Heterogeneity in susceptibility dictates the order of epidemic models

Prior exposure to pathogens augments host heterogeneity in susceptibility and has key epidemiological consequences

Pathogen epidemics are key threats to human and wildlife health. Across systems, host protection from pathogens following initial exposure is often incomplete, resulting in recurrent epidemics through partially-immune hosts. Variation in population-level protection has important consequences for epidemic dynamics, but how acquired protection influences inter-individual heterogeneity in susceptibility and its epidemiological consequences remains understudied. We experimentally investigated whether prior exposure (none, low-dose, or high-dose) to a bacterial pathogen alters host heterogeneity in susceptibility among songbirds. Hosts with no prior pathogen exposure had little variation in protection, but heterogeneity in susceptibility was significantly augmented by prior pathogen exposure, with the highest variability detected in hosts given high-dose prior exposure. An epidemiological model parameterized with experimental data found that heterogeneity in susceptibility from prior exposure more than halved epidemic sizes compared with a homogeneous population with identical mean protection. However, because infection-induced mortality was also greatly reduced in hosts with prior pathogen exposure, reductions in epidemic size were smaller than expected in hosts with prior exposure. These results highlight the importance of variable protection from prior exposure and/or vaccination in driving population-level heterogeneity and epidemiological dynamics.

Prior exposure to pathogens augments host heterogeneity in susceptibility and has key epidemiological consequences

Pathogen epidemics are key threats to human and wildlife health. Across systems, host protection from pathogens following initial exposure is often incomplete, resulting in recurrent epidemics through partially-immune hosts. Variation in population-level protection has important consequences for epidemic dynamics, but how acquired protection influences inter-individual heterogeneity in susceptibility and its epidemiological consequences remains understudied. We experimentally investigated whether prior exposure (none, low-dose, or high-dose) to a bacterial pathogen alters host heterogeneity in susceptibility among songbirds. Hosts with no prior pathogen exposure had little variation in protection, but heterogeneity in susceptibility was significantly augmented by prior pathogen exposure, with the highest variability detected in hosts given high-dose prior exposure. An epidemiological model parameterized with experimental data found that heterogeneity in susceptibility from prior exposure more than halved epidemic sizes compared with a homogeneous population with identical mean protection. However, because infection-induced mortality was also greatly reduced in hosts with prior pathogen exposure, reductions in epidemic size were smaller than expected in hosts with prior exposure. These results highlight the importance of variable protection from prior exposure and/or vaccination in driving population-level heterogeneity and epidemiological dynamics.

Modelling flock heterogeneity in the transmission of peste des petits ruminants virus and its impact on the effectiveness of vaccination for eradication

Peste des petits ruminants (PPR) is an acute infectious disease of small ruminants targeted for global eradication by 2030. The Global Strategy for Control and Eradication (GSCE) recommends mass vaccination targeting 70% coverage of small ruminant populations in PPR-endemic regions. These small ruminant populations are diverse with heterogeneous mixing patterns that may influence PPR virus (PPRV) transmission dynamics. This paper evaluates the impact of heterogeneous mixing on (i) PPRV transmission and (ii) the likelihood of different vaccination strategies achieving PPRV elimination, including the GSCE recommended strategy. We develop models simulating heterogeneous transmission between hosts, including a metapopulation model of PPRV transmission between villages in lowland Ethiopia fitted to serological data. Our results demonstrate that although heterogeneous mixing of small ruminant populations increases the instability of PPRV transmission-increasing the chance of fadeout in the absence of intervention-a vaccination coverage of 70% may be insufficient to achieve elimination if high-risk populations are not targeted. Transmission may persist despite very high vaccination coverage (>90% small ruminants) if vaccination is biased towards more accessible but lower-risk populations such as sedentary small ruminant flocks. These results highlight the importance of characterizing small ruminant mobility patterns and identifying high-risk populations for vaccination and support a move towards targeted, risk-based vaccination programmes in the next phase of the PPRV eradication programme. Our modelling approach also illustrates a general framework for incorporating heterogeneous mixing patterns into models of directly transmitted infectious diseases where detailed contact data are limited. This study improves understanding of PPRV transmission and elimination in heterogeneous small ruminant populations and should be used to inform and optimize the design of PPRV vaccination programmes.

Beyond classical theories: An integrative mathematical model of mating dynamics and parental care

Classical theories, such as Bateman's principle and Trivers' parental investment theory, attempted to explain the coevolution of sexual selection and parental care through simple verbal arguments. Since then, quantitative models have demonstrated that it is rarely that simple because many non-intuitive structures and non-linear relationships are actually at play. In this study, we propose a new standard for models of mating dynamics and parental care, emphasizing the clarity and use of mathematical and probabilistic arguments, the meaning of consistency conditions, and the key role of spatial densities and the law of mass action. We used adaptive dynamics to calculate the evolutionary trajectory of the total care duration. Our results clearly show how the outcomes of parental care evolution can be diverse, depending on the quantitative balance between a set of dynamical forces arising from relevant differences and conditions in the male and female populations. The intensity of sexual selection, synergy of care, care quality, and relative mortality rates during mating interactions and caring activities act as forces driving evolutionary transitions between uniparental and biparental care. Sexual selection reduces the care duration of the selected sex, uniparental care evolves in the sex that offers the higher care quality, higher mortality during mating interactions of one sex leads to more care by that sex, and higher mortality during caring activities of one sex favours the evolution of uniparental care in the other sex. Both synergy and higher overall mortality during mating interactions can stabilize biparental care when sexual selection reduces the care duration of the selected sex. We discuss how the interaction between these forces influences the evolution of care patterns, and how sex ratios can vary and be interpreted in these contexts. We also propose new directions for future developments of our integrative model, creating new comparable analyses that share the same underlying assumptions and dynamical frameworks.

