How population heterogeneity in susceptibility and infectivity influences epidemic 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 whether acquired protection influences host heterogeneity in susceptibility and its epidemiological consequences remains unexplored. 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 host heterogeneity and epidemiological dynamics.

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.

A note on variable susceptibility, the herd-immunity threshold and modeling of infectious diseases

The unfolding of the COVID-19 pandemic has been very difficult to predict using mathematical models for infectious diseases. While it has been demonstrated that variations in susceptibility have a damping effect on key quantities such as the incidence peak, the herd-immunity threshold and the final size of the pandemic, this complex phenomenon is almost impossible to measure or quantify, and it remains unclear how to incorporate it for modeling and prediction. In this work we show that, from a modeling perspective, variability in susceptibility on an individual level is equivalent with a fraction θ of the population having an "artificial" sterilizing immunity. We also derive novel formulas for the herd-immunity threshold and the final size of the pandemic, and show that these values are substantially lower than predicted by the classical formulas, in the presence of variable susceptibility. In the particular case of SARS-CoV-2, there is by now undoubtedly variable susceptibility due to waning immunity from both vaccines and previous infections, and our findings may be used to greatly simplify models. If such variations were also present prior to the first wave, as indicated by a number of studies, these findings can help explain why the magnitude of the initial waves of SARS-CoV-2 was relatively low, compared to what one may have expected based on standard models.

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.

Uncertainty quantification for ecological models with random parameters

There is often considerable uncertainty in parameters in ecological models. This uncertainty can be incorporated into models by treating parameters as random variables with distributions, rather than fixed quantities. Recent advances in uncertainty quantification methods, such as polynomial chaos approaches, allow for the analysis of models with random parameters. We introduce these methods with a motivating case study of sea ice algal blooms in heterogeneous environments. We compare Monte Carlo methods with polynomial chaos techniques to help understand the dynamics of an algal bloom model with random parameters. Modelling key parameters in the algal bloom model as random variables changes the timing, intensity and overall productivity of the modelled bloom. The computational efficiency of polynomial chaos methods provides a promising avenue for the broader inclusion of parametric uncertainty in ecological models, leading to improved model predictions and synthesis between models and data.

Strain wars 2: Binding constants, enthalpies, entropies, Gibbs energies and rates of binding of SARS-CoV-2 variants

SARS-CoV-2 virus is the cause of COVID-19 pandemic and belongs to RNA viruses, showing great tendency to mutate. Several dozens of mutations have been observed on the SARS-CoV-2 virus, during the last two years. Some of the mutated strains show a greater infectivity and are capable of suppressing the earlier strains, through interference. In this work, kinetic and thermodynamic properties were calculated for strains characterized by various numbers and locations of mutations. It was shown that mutations lead to changes in chemical composition, thermodynamic properties and infectivity. Through competition, the phenomenon of interference of various SARS-CoV-2 strains was explained, which results in suppression of the wild type by mutant strains. Standard Gibbs energy of binding and binding constant for the Omicron (B.1.1.529) strain were found to be Δ_BG⁰ = −45.96 kJ/mol and K_B = 1.13 ∙ 10⁺⁸ M⁻¹, respectively.

Stochastic formulation of multiwave pandemic: decomposition of growth into inherent susceptibility and external infectivity distributions

Many known and unknown factors play significant roles in the persistence of an infectious disease, but two that are often ignored in theoretical modelling are the distributions of (i) inherent susceptibility ( σ inh ) and (ii) external infectivity ( ι ext ), in a population. While the former is determined by the immunity of an individual towards a disease, the latter depends on the exposure of a susceptible person to the infection. We model the spatio-temporal propagation of a pandemic as a chemical reaction kinetics on a network using a modified SAIR (Susceptible-Asymptomatic-Infected-Removed) model to include these two distributions. The resulting integro-differential equations are solved using Kinetic Monte Carlo Cellular Automata (KMC-CA) simulations. Coupling between σ inh and ι ext are combined into a new parameter Ω, defined as Ω = σ inh × ι ext ; infection occurs only if the value of Ω is greater than a Pandemic Infection Parameter (PIP), Ω 0 . Not only does this parameter provide a microscopic viewpoint of the reproduction number R₀ advocated by the conventional SIR model, but it also takes into consideration the viral load experienced by a susceptible person. We find that the neglect of this coupling could compromise quantitative predictions and lead to incorrect estimates of the infections required to achieve the herd immunity threshold. The figure represents the network model for spread of infectious diseases considered in this work. It also shows the resultant multiwave infection graph by inclusion of inherent susceptibility and external infectivity distributions and migration of infected individuals.

