Modelling: Understanding pandemics and how to control them.

New disease challenges, societal demands and better or novel types of data, drive innovations in the structure, formulation and analysis of epidemic models. Innovations in modelling can lead to new insights into epidemic processes and better use of available data, yielding improved disease control and stimulating collection of better data and new data types. Here we identify key challenges for the structure, formulation, analysis and use of mathematical models of pathogen transmission relevant to current and future pandemics.

Commentary on the use of the reproduction number R during the COVID-19 pandemic.

Since the beginning of the COVID-19 pandemic, the reproduction number [Formula: see text] has become a popular epidemiological metric used to communicate the state of the epidemic. At its most basic, [Formula: see text] is defined as the average number of secondary infections caused by one primary infected individual. [Formula: see text] seems convenient, because the epidemic is expanding if [Formula: see text] and contracting if [Formula: see text]. The magnitude of [Formula: see text] indicates by how much transmission needs to be reduced to control the epidemic. Using [Formula: see text] in a naïve way can cause new problems. The reasons for this are threefold: (1) There is not just one definition of [Formula: see text] but many, and the precise definition of [Formula: see text] affects both its estimated value and how it should be interpreted. (2) Even with a particular clearly defined [Formula: see text], there may be different statistical methods used to estimate its value, and the choice of method will affect the estimate. (3) The availability and type of data used to estimate [Formula: see text] vary, and it is not always clear what data should be included in the estimation. In this review, we discuss when [Formula: see text] is useful, when it may be of use but needs to be interpreted with care, and when it may be an inappropriate indicator of the progress of the epidemic. We also argue that careful definition of [Formula: see text], and the data and methods used to estimate it, can make [Formula: see text] a more useful metric for future management of the epidemic.

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 regions differently. The current disease-induced immunity level î in a region approximately equals the cumulative fraction infected, which primarily depends on two factors: (i) the initial potential for COVID-19 in the region (R 0), and (ii) the preventive measures put in place. Using 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 are investigated. Focus lies on quantifying the minimal overall effect of preventive measures p Min needed to prevent a future outbreak. It is shown that î plays a more influential roll than when immunity is obtained from vaccination. Secondly, by comparing regions with different R 0 and î it is shown that regions with lower R 0 and low î may need higher preventive measures (p Min) compared with regions having higher R 0 but also higher î, even when such immunity levels are far from herd immunity. Our results are illustrated on different regions but these comparisons contain lots of uncertainty due to simplistic model assumptions and insufficient data fitting, and should accordingly be interpreted with caution.

Key questions for modelling COVID-19 exit strategies.

Combinations of intense non-pharmaceutical interventions (lockdowns) were introduced worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement exit strategies that relax restrictions while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute 'Models for an exit strategy' workshop (11-15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, would allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. This roadmap requires a global collaborative effort from the scientific community and policymakers, and has three parts: (i) improve estimation of key epidemiological parameters; (ii) understand sources of heterogeneity in populations; and (iii) focus on requirements for data collection, particularly in low-to-middle-income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.

A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2.

Despite various levels of preventive measures, in 2020, many countries have suffered severely from the coronavirus disease 2019 (COVID-19) pandemic caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus. Using a model, we show that population heterogeneity can affect disease-induced immunity considerably because the proportion of infected individuals in groups with the highest contact rates is greater than that in groups with low contact rates. We estimate that if R 0 = 2.5 in an age-structured community with mixing rates fitted to social activity, then the disease-induced herd immunity level can be ~43%, which is substantially less than the classical herd immunity level of 60% obtained through homogeneous immunization of the population. Our estimates should be interpreted as an illustration of how population heterogeneity affects herd immunity rather than as an exact value or even a best estimate.

SIR epidemics and vaccination on random graphs with clustering.

In this paper we consider Susceptible [Formula: see text] Infectious [Formula: see text] Recovered (SIR) epidemics on random graphs with clustering. To incorporate group structure of the underlying social network, we use a generalized version of the configuration model in which each node is a member of a specified number of triangles. SIR epidemics on this type of graph have earlier been investigated under the assumption of homogeneous infectivity and also under the assumption of Poisson transmission and recovery rates. We extend known results from literature by relaxing the assumption of homogeneous infectivity both in individual infectivity and between different kinds of neighbours. An important special case of the epidemic model analysed in this paper is epidemics in continuous time with arbitrary infectious period distribution. We use branching process approximations of the spread of the disease to provide expressions for the basic reproduction number [Formula: see text], the probability of a major outbreak and the expected final size. In addition, the impact of random vaccination with a perfect vaccine on the final outcome of the epidemic is investigated. We find that, for this particular model, [Formula: see text] equals the perfect vaccine-associated reproduction number. Generalizations to groups larger than three are discussed briefly.

Who is the infector? Epidemic models with symptomatic and asymptomatic cases.

