arXiv subjects

Frank Ball

Publications and source records attributed to Frank Ball.

At least 19 recordsLinked to original sources

arXiv2026

Bounds on $R_0$ and final epidemic size when the next-generation matrix $M$ is only partially known

We study a multitype SIR epidemic model where individuals are categorized into different types, and where infection spread is characterized by a next-generation matrix $M=\{m_{ij}\}$ with community fractions $\{\pi_j\}$ for the different types of individuals. We analyse two key quantities: the basic reproduction number $R_0$ and the final epidemic outcome of the different types $\{\tau_i\}$. We consider the situation where $M$ is only partly known, through the row sums $\{r_i\}$ or the column sums $\{c_j\}$, and treat both a general $M$ and the special but common situation where $M$ is proportional to a contact matrix satisfying detailed balance. For a general $M$, which is partially observed through $\{r_i\}$ or $\{c_j\}$, we obtain sharp upper and lower bounds of $R_0$ and $\{\tau_i\}$, but for the case where $M$ satisfies detailed balance the problem is harder: our obtained bounds for $R_0$ are narrower than the general case but still not sharp, and bounds for the final size are only obtained when there are two types of individual.

q-bio.PE↗

arXiv2025

Household epidemic models revisited

We analyse a generalized stochastic household epidemic model defined by a bivariate random variable $(X_G, X_L)$, representing the number of global and local infectious contacts that an infectious individual makes during their infectious period. Each global contact is selected uniformly among all individuals and each local contact is selected uniformly among all other household members. The main focus is when all households have the same size $h \geq 2$, and the number of households is large. Large population properties of the model are derived including a central limit theorem for the final size of a major epidemic, the proof of which utilises an enhanced embedding argument. A modification of the epidemic model is considered where local contacts are replaced by global contacts independently with probability $p$. We then prove monotonicity results for the probability of the major outbreak and the limiting final fraction infected $z$ (conditioned on a major outbreak). a) The probability of a major outbreak is shown to be increasing in both $h$ and $p$ for any distribution of $X_L$. b) The final size $z$ increases monotonically with both $h$ and $p$ if the probability generating function (pgf) of $X_L$ is log-convex, which is satisfied by traditional household epidemic models where $X_L$ has a mixed-Poisson distribution. Additionally, we provide counter examples to b) when the pgf of $X_L$ is not log-convex.

math.PR↗

arXiv2024

Modelling the spread of two successive SIR epidemics on a configuration model network

We present a stochastic model for two successive SIR (Susceptible, Infectious, Recovered) epidemics in the same network structured population. Individuals infected during the first epidemic might have (partial) immunity for the second one. The first epidemic is analysed through a bond percolation model, while the second epidemic is approximated by a three-type branching process in which the types of individuals depend on their position in the percolation clusters used for the first epidemic. This branching process approximation enables us to calculate, in the large population limit and conditional upon a large outbreak in the first epidemic, a threshold parameter and the probability of a large outbreak for the second epidemic. A second branching process approximation enables us to calculate the fraction of the population that are infected by such a second large outbreak. We illustrate our results through some specific cases which have appeared previously in the literature and show that our asymptotic results give good approximations for finite populations.

q-bio.PE↗

arXiv2024

An epidemic model on a network having two group structures with tunable overlap

A network epidemic model is studied. The underlying social network has two different types of group structures, households and workplaces, such that each individual belongs to exactly one household and one workplace. The random network is constructed such that a parameter $\theta$ controls the degree of overlap between the two group structures: $\theta=0$ corresponding to all household members belonging to the same workplace and $\theta=1$ to all household members belonging to distinct workplaces. On the network a stochastic SIR epidemic is defined, having an arbitrary but specified infectious period distribution, with global (community), household and workplace infectious contacts. The stochastic epidemic model is analysed as the population size $n\to\infty$ with the (asymptotic) probability, and size, of a major outbreak obtained. These results are proved in greater generality than existing results in the literature by allowing for any fixed $0 \leq \theta \leq 1$, a non-constant infectious period distribution, the presence or absence of global infection and potentially (asymptotically) infinite local outbreaks.