Time-dependent heterogeneity leads to transient suppression of the COVID-19 epidemic, not herd immunity

Epidemics generally spread through a succession of waves that reflect factors on multiple timescales. On short timescales, superspreading events lead to burstiness and overdispersion, whereas long-term persistent heterogeneity in susceptibility is expected to lead to a reduction in both the infection peak and the herd immunity threshold (HIT). Here, we develop a general approach to encompass both timescales, including time variations in individual social activity, and demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models through reparameterization. We derive a nonlinear dependence of the effective reproduction number [Formula: see text] on the susceptible population fraction S. We show that a state of transient collective immunity (TCI) emerges well below the HIT during early, high-paced stages of the epidemic. However, this is a fragile state that wanes over time due to changing levels of social activity, and so the infection peak is not an indication of long-lasting herd immunity: Subsequent waves may emerge due to behavioral changes in the population, driven by, for example, seasonal factors. Transient and long-term levels of heterogeneity are estimated using empirical data from the COVID-19 epidemic and from real-life face-to-face contact networks. These results suggest that the hardest hit areas, such as New York City, have achieved TCI following the first wave of the epidemic, but likely remain below the long-term HIT. Thus, in contrast to some previous claims, these regions can still experience subsequent waves.

Modelling the epidemiology of residual Plasmodium vivax malaria in a heterogeneous host population: A case study in the Amazon Basin

The overall malaria burden in the Americas has decreased dramatically over the past two decades, but residual transmission pockets persist across the Amazon Basin, where Plasmodium vivax is the predominant infecting species. Current elimination efforts require a better quantitative understanding of malaria transmission dynamics for planning, monitoring, and evaluating interventions at the community level. This can be achieved with mathematical models that properly account for risk heterogeneity in communities approaching elimination, where few individuals disproportionately contribute to overall malaria prevalence, morbidity, and onwards transmission. Here we analyse demographic information combined with routinely collected malaria morbidity data from the town of Mâncio Lima, the main urban transmission hotspot of Brazil. We estimate the proportion of high-risk subjects in the host population by fitting compartmental susceptible-infected-susceptible (SIS) transmission models simultaneously to age-stratified vivax malaria incidence densities and the frequency distribution of P. vivax malaria attacks experienced by each individual over 12 months. Simulations with the best-fitting SIS model indicate that 20% of the hosts contribute 86% of the overall vivax malaria burden. Despite the low overall force of infection typically found in the Amazon, about one order of magnitude lower than that in rural Africa, high-risk individuals gradually develop clinical immunity following repeated infections and eventually constitute a substantial infectious reservoir comprised of asymptomatic parasite carriers that is overlooked by routine surveillance but likely fuels onwards malaria transmission. High-risk individuals therefore represent a priority target for more intensive and effective interventions that may not be readily delivered to the entire community.

Susceptibility to organophosphates pesticides and the development of infectious-contagious respiratory diseases

In this paper we develop an SIRS compartmental model to investigate the dynamic interplay between pesticide intoxication and the spread of infectious-contagious respiratory diseases. We are particularly interested in investigating three levels of genetic susceptibility to pesticide intoxication. The genotypic distribution of susceptibility to pesticide intoxication, is proposed and parameterized according to ethnic variation using real population data from published studies, and we assume that pesticide intoxication increases susceptibility to infection with a respiratory pathogen. We use mathematical models to illustrate the impact of this distribution on the spread of hypothetical respiratory disease in a population exposed to the organophosphate pesticide. In this context, we show how an initial basic reproductive number below the epidemic threshold of 1.0 could be enhanced to support epidemic outbreaks in agricultural populations that employ chlorpyrifos pesticides. We further illustrate our modeling framework to study the effect of ethnic group variation in Singapore (Malay, Indian and Chinese) using genetic distribution data from published studies.