What role do asymptomatically infected individuals play in the transmission dynamics? There are many diseases, such as norovirus and influenza, where some infected hosts show symptoms of the disease while others are asymptomatically infected, i.e. do not show any symptoms. The current paper considers a class of epidemic models following an SEIR (Susceptible  →  Exposed  →  Infectious  →  Recovered) structure that allows for both symptomatic and asymptomatic cases. The following question is addressed: what fraction ρ of those individuals getting infected are infected by symptomatic (asymptomatic) cases? This is a more complicated question than the related question for the beginning of the epidemic: what fraction of the expected number of secondary cases of a typical newly infected individual, i.e. what fraction of the basic reproduction number R0, is caused by symptomatic individuals? The latter fraction only depends on the type-specific reproduction numbers, while the former fraction ρ also depends on timing and hence on the probabilistic distributions of latent and infectious periods of the two types (not only their means). Bounds on ρ are derived for the situation where these distributions (and even their means) are unknown. Special attention is given to the class of Markov models and the class of continuous-time Reed-Frost models as two classes of distribution functions for latent and infectious periods. We show how these two classes of models can exhibit very different behaviour.

Branching process approach for epidemics in dynamic partnership network.

We study the spread of sexually transmitted infections (STIs) and other infectious diseases on a dynamic network by using a branching process approach. The nodes in the network represent the sexually active individuals, while connections represent sexual partnerships. This network is dynamic as partnerships are formed and broken over time and individuals enter and leave the sexually active population due to demography. We assume that individuals enter the sexually active network with a random number of partners, chosen according to a suitable distribution and that the maximal number of partners that an individual can have at a time is finite. We discuss two different branching process approximations for the initial stages of an outbreak of the STI. In the first approximation we ignore some dependencies between infected individuals. We compute the offspring mean of this approximating branching process and discuss its relation to the basic reproduction number [Formula: see text]. The second branching process approximation is asymptotically exact, but only defined if individuals can have at most one partner at a time. For this model we compute the probability of a minor outbreak of the epidemic starting with one or few initial cases. We illustrate complications caused by dependencies in the epidemic model by showing that if individuals have at most one partner at a time, the probabilities of extinction of the two approximating branching processes are different. This implies that ignoring dependencies in the epidemic model leads to a wrong prediction of the probability of a large outbreak. Finally, we analyse the first branching process approximation if the number of partners an individual can have at a given time is unbounded. In this model we show that the branching process approximation is asymptomatically exact as the population size goes to infinity.

Inferring R0 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 R0 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 R0 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.

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 → infective → 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 R0 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 Rr, which is loosely defined as the estimate of R0 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 Rr ≥ R0 when R0 ≥ 1 and Rr ≤ R0 when R0 ≤ 1. We show that, in contrast to models without social structure, vaccination of a fraction 1-1/R0 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.

Stochastic SIR epidemics in a population with households and schools.

We study the spread of stochastic SIR (Susceptible [Formula: see text] Infectious [Formula: see text] Recovered) epidemics in two types of structured populations, both consisting of schools and households. In each of the types, every individual is part of one school and one household. In the independent partition model, the partitions of the population into schools and households are independent of each other. This model corresponds to the well-studied household-workplace model. In the hierarchical model which we introduce here, members of the same household are also members of the same school. We introduce computable branching process approximations for both types of populations and use these to compare the probabilities of a large outbreak. The branching process approximation in the hierarchical model is novel and of independent interest. We prove by a coupling argument that if all households and schools have the same size, an epidemic spreads easier (in the sense that the number of individuals infected is stochastically larger) in the independent partition model. We also show by example that this result does not necessarily hold if households and/or schools do not all have the same size.

Five challenges for stochastic epidemic models involving global transmission.

The most basic stochastic epidemic models are those involving global transmission, meaning that infection rates depend only on the type and state of the individuals involved, and not on their location in the population. Simple as they are, there are still several open problems for such models. For example, when will such an epidemic go extinct and with what probability (questions depending on the population being fixed, changing or growing)? How can a model be defined explaining the sometimes observed scenario of frequent mid-sized epidemic outbreaks? How can evolution of the infectious agent transmission rates be modelled and fitted to data in a robust way?

Eight challenges for network epidemic models.

Networks offer a fertile framework for studying the spread of infection in human and animal populations. However, owing to the inherent high-dimensionality of networks themselves, modelling transmission through networks is mathematically and computationally challenging. Even the simplest network epidemic models present unanswered questions. Attempts to improve the practical usefulness of network models by including realistic features of contact networks and of host-pathogen biology (e.g. waning immunity) have made some progress, but robust analytical results remain scarce. A more general theory is needed to understand the impact of network structure on the dynamics and control of infection. Here we identify a set of challenges that provide scope for active research in the field of network epidemic models.

