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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., Comment: 42 pages, 4 figures

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., Comment: Supplementary material not included. Contact Tom Britton if you are interested

We consider an SEIR epidemic model on a network also allowing random contacts, where recovered individuals could either recover naturally or be diagnosed. Upon diagnosis, manual contact tracing is triggered such that each infected network contact is reported, tested and isolated with some probability and after a random delay. Additionally, digital tracing (based on a tracing app) is triggered if the diagnosed individual is an app-user, and then all of its app-using infectees are immediately notified and isolated. The early phase of the epidemic with manual and/or digital tracing is approximated by different multi-type branching processes, and three respective reproduction numbers are derived. The effectiveness of both contact tracing mechanisms is numerically quantified through the reduction of the reproduction number. This shows that app-using fraction plays an essential role in the overall effectiveness of contact tracing. The relative effectiveness of manual tracing compared to digital tracing increases if: more of the transmission occurs on the network, when the tracing delay is shortened, and when the network degree distribution is heavy-tailed. For realistic values, the combined tracing case can reduce $R_0$ by $20-30\%$, so other preventive measures are needed to reduce the reproduction number down to $1.2-1.4$ for contact tracing to make it successful in avoiding big outbreaks.

We analyse an extended SIRS epidemic model in which immunity at the individual level wanes gradually at exponential rate, but where the waning rate may differ between individuals, for instance as an effect of differences in immune systems. The model also includes vaccination schemes aimed to reach and maintain herd immunity. We consider both the informed situation where the individual waning parameters are known, thus allowing selection of vaccinees being based on both time since last vaccination as well as on the individual waning rate, and the more likely uninformed situation where individual waning parameters are unobserved, thus only allowing vaccination schemes to depend on time since last vaccination. The optimal vaccination policies for both the informed and uniformed heterogeneous situation are derived and compared with the homogeneous waning model (meaning all individuals have the same immunity waning rate), as well as to the classic SIRS model where immunity at the individual level drops from complete immunity to complete susceptibility in one leap. It is shown that the classic SIRS model requires least vaccines, followed by the SIRS with homogeneous gradual waning, followed by the informed situation for the model with heterogeneous gradual waning. The situation requiring most vaccines for herd immunity is the most likely scenario, that immunity wanes gradually with unobserved individual heterogeneity. For parameter values chosen to mimic COVID-19 and assuming perfect initial immunity and cumulative immunity of 12 months, the classic homogeneous SIRS epidemic suggests that vaccinating individuals every 15 months is sufficient to reach and maintain herd immunity, whereas the uninformed case for exponential waning with rate heterogeneity corresponding to a coefficient of variation being 0.5, requires that individuals instead need to be vaccinated every 4.4 months.

We consider the Markovian SIR epidemic model, but where individuals either recover naturally or are diagnosed, the latter implying isolation and subject to being contact traced. We focus on the novel digital contact tracing based on individuals using a tracing app and consider both the situation when this is the only form of contact tracing, and when combined with manual contact tracing. It is proven that the epidemic process converges to a limiting process as the population size $n$ grows large. The limiting process is not a branching process (as is common for epidemic processes) because infectees may be contact traced "from" their infectors, thus creating dependencies. But by instead treating to-be-traced components as macro-individuals allows a multi-type branching process interpretation, thus giving an expression for the reproduction number $R$. This reproduction number (for the digital-only case) is then converted to an individual reproduction number $R^{(ind)}$ being easier to interpret. It is proven that, contrary to $R$, $R^{(ind)}$ decays monotonically with $\pi$ (the fraction of app-users), and the two reproduction numbers are proven to have the same threshold at 1. The digital (only) contact tracing is then compared with earlier results for manual (only) contact tracing where a fraction $p$ of all contacts are traced. It is proven that the critical fraction app-users $\pi_c$ for which digital contact tracing is reduced to $R=1$, and the critical fraction manually contact traced $p_c$ to reach $R=1$, satisfy $\pi_c>p_c$.

SIRS epidemic models assume that individual immunity (from infection and vaccination) wanes in one big leap, from complete immunity to complete susceptibility. For many diseases immunity on the contrary wanes gradually, something that's become even more evident during COVID-19 pandemic where also recently infected have a reinfection risk, and where booster vaccines are given to increase immunity. This paper considers an epidemic model allowing for such gradual waning of immunity (either linear or exponential waning) thereby extending SIRS epidemics, and also incorporates vaccination. The two versions for gradual waning of immunity are compared with the classic SIRS epidemic, where the three models are calibrated by having the same \emph{average cumulative immunity}. All models are shown to have identical basic reproduction number $R_0$. However, if no prevention is put in place, the exponential waning model has highest prevalence and the classic SIRS model has lowest. Similarly, the amount of vaccine supply needed to reach and maintain herd immunity is highest for the model with exponential decay of immunity and lowest for the classic SIRS model. consequently, if truth lies close to exponential (or linear) decay of immunity, expressions based on the SIRS epidemic will underestimate the endemic level and the critical vaccine supply will not be sufficient to reach and maintain herd immunity. For parameter choices fitting to COVID-19, the critical amount of vaccine supply is about 50% higher if immunity wanes linearly, and more than 150% higher when immunity wanes exponentially, as compared to the classic SIRS epidemic model.