Quantifying individual-level heterogeneity in infectiousness and susceptibility through household studies

The spread of SARS-CoV-2, like that of many other pathogens, is governed by heterogeneity. "Superspreading," or "over-dispersion," is an important factor in transmission, yet it is hard to quantify. Estimates from contact tracing data are prone to potential biases due to the increased likelihood of detecting large clusters of cases, and may reflect variation in contact behavior more than biological heterogeneity. In contrast, the average number of secondary infections per contact is routinely estimated from household surveys, and these studies can minimize biases by testing all members of a household. However, the models used to analyze household transmission data typically assume that infectiousness and susceptibility are the same for all individuals or vary only with predetermined traits such as age. Here we develop and apply a combined forward simulation and inference method to quantify the degree of inter-individual variation in both infectiousness and susceptibility from observations of the distribution of infections in household surveys. First, analyzing simulated data, we show our method can reliably ascertain the presence, type, and amount of these heterogeneities given data from a sufficiently large sample of households. We then analyze a collection of household studies of COVID-19 from diverse settings around the world, and find strong evidence for large heterogeneity in both the infectiousness and susceptibility of individuals. Our results also provide a framework to improve the design of studies to evaluate household interventions in the presence of realistic heterogeneity between individuals.

Epidemic modeling with heterogeneity and social diffusion

We propose and analyze a family of epidemiological models that extend the classic Susceptible-Infectious-Recovered/Removed (SIR)-like framework to account for dynamic heterogeneity in infection risk. The family of models takes the form of a system of reaction-diffusion equations given populations structured by heterogeneous susceptibility to infection. These models describe the evolution of population-level macroscopic quantities S, I, R as in the classical case coupled with a microscopic variable f, giving the distribution of individual behavior in terms of exposure to contagion in the population of susceptibles. The reaction terms represent the impact of sculpting the distribution of susceptibles by the infection process. The diffusion and drift terms that appear in a Fokker-Planck type equation represent the impact of behavior change both during and in the absence of an epidemic. We first study the mathematical foundations of this system of reaction-diffusion equations and prove a number of its properties. In particular, we show that the system will converge back to the unique equilibrium distribution after an epidemic outbreak. We then derive a simpler system by seeking self-similar solutions to the reaction-diffusion equations in the case of Gaussian profiles. Notably, these self-similar solutions lead to a system of ordinary differential equations including classic SIR-like compartments and a new feature: the average risk level in the remaining susceptible population. We show that the simplified system exhibits a rich dynamical structure during epidemics, including plateaus, shoulders, rebounds and oscillations. Finally, we offer perspectives and caveats on ways that this family of models can help interpret the non-canonical dynamics of emerging infectious diseases, including COVID-19.

A systematic procedure for incorporating separable static heterogeneity into compartmental epidemic models

In this paper, we show how to modify a compartmental epidemic model, without changing the dimension, such that separable static heterogeneity is taken into account. The derivation is based on the Kermack-McKendrick renewal equation.

Quantifying individual-level heterogeneity in infectiousness and susceptibility through household studies

The spread of SARS-CoV-2, like that of many other pathogens, is governed by heterogeneity. "Superspreading," or "over-dispersion," is an important factor in transmission, yet it is hard to quantify. Estimates from contact tracing data are prone to potential biases due to the increased likelihood of detecting large clusters of cases, and may reflect variation in contact behavior more than biological heterogeneity. In contrast, the average number of secondary infections per contact is routinely estimated from household surveys, and these studies can minimize biases by testing all members of a household. However, the models used to analyze household transmission data typically assume that infectiousness and susceptibility are the same for all individuals or vary only with predetermined traits such as age. Here we develop and apply a combined forward simulation and inference method to quantify the degree of inter-individual variation in both infectiousness and susceptibility from observations of the distribution of infections in household surveys. First, analyzing simulated data, we show our method can reliably ascertain the presence, type, and amount of these heterogeneities with data from a sufficiently large sample of households. We then analyze a collection of household studies of COVID-19 from diverse settings around the world, and find strong evidence for large heterogeneity in both the infectiousness and susceptibility of individuals. Our results also provide a framework to improve the design of studies to evaluate household interventions in the presence of realistic heterogeneity between individuals.