Five challenges for spatial epidemic models.

Infectious disease incidence data are increasingly available at the level of the individual and include high-resolution spatial components. Therefore, we are now better able to challenge models that explicitly represent space. Here, we consider five topics within spatial disease dynamics: the construction of network models; characterising threshold behaviour; modelling long-distance interactions; the appropriate scale for interventions; and the representation of population heterogeneity.

Inferring global network properties from egocentric data with applications to epidemics.

Social networks are often only partly observed, and it is sometimes desirable to infer global properties of the network from 'egocentric' data. In the current paper, we study different types of egocentric data, and show which global network properties are consistent with data. Two global network properties are considered: the size of the largest connected component (the giant) and the size of an epidemic outbreak taking place on the network. The main conclusion is that, in most cases, egocentric data allow for a large range of possible sizes of the giant and the outbreak, implying that egocentric data carry very little information about these global properties. The asymptotic size of the giant and the outbreak is also characterized, assuming the network is selected uniformly among networks with prescribed egocentric data.

Stochastic epidemics in growing populations.

Consider a uniformly mixing population which grows as a super-critical linear birth and death process. At some time an infectious disease (of SIR or SEIR type) is introduced by one individual being infected from outside. It is shown that three different scenarios may occur: (i) an epidemic never takes off, (ii) an epidemic gets going and grows but at a slower rate than the community thus still being negligible in terms of population fractions, or (iii) an epidemic takes off and grows quicker than the community eventually leading to an endemic equilibrium. Depending on the parameter values, either scenario (i) is the only possibility, both scenarios (i) and (ii) are possible, or scenarios (i) and (iii) are possible.

Reproduction numbers for epidemic models with households and other social structures. I. Definition and calculation of R0.

The basic reproduction number R(0) is one of the most important quantities in epidemiology. However, for epidemic models with explicit social structure involving small mixing units such as households, its definition is not straightforward and a wealth of other threshold parameters has appeared in the literature. In this paper, we use branching processes to define R(0), we apply this definition to models with households or other more complex social structures and we provide methods for calculating it.

The nosocomial transmission rate of animal-associated ST398 meticillin-resistant Staphylococcus aureus.

The global epidemiology of meticillin-resistant Staphylococcus aureus (MRSA) is characterized by different clonal lineages with different epidemiological behaviour. There are pandemic hospital clones (hospital-associated (HA-)MRSA), clones mainly causing community-acquired infections (community-associated (CA-)MRSA, mainly USA300) and an animal-associated clone (ST398) emerging in European and American livestock with subsequent spread to humans. Nosocomial transmission capacities (R(A)) of these different MRSA types have never been quantified. Using two large datasets from MRSA outbreaks in Dutch hospitals (dataset 1, the UMC Utrecht for 144 months; dataset 2, 51 hospitals for six months) and a recently developed mathematical model, we determined the genotype-specific R(A) for ST398 and non-ST398 isolates (categorized as HA-MRSA), using observational data, the detection rate of MRSA carriage and the discharge rate from hospital as the input. After detection of 42 MRSA index cases in dataset 1 (all non-ST398 MRSA) 5076 people were screened, yielding 30 secondary cases. In dataset 2, 75 index cases (51 non-ST398 MRSA and 24 ST398) resulted in 7892 screened individuals and 56 and three secondary cases for non-ST398 MRSA and ST398, respectively. The ratio between discharge and the detection rate was 2.7. R(A) values (95% confidence interval (CI)) were 0.68 (0.47-0.95) for non-ST398 MRSA in dataset 1, 0.93 (0.71-1.21) for non-ST398 MRSA in dataset 2 and 0.16 (0.04-0.40) for ST398. The R(A) ratio between non-ST398 MRSA and ST398 was 5.90 (95% CI 2.24-23.81). ST398 is 5.9 times less transmissible than non-ST398 MRSA in Dutch hospitals, which may allow less stringent transmission-control measures for ST398 MRSA.

Analysis of a stochastic SIR epidemic on a random network incorporating household structure.

This paper is concerned with a stochastic SIR (susceptible-->infective-->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. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.

A useful relationship between epidemiology and queueing theory: the distribution of the number of infectives at the moment of the first detection.

In this paper we establish a relation between the spread of infectious diseases and the dynamics of so called M/G/1 queues with processor sharing. The relation between the spread of epidemics and branching processes, which is well known in epidemiology, and the relation between M/G/1 queues and birth death processes, which is well known in queueing theory, will be combined to provide a framework in which results from queueing theory can be used in epidemiology and vice versa. In particular, we consider the number of infectious individuals in a standard SIR epidemic model at the moment of the first detection of the epidemic, where infectious individuals are detected at a constant per capita rate. We use a result from the literature on queueing processes to show that this number of infectious individuals is geometrically distributed.