Sub-national variation in measles vaccine coverage and outbreak risk: a case study from a 2010 outbreak in Malawi

BACKGROUND

Despite progress towards increasing global vaccination coverage, measles continues to be one of the leading, preventable causes of death among children worldwide. Whether and how to target sub-national areas for vaccination campaigns continues to remain a question. We analyzed three metrics for prioritizing target areas: vaccination coverage, susceptible birth cohort, and the effective reproductive ratio (R_E) in the context of the 2010 measles epidemic in Malawi.

METHODS

Using case-based surveillance data from the 2010 measles outbreak in Malawi, we estimated vaccination coverage from the proportion of cases reporting with a history of prior vaccination at the district and health facility catchment scale. Health facility catchments were defined as the set of locations closer to a given health facility than to any other. We combined these estimates with regional birth rates to estimate the size of the annual susceptible birth cohort. We also estimated the effective reproductive ratio, R_E, at the health facility polygon scale based on the observed rate of exponential increase of the epidemic. We combined these estimates to identify spatial regions that would be of high priority for supplemental vaccination activities.

RESULTS

The estimated vaccination coverage across all districts was 84%, but ranged from 61 to 99%. We found that 8 districts and 354 health facility catchments had estimated vaccination coverage below 80%. Areas that had highest birth cohort size were frequently large urban centers that had high vaccination coverage. The estimated R_E ranged between 1 and 2.56. The ranking of districts and health facility catchments as priority areas varied depending on the measure used.

CONCLUSIONS

Each metric for prioritization may result in discrete target areas for vaccination campaigns; thus, there are tradeoffs to choosing one metric over another. However, in some cases, certain areas may be prioritized by all three metrics. These areas should be treated with particular concern. Furthermore, the spatial scale at which each metric is calculated impacts the resulting prioritization and should also be considered when prioritizing areas for vaccination campaigns. These methods may be used to allocate effort for prophylactic campaigns or to prioritize response for outbreak response vaccination.

The effects of heterogeneity on stochastic cycles in epidemics

Models of biological processes are often subject to different sources of noise. Developing an understanding of the combined effects of different types of uncertainty is an open challenge. In this paper, we study a variant of the susceptible-infective-recovered model of epidemic spread, which combines both agent-to-agent heterogeneity and intrinsic noise. We focus on epidemic cycles, driven by the stochasticity of infection and recovery events, and study in detail how heterogeneity in susceptibilities and propensities to pass on the disease affects these quasi-cycles. While the system can only be described by a large hierarchical set of equations in the transient regime, we derive a reduced closed set of equations for population-level quantities in the stationary regime. We analytically obtain the spectra of quasi-cycles in the linear-noise approximation. We find that the characteristic frequency of these cycles is typically determined by population averages of susceptibilities and infectivities, but that their amplitude depends on higher-order moments of the heterogeneity. We also investigate the synchronisation properties and phase lag between different groups of susceptible and infected individuals.

Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity

The paper presents a model for the evolution of an infectious disease in a population with individual-specific immunity. The immune state of an individual varies with time according to its own dynamics, depending on whether the individual is infected or not. The model involves a system of size-structured (first-order) PDEs that capture both the dynamics of the immune states and the transition between compartments consisting of infected, susceptible, etc.

INDIVIDUALS

Due to the unavailability of precise data about the immune states of the individuals, the main focus in the paper is on developing a technique for set-membership estimations of aggregated quantities of interest. The technique involves solving specific optimization problems for the underlying PDE system and is developed up to a numerical method. Results of numerical simulations are presented for a benchmark model of SIS-type, potentially applicable to diseases like influenza and to various sexually transmitted diseases.

Heterogeneous population dynamics and scaling laws near epidemic outbreaks

In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping points). This theory shows that one may be able to anticipate whether a bifurcation point is close before it happens. We use numerical simulations of a stochastic fast-slow heterogeneous population SIS model and show various aspects of heterogeneity have crucial influences on the scaling laws that are used as early-warning signs for the homogeneous system. Thus, although the basic structural qualitative dynamical properties are the same for both systems, the quantitative features for epidemic prediction are expected to change and care has to be taken to interpret potential warning signs for disease outbreaks correctly.