A COVID-19 model incorporating variants, vaccination, waning immunity, and population behavior

Vaccines for COVID-19 have allowed countries to combat the spread of the disease. However, new variants have resulted in significant spikes in cases and raised severe health and economic concerns. We present a COVID-19 model to predict coupled effects of vaccine multiple-dose roll-out strategies, vaccine efficacy, waning immunity, population level of caution, sense of safety, under-reporting of cases, and highly prevalent variants such as the Delta (B.1.617.2) and Omicron (B.1.1.529). The modeling framework can incorporate new variants as they emerge to give critical insights into the new cases and guide public policy decision-making concerning vaccine roll-outs and reopening strategies. The model is shown to recreate the history of COVID-19 for five countries (Germany, India, Japan, South Africa, and the United States). Parameters for crucial aspects of the pandemic, such as population behavior, new variants, vaccination, and waning immunity, can be adjusted to predict pandemic scenarios. The model was used to conduct trend analysis to simulate pandemic dynamics taking into account the societal level of caution, societal sense of safety, and the proportions of individuals vaccinated with first, second, and booster doses. We used the results of serological testing studies to estimate the actual number of cases across countries. The model allows quantification of otherwise hard to quantify aspects such as the infectious power of variants and the effectiveness of government mandates and population behavior. Some example cases are presented by investigating the competitive nature of COVID variants and the effect of different vaccine distribution strategies between immunity groups.

Herd immunity under individual variation and reinfection

We study a susceptible-exposed-infected-recovered (SEIR) model considered by Aguas et al. (In: Herd immunity thresholds for SARS-CoV-2 estimated from unfolding epidemics, 2021), Gomes et al. (In: J Theor Biol. 540:111063, 2022) where individuals are assumed to differ in their susceptibility or exposure to infection. Under this heterogeneity assumption, epidemic growth is effectively suppressed when the percentage of the population having acquired immunity surpasses a critical level - the herd immunity threshold - that is lower than in homogeneous populations. We derive explicit formulas to calculate herd immunity thresholds and stable configurations, especially when susceptibility or exposure are gamma distributed, and explore extensions of the model.

SARS-CoV-2 transmission and impacts of unvaccinated-only screening in populations of mixed vaccination status

Screening programs that test only the unvaccinated population have been proposed and implemented to mitigate SARS-CoV-2 spread, implicitly assuming that the unvaccinated population drives transmission. To evaluate this premise and quantify the impact of unvaccinated-only screening programs, we introduce a model for SARS-CoV-2 transmission through which we explore a range of transmission rates, vaccine effectiveness scenarios, rates of prior infection, and screening programs. We find that, as vaccination rates increase, the proportion of transmission driven by the unvaccinated population decreases, such that most community spread is driven by vaccine-breakthrough infections once vaccine coverage exceeds 55% (omicron) or 80% (delta), points which shift lower as vaccine effectiveness wanes. Thus, we show that as vaccination rates increase, the transmission reductions associated with unvaccinated-only screening decline, identifying three distinct categories of impact on infections and hospitalizations. More broadly, these results demonstrate that effective unvaccinated-only screening depends on population immunity, vaccination rates, and variant.

SARS-CoV-2 Transmission and Impacts of Unvaccinated-Only Screening in Populations of Mixed Vaccination Status

Community testing programs focused on the unvaccinated population are being enacted in populations with mixed vaccination status to mitigate SARS-CoV-2 spread. Presumably, these policies assume that the unvaccinated are driving transmission, though it is not well understood how viral spread occurs in mixed-status populations. Here, we analyze a model of transmission in which a variable fraction of the population is vaccinated, with unvaccinated individuals proactively screened for infection. By exploring a range of transmission rates, vaccine effectiveness (VE) scenarios, and rates of prior infection, this analysis reveals principles of viral spread in communities of mixed vaccination status, with implications for screening policies. As vaccination rates increase, the proportion of transmission driven by the unvaccinated population decreases, such that most community spread is driven by breakthrough infections once vaccine coverage exceeds 55% (omicron) or 80% (delta), with additional variation dependent on waning or boosted VE. More broadly, the potential impacts of unvaccinated-only screening fall into three distinct parameter regions: (I) "flattening the curve" with little impact on cumulative infections, (II) effectively suppressing transmission, and (III) negligible impact because herd immunity is reached without screening. By evaluating a wide range of scenarios, this work finds broadly that effective mitigation of SARS-CoV-2 transmission by unvaccinated-only screening is highly dependent on vaccination rate, population-level immunity, screening compliance, and vaccine effectiveness against the current variant.

Stochastic social behavior coupled to COVID-19 dynamics leads to waves, plateaus, and an endemic state

It is well recognized that population heterogeneity plays an important role in the spread of epidemics. While individual variations in social activity are often assumed to be persistent, that is, constant in time, here we discuss the consequences of dynamic heterogeneity. By integrating the stochastic dynamics of social activity into traditional epidemiological models, we demonstrate the emergence of a new long timescale governing the epidemic, in broad agreement with empirical data. Our stochastic social activity model captures multiple features of real-life epidemics such as COVID-19, including prolonged plateaus and multiple waves, which are transiently suppressed due to the dynamic nature of social activity. The existence of a long timescale due to the interplay between epidemic and social dynamics provides a unifying picture of how a fast-paced epidemic typically will transition to an endemic state.