9, 2.8) in the second disease generation, assuming a gamma-distributed generation interval with the mean of 14 days and standard deviation of 2 days. The generalized-Richards model outperformed the logistic growth model and provided forecasts within 22% of the actual epidemic size based on an assessment 30 days into the epidemic, with the epidemic peaking on day 36.Conclusion: Phenomenological models represent promising tools to generate early forecasts of epidemic impact particularly in the context of substantial uncertainty in epidemiological parameters. Our findings underscore the need to treat the reproduction number as a dynamic quantity even during the early growth phase, and emphasize the sensitivity of reproduction number estimates to assumptions on the generation interval distribution.

Age of infection (the time lapsed since infection) is an important factor to consider when modeling the transmission dynamics of influenza under the influence of antiviral treatment and drug-resistance. In this paper, we consider an influenza model which includes an age of infection. The model includes partial differential equations (PDEs) in order to describe the variable infectiousness and effect of antivirals during the infectious period. We derived the formulas for various reproduction numbers (RN) including \({\mathcal R_{SC}}\) (the controlled RN by one sensitive case), \({\mathcal R_{TC}}\) (the controlled total RN by one sensitive case), and \({\mathcal R_R}\) (the RN by one resistant case). The model analysis shows that \({\mathcal R_{SC}}\) and \({\mathcal R_R}\) determine both the global stability of the disease free equilibrium and the existence of the non-trivial equilibria. Local stabilities of the non-trivial equilibria are also discussed. Numerical simulations are conducted to not only confirm or extend the analytic results on qualitative behaviors of the system, but also reveal important quantitative properties of the disease dynamics influenced by antiviral treatment. These results are then used to assess the effectiveness of treatment programs in terms of both the RNs and the epidemic size. Our findings illustrate possibility that a higher level of antiviral use may lead to an increase of the epidemic size, despite the fact that there is a fitness cost of the drug-resistant strains. This suggests that programs for antiviral use should be designed carefully to avoid the adverse effect.

Seasonal variability and stochastic branching process in malaria outbreak probability

We investigate the stochastic susceptible-infected-recovered (SIR) model of infectious disease dynamics in the Fock-space approach. In contrast to conventional SIR models based on ordinary differential equationsfor the subpopulation sizes of S, I, and R individuals, the stochastic SIR model is driven by a master equationgoverning the transition probabilities among the system's states defined by SIR occupation numbers. In the Fock-space approach the master equationis recast in the form of a real-valued Schrödinger-type equationwith a second quantization Hamiltonian-like operator describing the infection and recovery processes. We find exact analytic expressions for the Hamiltonian eigenvalues for any population size N. We present small- and large-N results for the average numbers of SIR individuals and basic reproduction number. For small N we also obtain the probability distributions of SIR states, epidemic sizes and durations, which cannot be found from deterministic SIR models. Our Fock-space approach to stochastic SIR models introduces a powerful set of tools to calculate central quantities of epidemic processes, especially for relatively small populations where statistical fluctuations not captured by conventional deterministic SIR models play a crucial role.

The basic reproductive number, R(0), and the effective reproductive number, R, are commonly used in mathematical epidemiology as summary statistics for the size and controllability of epidemics. However, these commonly used reproductive numbers can be misleading when applied to predict pathogen evolution because they do not incorporate the impact of the timing of events in the life-history cycle of the pathogen. To study evolution problems where the host population size is changing, measures like the ultimate proliferation rate must be used. A third measure of reproductive success, which combines properties of both the basic reproductive number and the ultimate proliferation rate, is the discounted reproductive number R(d). The discounted reproductive number is a measure of reproductive success that is an individual's expected lifetime offspring production discounted by the background population growth rate. Here, we draw attention to the discounted reproductive number by providing an explicit definition and a systematic application framework. We describe how the discounted reproductive number overcomes the limitations of both the standard reproductive numbers and proliferation rates, and show that R(d) is closely connected to Fisher's reproductive values for different life-history stages.

Estimation of the reproduction number and identification of periodicity for HFMD infections in northwest China

The Covid-19 outbreak in Spain. A simple dynamics model, some lessons, and a theoretical framework for control response

Flu, a common respiratory disease is caused mainly by the influenza virus. The Avian influenza (H5N1) outbreaks, as well as the 2009 H1N1 pandemic, have heightened global concerns about the emergence of a lethal influenza virus capable of causing a catastrophic pandemic. During the early stages of an epidemic a favourable change in the behaviour of people can be of utmost importance. An economic status-based (higher and lower economic class) structured model is formulated to examine the behavioural effect in controlling influenza. Following that, we have introduced controls into the model to analyse the efficacy of antiviral treatment in restraining infections in both economic classes and examined an optimal control problem. We have obtained the reproduction number along with the final epidemic size for both the strata and the relation between reproduction number and epidemic size. Through numerical simulation and global sensitivity analysis, we have shown the importance of the parameters and on reproduction number. Our result shows that by increasing , and by decreasing , and , we can reduce the infection in both the economic group. As a result of our analysis, we have found that the reduction of infections and their level of adversity is directly influenced by positive behavioural patterns or changes as without control susceptible population is increased by , the infective population is decreased by and the recovered population is increased by in the higher economic group who opted changed behaviour as compared to the lower the economic group (people living with normal behaviour). Thus normal behaviour contributes to the spread and growth of viruses and adds to the hassle. We also examined how antiviral drug control impacts both economic strata and found that in the higher economic strata, the susceptible population increased by , the infective population decreased by and the recovered population improved by as compared to the lower economic group, the susceptible population has increased by , the infective population is decreased by and the recovered population is improved by . Our results enlighten the role that how different behaviour in separate socio-economic class plays an important role in changing the dynamics of the system and also affects the basic reproduction number. The results of our study suggest that it is important to adopt a modified behaviour like social distancing, wearing masks accompanying the time-dependent controls in the form of an antiviral drug's effectiveness to combat infections and increasing the proportion of the susceptible population.