math.PR↗

arXiv2024

Improving the use of social contact studies in epidemic modelling

Social contact studies, investigating social contact patterns in a population sample, have been an important contribution for epidemic models to better fit real life epidemics. A contact matrix $M$, having the \emph{mean} number of contacts between individuals of different age groups as its elements, is estimated and used in combination with a multitype epidemic model to produce better data fitting and also giving more appropriate expressions for $R_0$ and other model outcomes. However, $M$ does not capture \emph{variation} in contacts \emph{within} each age group, which is often large in empirical settings. Here such variation within age groups is included in a simple way by dividing each age group into two halves: the socially active and the socially less active. The extended contact matrix, and its associated epidemic model, empirically show that acknowledging variation in social activity within age groups has a substantial impact on modelling outcomes such as $R_0$ and the final fraction $\tau$ getting infected. In fact, the variation in social activity within age groups is often more important for data fitting than the division into different age groups itself. However, a difficulty with heterogeneity in social activity is that social contact studies typically lack information on if mixing with respect to social activity is assortative or not, i.e.\ do socially active tend to mix more with other socially active or more with socially less active? The analyses show that accounting for heterogeneity in social activity improves the analyses irrespective of if such mixing is assortative or not, but the different assumptions gives rather different output. Future social contact studies should hence also try to infer the degree of assortativity of contacts with respect to social activity.

q-bio.PE↗

arXiv2022

SIR epidemics in populations with large sub-communities

We investigate final outcome properties of an SIR (susceptible $\to$ infective $\to$ recovered) epidemic model defined on a population of large sub-communities in which there is stronger disease transmission within the communities than between them. Our analysis involves approximation of the epidemic process by a chain of within-community large outbreaks spreading between the communities. We derive law of large numbers and central limit type results for the number of individuals and the number of communities affected and the so-called severity of the outbreak. These results are valid as the size of communities tends to infinity, with the number of communities either fixed or also tending to infinity. The weaker between-community connections lead to randomness even in the law of large numbers type limit. As part of our proofs we also obtain a new result concerning the rate of convergence of the expected fraction infected in a standard SIR epidemic to its large-population limit.

math.PR↗

arXiv2020

The risk for a new COVID-19 wave -- and how it depends on $R_0$, the current immunity level and current restrictions

The COVID-19 pandemic has hit different parts of the world differently: some regions are still in the rise of the first wave, other regions are now facing a decline after a first wave, and yet other regions have started to see a second wave. The current immunity level $\hat i$ in a region is closely related to the cumulative fraction infected, which primarily depends on two factors: a) the initial potential for COVID-19 in the region (often quantified by the basic reproduction number $R_0$), and b) the timing, amount and effectiveness of preventive measures put in place. By means of a mathematical model including heterogeneities owing to age, social activity and susceptibility, and allowing for time-varying preventive measures, the risk for a new epidemic wave and its doubling time, and how they depend on $R_0$, $\hat i$ and the overall effect of the current preventive measures, are investigated. Focus lies on quantifying the minimal overall effect of preventive measures $p_{Min}$ needed to prevent a future outbreak. The first result shows that the current immunity level $\hat i$ plays a more influential roll than when immunity is obtained from vaccination. Secondly, by comparing regions with different $R_0$ and $\hat i$ it is shown that regions with lower $R_0$ and low $\hat i$ may now need higher preventive measures ($p_{Min}$) compared with other regions having higher $R_0$ but also higher $\hat i$, even when such immunity levels are far from herd immunity.

q-bio.PE↗

arXiv2020

Epidemics on networks with preventive rewiring

A stochastic SIR (susceptible $\to$ infective $\to$ recovered) epidemic model defined on a social network is analysed. The underlying social network is described by an Erdős-Rényi random graph but, during the course of the epidemic, susceptible individuals connected to infectious neighbours may drop or rewire such connections. Large population limits of the model are derived giving both convergence results for the early branching process-like behaviour, and, assuming a major outbreak, the main phase of the epidemic process which converges to a deterministic model that is equivalent to a certain pair approximation model. Law of large numbers results are also obtained for the final size (i.e. total number of individuals infected) of a major outbreak. Two results stand out (valid for a range of parameter set-ups): (i) the limiting final fraction infected may be discontinuous in the infection rate $λ$ at its threshold $λ_c$ (thus making a discrete jump from $0$ to a strictly positive number) and (ii) for the situation when rewiring is necessarily to uninfected individuals, if it is discontinuous, the limiting final fraction infected jumps from $0$ to $1$ as $λ$ passes through $λ_c$.

math.PR↗

arXiv2020

Summer vacation and COVID-19: effects of metropolitan people going to summer provinces

Many countries are now investigating what the effects of summer vacation might be on the COVID-19 pandemic. Here one particular such question is addressed: what will happen if large numbers of metropolitan people visit a less populated province during the summer vacation? By means of a simple epidemic model, allowing for both short and long-term visitors to the province, it is studied which features are most influential in determining if such summer movements will result in large number of infections among the province population. The method is applied to the island of Gotland off the South East coast of Sweden. It is shown that the amount of mixing between the metropolitan and province groups and the fraction of metropolitan people being infectious upon arrival are most influential. Consequently, minimizing events gathering both the province and metropolitan groups and/or reducing the number of short-term visitors could substantially decrease spreading, as could measures to lower the fraction initially infectious upon arrival.