Set-membership estimations for the evolution of infectious diseases in heterogeneous populations

The paper presents an approach for set-membership estimation of the state of a heterogeneous population in which an infectious disease is spreading. The population state may consist of susceptible, infected, recovered, etc. groups, where the individuals are heterogeneous with respect to traits, relevant to the particular disease. Set-membership estimations in this context are reasonable, since only vague information about the distribution of the population along the space of heterogeneity is available in practice. The presented approach comprises adapted versions of methods which are known in estimation and control theory, and involve solving parametrized families of optimization problems. Since the models of disease spreading in heterogeneous populations involve distributed systems (with non-local dynamics and endogenous boundary conditions), these problems are non-standard. The paper develops the needed theoretical instruments and a solution scheme. SI and SIR models of epidemic diseases are considered as case studies and the results reveal qualitative properties that may be of interest.

Applying the stochastic Galerkin method to epidemic models with uncertainty in the parameters

Parameters in modelling are not always known with absolute certainty. In epidemic modelling, this is true of many of the parameters. It is important for this uncertainty to be included in any model. This paper looks at using the stochastic Galerkin method to solve an SIR model with uncertainty in the parameters. The results obtained from the stochastic Galerkin method are then compared with results obtained through Monte Carlo sampling. The computational cost of each method is also compared. It is shown that the stochastic Galerkin method produces good results, even at low order expansions, that are much less computationally expensive than Monte Carlo sampling. It is also shown that the stochastic Galerkin method does not always converge and this non-convergence is explored.

Aggregation and asymptotic analysis of an SI-epidemic model for heterogeneous populations

The paper investigates a version of a simple epidemiological model involving only susceptible and infected individuals, where the heterogeneity of the population with respect to susceptibility/infectiousness is taken into account. A comprehensive analysis of the asymptotic behaviour of the disease is given, based on an explicit aggregation of the model. The results are compared with those of a homogeneous version of the model to highlight the influence of the heterogeneity on the asymptotics. Moreover, the performed analysis reveals in which cases incomplete information about the heterogeneity of the population is sufficient in order to determine the long-run outcome of the disease. Numerical simulation is used to emphasize that, for a given level of prevalence, the evolution of the disease under the influence of heterogeneity may in the long run qualitatively differ from the one 'predicted' by the homogeneous model. Furthermore, it is shown that, in a closed population, the indicator for the survival of the population is in the presence of heterogeneity distinct from the basic reproduction number.

Threshold analysis for epidemic models with high-risk immunization on networks

In this paper, an SIRS epidemic model with high-risk immunization was investigated, where a susceptible neighbor of an infected node is immunized with rate h. Through analyzing the discrete-time model, we found that the epidemic threshold above which an epidemic can prevail and persist in a population is inversely proportional to 1 - h value. We also studied the continuous-time epidemic model and obtained a different result: the epidemic threshold does not depend on the immunization parameter h. Our results suggest that the difference between the discrete-time epidemic model and the continuous-time epidemic model exists in the high-risk immunization.

On the correlation between variance in individual susceptibilities and infection prevalence in populations

The hypothesis that infection prevalence in a population correlates negatively with variance in the susceptibility of its individuals has support from experimental, field, and theoretical studies. However, its generality has never been formally demonstrated. Here we formulate an endemic SIS model with individual susceptibility distributed according to a discrete or continuous probability function to assess the generality of such hypothesis. We introduce an ordering among susceptibility distributions with the same mean, analogous to that considered in Katriel (J Math Biol 65:237-262, 2012) to order the attack rates in an epidemic SIR model with heterogeneity. It turns out that if one distribution dominates another in this order then it has greater variance and corresponds to a lower infection prevalence for R0 varying in a suitable maximal interval of the form ]1, R0*]. We show that in both the discrete and continuous frameworks R0* can be finite, so that the expected correlation among variance and prevalence does not always hold. For discrete distributions this fact is demonstrated analytically, and the proof introduces a constructive procedure to find ordered pairs for which R0* is arbitrarily close to 1. For continuous distributions our conclusion is based on numerical studies with the beta distribution. Finally, we present explicit partial orderings among discrete susceptibility distributions and among symmetric beta distributions which guarantee that R0* = +∞.