This article considers the minimization of the total number of infected individuals over the course of an epidemic in which the rate of infectious contacts can be reduced by time-dependent nonpharmaceutical interventions. The societal and economic costs of interventions are taken into account using a linear budget constraint which imposes a trade-off between short-term heavy interventions and long-term light interventions. We search for an optimal intervention strategy in an infinite-dimensional space of controls containing multiple consecutive lockdowns, gradually imposed and lifted restrictions, and various heuristic controls based for example on tracking the effective reproduction number. Mathematical analysis shows that among all such strategies, the global optimum is achieved by a single constant-level lockdown of maximum possible magnitude. Numerical simulations highlight the need of careful timing of such interventions, and illustrate their benefits and disadvantages compared to strategies designed for minimizing peak prevalence. Rather counterintuitively, adding restrictions prior to the start of a well-planned intervention strategy may even increase the total incidence., Comment: 22 pages, 5 figures

The real-time analysis of infectious disease surveillance data, e.g., in the form of a time-series of reported cases or fatalities, is essential in obtaining situational awareness about the current dynamics of an adverse health event such as the COVID-19 pandemic. This real-time analysis is complicated by reporting delays that lead to underreporting of the number of events for the most recent time points (e.g., days or weeks). This can lead to misconceptions by the interpreter, e.g., the media or the public, as was the case with the time-series of reported fatalities during the COVID-19 pandemic in Sweden. Nowcasting methods provide real-time estimates of the complete number of events using the incomplete time-series of currently reported events by using information about the reporting delays from the past. Here, we consider nowcasting the number of COVID-19-related fatalities in Sweden. We propose a flexible Bayesian approach, extending existing nowcasting methods by incorporating regression components to accommodate additional information provided by leading indicators such as time-series of the number of reported cases and ICU admissions. By a retrospective evaluation, we show that the inclusion of ICU admissions as a leading signal improved the nowcasting performance of case fatalities for COVID-19 in Sweden compared to existing methods.

We present a stochastic epidemic model to study the effect of various preventive measures, such as uniform reduction of contacts and transmission, vaccination, isolation, screening and contact tracing, on a disease outbreak in a homogeneously mixing community. The model is based on an infectivity process, which we define through stochastic contact and infectiousness processes, so that each individual has an independent infectivity profile. In particular, we monitor variations of the reproduction number and of the distribution of generation times. We show that some interventions, i.e. uniform reduction and vaccination, affect the former while leaving the latter unchanged, whereas other interventions, i.e. isolation, screening and contact tracing, affect both quantities. We provide a theoretical analysis of the variation of these quantities, and we show that, in practice, the variation of the generation time distribution can be significant and that it can cause biases in the estimation of reproduction numbers. The framework, because of its general nature, captures the properties of many infectious diseases, but particular emphasis is on COVID-19, for which numerical results are provided., Comment: 26 pages, 4 figures, 5 tables

Consider a Markovian SIR epidemic model in a homogeneous community. To this model we add a rate at which individuals are tested, and once an infectious individual tests positive it is isolated and each of their contacts are traced and tested independently with some fixed probability. If such a traced individual tests positive it is isolated, and the contact tracing is iterated. This model is analysed using large population approximations, both for the early stage of the epidemic when the "to-be-traced components" of the epidemic behaves like a branching process, and for the main stage of the epidemic where the process of to-be-traced components converges to a deterministic process defined by a system of differential equations. These approximations are used to quantify the effect of testing and of contact tracing on the effective reproduction numbers (for the components as well as for the individuals), the probability of a major outbreak, and the final fraction getting infected. Using numerical illustrations when rates of infection and natural recovery are fixed, it is shown that Test-and-Trace strategy is effective in reducing the reproduction number. Surprisingly, the reproduction number for the branching process of components is not monotonically decreasing in the tracing probability, but the individual reproduction number is conjectured to be monotonic as expected. Further, in the situation where individuals also self-report for testing, the tracing probability is more influential than the screening rate (measured by the fraction infected being screened).