q-bio.PE↗

arXiv2020

The disease-induced herd immunity level for Covid-19 is substantially lower than the classical herd immunity level

Most countries are suffering severely from the ongoing covid-19 pandemic despite various levels of preventive measures. A common question is if and when a country or region will reach herd immunity $h$. The classical herd immunity level $h_C$ is defined as $h_C=1-1/R_0$, where $R_0$ is the basic reproduction number, for covid-19 estimated to lie somewhere in the range 2.2-3.5 depending on country and region. It is shown here that the disease-induced herd immunity level $h_D$, after an outbreak has taken place in a country/region with a set of preventive measures put in place, is actually substantially smaller than $h_C$. As an illustration we show that if $R_0=2.5$ in an age-structured community with mixing rates fitted to social activity studies, and also categorizing individuals into three categories: low active, average active and high active, and where preventive measures affect all mixing rates proportionally, then the disease-induced herd immunity level is $h_D=43\%$ rather than $h_C=1-1/2.5=60\%$. Consequently, a lower fraction infected is required for herd immunity to appear. The underlying reason is that when immunity is induced by disease spreading, the proportion infected in groups with high contact rates is greater than that in groups with low contact rates. Consequently, disease-induced immunity is stronger than when immunity is uniformly distributed in the community as in the classical herd immunity level.

q-bio.PE↗

arXiv2019

A stochastic SIR network epidemic model with preventive dropping of edges

A Markovian SIR (Susceptible-Infectious-Recovered) model is considered for the spread of an epidemic on a configuration model network, in which susceptible individuals may take preventive measures by dropping edges to infectious neighbours. An effective degree formulation of the model is used in conjunction with the theory of density dependent population processes to obtain a law of large numbers and a functional central limit theorem for the epidemic as the population size $N \to \infty$, assuming that the degrees of individuals are bounded. A central limit theorem is conjectured for the final size of the epidemic. The results are obtained for both the Molloy-Reed (in which the degrees of individuals are deterministic) and Newman-Strogatz-Watts (in which the degrees of individuals are independent and identically distributed) versions of the configuration model. The two versions yield the same limiting deterministic model but the asymptotic variances in the central limit theorema are greater in the Newman-Strogatz-Watts version. The basic reproduction number $R_0$ and the process of susceptible individuals in the limiting deterministic model, for the model with dropping of edges, are the same as for a corresponding SIR model without dropping of edges but an increased recovery rate, though, when $R_0>1$, the probability of a major outbreak is greater in the model with dropping of edges. The results are specialised to the model without dropping of edges to yield conjectured central limit theorems for the final size of Markovian SIR epidemics on configuration-model networks, and for the giant components of those networks. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations are good, even for moderate $N$.

math.PR↗

arXiv2018

Central limit theorems for SIR epidemics and percolation on configuration model random graphs

We consider a stochastic SIR (susceptible $\to$ infective $\to$ recovered) epidemic defined on a configuration model random graph, in which infective individuals can infect only their neighbours in the graph during an infectious period which has an arbitrary but specified distribution. Central limit theorems for the final size (number of initial susceptibles that become infected) of such an epidemic as the population size $n$ tends to infinity, with explicit, easy to compute expressions for the asymptotic variance, are proved assuming that the degrees are bounded. The results are obtained for both the Molloy-Reed random graph, in which the degrees of individuals are deterministic, and the Newman-Strogatz-Watts random graph, in which the degrees are independent and identically distributed. The central limit theorems cover the cases when the number of initial infectives either (a) tends to infinity or (b) is held fixed as $n \to \infty$. In (a) it is assumed that the fraction of the population that is initially infected converges to a limit (which may be $0$) as $n \to \infty$, while in (b) the central limit theorems are conditional upon the occurrence of a large outbreak (more precisely one of size at least $\log n$). Central limit theorems for the size of the largest cluster in bond percolation on Molloy-Reed and Newman-Strogatz-Watts random graphs follow immediately from our results, as do central limit theorems for the size of the giant component of those graphs. Corresponding central limit theorems for site percolation on those graphs are also proved.

math.PR↗

arXiv2018

Individual preventive measures during an epidemic may have negative population-level outcomes

The outbreak of an infectious disease in a human population can lead to individuals responding with preventive measures in an attempt to avoid getting infected. This leads to changes in contact patterns. However, as we show in this paper, rational behaviour at the individual level, such as social distancing from infectious contacts, may not always be beneficial for the population as a whole. We use epidemic network models to demonstrate the potential negative consequences at the population level. We take into account the social structure of the population through several network models. As the epidemic evolves, susceptible individuals may distance themselves from their infectious contacts. Some individuals replace their lost social connections by seeking new ties. We show that social distancing can worsen the disease outcome both in the initial phase of an outbreak and the final epidemic size. Moreover, the same negative effect can arise in real-world networks. Our results suggest that one needs to be careful when targeting behavioural changes as they could potentially worsen the epidemic outcome. Furthermore, network structure crucially influences the way that individual-level measures impact the epidemic at the population level. These findings highlight the importance of careful analysis of preventive measures in epidemic models.

q-bio.PE↗

arXiv2017

Heterogeneous network epidemics: real-time growth, variance and extinction of infection

Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multi-type branching process. We find closed-form analytic expressions for the mean and variance of the number of infectious individuals as a function of time and the degree of the initially infected individual(s), and write down a system of differential equations for the probability of extinction that are numerically fast compared to Monte Carlo simulation. We show that these quantities are all sensitive to the degree distribution - in particular we confirm that the mean prevalence of infection depends on the first two moments of the degree distribution and the variance in prevalence depends on the first three moments of the degree distribution. In contrast to most existing analytic approaches, the accuracy of these results does not depend on having a large number of infectious individuals, meaning that in the large population limit they would be asymptotically exact even for one initial infectious individual.

q-bio.PE↗

arXiv2016

Evaluation of vaccination strategies for SIR epidemics on random networks incorporating household structure

This paper is concerned with the analysis of vaccination strategies in a stochastic SIR (susceptible $\to$ infected $\to$ removed) model for the spread of an epidemic amongst a population of individuals with a random network of social contacts that is also partitioned into households. Under various vaccine action models, we consider both household-based vaccination schemes, in which the way in which individuals are chosen for vaccination depends on the size of the households in which they reside, and acquaintance vaccination, which targets individuals of high degree in the social network. For both types of vaccination scheme, assuming a large population with few initial infectives, we derive a threshold parameter which determines whether or not a large outbreak can occur and also the probability and fraction of the population infected by such an outbreak. The performance of these schemes is studied numerically, focusing on the influence of the household size distribution and the degree distribution of the social network. We find that acquaintance vaccination can significantly outperform the best household-based scheme if the degree distribution of the social network is heavy-tailed. For household-based schemes, when the vaccine coverage is insufficient to prevent a major outbreak and the vaccine is imperfect, we find situations in which both the probability and size of a major outbreak under the scheme which minimises the threshold parameter are \emph{larger} than in the scheme which maximises the threshold parameter.

physics.soc-ph↗

arXiv2016

Inferring $R_0$ in emerging epidemics - the effect of common population structure is small

When controlling an emerging outbreak of an infectious disease it is essential to know the key epidemiological parameters, such as the basic reproduction number $R_0$ and the control effort required to prevent a large outbreak. These parameters are estimated from the observed incidence of new cases and information about the infectious contact structures of the population in which the disease spreads. However, the relevant infectious contact structures for new, emerging infections are often unknown or hard to obtain. Here we show that for many common true underlying heterogeneous contact structures, the simplification to neglect such structures and instead assume that all contacts are made homogeneously in the whole population, results in conservative estimates for $R_0$ and the required control effort. This means that robust control policies can be planned during the early stages of an outbreak, using such conservative estimates of the required control effort.

q-bio.PE↗

arXiv2015

Reproduction numbers for epidemic models with households and other social structures II: comparisons and implications for vaccination

In this paper we consider epidemic models of directly transmissible SIR (susceptible $\to$ infective $\to$ recovered) and SEIR (with an additional latent class) infections in fully-susceptible populations with a social structure, consisting either of households or of households and workplaces. We review most reproduction numbers defined in the literature for these models, including the basic reproduction number $R_0$ introduced in the companion paper of this, for which we provide a simpler, more elegant derivation. Extending previous work, we provide a complete overview of the inequalities among these reproduction numbers and resolve some open questions. Special focus is put on the exponential-growth-associated reproduction number $R_r$, which is loosely defined as the estimate of $R_0$ based on the observed exponential growth of an emerging epidemic obtained when the social structure is ignored. We show that for the vast majority of the models considered in the literature $R_r \geq R_0$ when $R_0 \ge 1$ and $R_r \leq R_0$ when $R_0 \le 1$. We show that, in contrast to models without social structure, vaccination of a fraction $1-1/R_0$ of the population, chosen uniformly at random, with a perfect vaccine is usually insufficient to prevent large epidemics. In addition, we provide significantly sharper bounds than the existing ones for bracketing the critical vaccination coverage between two analytically tractable quantities, which we illustrate by means of extensive numerical examples.

q-bio.PE↗

arXiv2015

An epidemic in a dynamic population with importation of infectives

Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_0$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $ n \to \infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{ S(t);t\ge 0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the previous jump. Properties of the process $S$, including the jump size and stationary distributions, are determined.

math.PR↗