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.

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 Erd\H{o}s-R\'{e}nyi 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 $\lambda$ at its threshold $\lambda_c$ (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 $\lambda$ passes through $\lambda_c$., Comment: 49 pages, 5 figures

Combinations of intense non-pharmaceutical interventions ('lockdowns') were introduced in countries worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement lockdown exit strategies that allow restrictions to be relaxed 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, will 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. The roadmap requires a global collaborative effort from the scientific community and policy-makers, and is made up of three parts: i) improve estimation of key epidemiological parameters; ii) understand sources of heterogeneity in populations; 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.

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.

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.

Consider stochastic models for the spread of an infection in a structured community, where this structured community is itself described by a random network model. Some common network models and transmission models are defined and large population proporties of them are presented. Focus is then shifted to statistical methodology: what can be estimated and how, depending on the underlying network, transmission model and the available data? This survey paper discusses several different scenarios, also giving references to publications where more details can be found., Comment: 20 pages, 4 figures

The directed preferential attachment model is revisited. A new exact characterization of the limiting in- and out-degree distribution is given by two \emph{independent} pure birth processes that are observed at a common exponentially distributed time $T$ (thus creating dependence between in- and out-degree). The characterization gives an explicit form for the joint degree distribution, and this confirms previously derived tail probabilities for the two marginal degree distributions. The new characterization is also used to obtain an explicit expression for tail probabilities in which both degrees are large. A new generalised directed prefererantial attachment model is then defined and analysed using similar methods. The two extensions, motivated by empirical evidence, are to allow double-directed (i.e.\ undirected) edges in the network, and to allow the probability to connect an ingoing (outgoing) edge to a specified node to also depend on the out-degree (in-degree) of that node.

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 $N$., Comment: v1: Submitted. v2: Revised in accordance with journal refereeing; some aspects of presentation changed but no changes to main results/conclusions

These notes describe stochastic epidemics in a homogenous community. Our main concern is stochastic compartmental models (i.e. models where each individual belongs to a compartment, which stands for its status regarding the epidemic under study : S for susceptible, E for exposed, I for infectious, R for recovered) for the spread of an infectious disease. In the present notes we restrict ourselves to homogeneously mixed communities. We present our general model and study the early stage of the epidemic in chapter 1. Chapter 2 studies the particular case of Markov models, especially in the asymptotic of a large population, which leads to a law of large numbers and a central limit theorem. Chapter 3 considers the case of a closed population, and describes the final size of the epidemic (i.e. the total number of individuals who ever get infected). Chapter 4 considers models with a constant influx of susceptibles (either by birth, immigration of loss of immunity of recovered individuals), and exploits the CLT and Large Deviations to study how long it takes for the stochastic disturbances to stop an endemic situation which is stable for the deterministic epidemic model. The document ends with an Appendix which presents several mathematical notions which are used in these notes, as well as solutions to many of the exercises which are proposed in the various chapters., Comment: Part I of "Stochastic Epidemic Models with Inference", T. Britton & E. Pardoux eds., Lecture Notes in Mathematics 2255, Springer 2019

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.

When analysing new emerging infectious disease outbreaks one typically has observational data over a limited period of time and several parameters to estimate, such as growth rate, R0, serial or generation interval distribution, latent and incubation times or case fatality rates. Also parameters describing the temporal relations between appearance of symptoms, notification, death and recovery/discharge will be of interest. These parameters form the basis for predicting the future outbreak, planning preventive measures and monitoring the progress of the disease. We study the problem of making inference during the emerging phase of an outbreak and point out potential sources of bias related to contact tracing, replacing generation times by serial intervals, multiple potential infectors or truncation effects amplified by exponential growth. These biases directly affect the estimation of e.g. the generation time distribution and the case fatality rate, but can then propagate to other estimates, e.g. of R0 and growth rate. Many of the traditionally used estimation methods in disease epidemiology may suffer from these biases when applied to the emerging disease outbreak situation. We show how to avoid these biases based on proper statistical modelling. We illustrate the theory by numerical examples and simulations based on the recent 2014-15 Ebola outbreak to quantify possible estimation biases, which may be up to 20% underestimation of R0, if the epidemic growth rate is fitted to observed data or, conversely, up to 62% overestimation of the growth rate if the correct R0 is used in conjunction with the Euler-Lotka equation.

The current survey paper concerns stochastic mathematical models for the spread of infectious diseases. It starts with the simplest setting of a homogeneous population in which a transmittable disease spreads during a short outbreak. Assuming a large population some important features are presented: branching process approximation, basic reproduction number $R_0$, and final size of an outbreak. Some extensions towards realism are then discussed: models for endemicity, various heterogeneities, and prior immmunity. The focus is then shifted to statistical inference. What can be estimated for these models for various levels of detailed data and with what precision? The paper ends by describing how the inference results may be used for determining successful vaccination strategies. This paper will appear as a chapter of a forthcoming book entitled \emph{Handbook of Infectious Disease Epidemiology}.

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 $\to$ Exposed $\to$ Infectious $\to$ Recovered) structure that allows for both symptomatic and asymptomatic cases. The following question is addressed: what fraction $\rho$ 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 $R_0$, is caused by symptomatic individuals? The latter fraction only depends on the type-specific reproduction numbers, while the former fraction $\rho$ 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 $\rho$ 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. We show how these two classes of models can exhibit very different behaviour.

We couple a multi-type stochastic epidemic process with a directed random graph, where edges have random lengths. This random graph representation is used to characterise the fractions of individuals infected by the different types of vertices among all infected individuals in the large population limit. For this characterisation we rely on theory of multi-type real-time branching processes. We identify a special case of the two-type model, in which the fraction of individuals of a certain type infected by individuals of the same type, is maximised among all two-type epidemics approximated by branching processes with the same mean offspring matrix.

We introduce the effect of site contamination in a model for spatial epidemic spread and show that the presence of site contamination may have a strict effect on the model in the sense that it can make an otherwise subcritical process supercritical. Each site on $\mathbb{Z}^d$ is independently assigned a random number of particles and these then perform random walks restricted to bounded regions around their home locations. At time 0, the origin is infected along with all its particles. The infection then spread in that an infected particle that jumps to a new site causes the site along with all particles located there to be infected. Also, a healthy particle that jumps to a site where infection is presents, either in that the site is infected or in the presence of infected particles, becomes infected. Particles and sites recover at rate $\lambda$ and $\gamma$, respectively, and then become susceptible to the infection again. We show that, for each given value of $\lambda$, there is a positive probability that the infection survives indefinitely if $\gamma$ is sufficiently small, and that, for each given value of $\gamma$, the infection dies out almost surely if $\lambda$ is large enough. Several open problems and modifications of the model are discussed, and some natural conjectures are supported by simulations.

An SEIRS epidemic with disease fatalities is introduced in a growing population (modelled as a super-critical linear birth and death process). The study of the initial phase of the epidemic is stochastic, while the analysis of the major outbreaks is deterministic. Depending on the values of the parameters, the following scenarios are possible. i) The disease dies out quickly, only infecting few; ii) the epidemic takes off, the \textit{number} of infected individuals grows exponentially, but the \textit{fraction} of infected individuals remains negligible; iii) the epidemic takes off, the \textit{number} of infected grows initially quicker than the population, the disease fatalities diminish the growth rate of the population, but it remains super critical, and the \emph{fraction} of infected go to an endemic equilibrium; iv) the epidemic takes off, the \textit{number} of infected individuals grows initially quicker than the population, the diseases fatalities turn the exponential growth of the population to an exponential decay.

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.

This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate $\omega$ (and reconnect to non-infectious individuals with probability $\alpha$ or else simply drop the edge if $\alpha=0$), so-called preventive rewiring. The models are denoted SIR-$\omega$ and SEIR-$\omega$, and we focus attention on the early stages of an outbreak, where we derive expression for the basic reproduction number $R_0$ and the expected degree of the infectious nodes $E(D_I)$ using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIR-$\omega$ and SEIR-$\omega$ epidemics on Poisson and scale-free networks. Without rewiring of exposed nodes, the two approaches predict the same epidemic threshold and the same $E(D_I)$ for both types of epidemics, the latter being very close to the mean degree obtained from simulated epidemics over Poisson networks. Above the epidemic threshold, pairwise models overestimate the value of $R_0$ computed from simulations, which turns out to be very close to the one predicted by the branching process approximation. When exposed individuals also rewire with $\alpha > 0$ (perhaps unaware of being infected), the two approaches give different epidemic thresholds, with the branching process approximation being more in agreement with simulations., Comment: 25 pages, 7 figures

Let $F$ be a probability distribution with support on the non-negative integers. Four methods for generating a simple undirected graph with (approximate) degree distribution $F$ are described and compared. Two methods are based on the so called configuration model with modifications ensuring a simple graph, one method is an extension of the classical Erd\H{o}s-R\'{e}nyi graph where the edge probabilities are random variables, and the last method starts with a directed random graph which is then modified to a simple undirected graph. All methods are shown to give the correct distribution in the limit of large graph size, but under different assumptions on the degree distribution $F$ and also using different order of operations.