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  • 1. Ball, Frank
    et al.
    Britton, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    AN EPIDEMIC IN A DYNAMIC POPULATION WITH IMPORTATION OF INFECTIVES2017In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 27, no 1, p. 242-274Article in journal (Refereed)
    Abstract [en]

    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 -> infective -> recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where n -> infinity, 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 -> infinity, 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 >= 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 start of the regenerative cycle. Properties of the process S, including the jump size and stationary distributions, are determined.

  • 2. Ball, Frank G.
    et al.
    Sirl, David J.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    EPIDEMICS ON RANDOM INTERSECTION GRAPHS2014In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 24, no 3, p. 1081-1128Article in journal (Refereed)
    Abstract [en]

    In this paper we consider a model for the spread of a stochastic SIR (Susceptible -> Infectious -> Recovered) epidemic on a network of individuals described by a random intersection graph. Individuals belong to a random number of cliques, each of random size, and infection can be transmitted between two individuals if and only if there is a clique they both belong to. Both the clique sizes and the number of cliques an individual belongs to follow mixed Poisson distributions. An infinite-type branching process approximation (with type being given by the length of an individual's infectious period) for the early stages of an epidemic is developed and made fully rigorous by proving an associated limit theorem as the population size tends to infinity. This leads to a threshold parameter R-*, so that in a large population an epidemic with few initial infectives can give rise to a large outbreak if and only if R-*>1. A functional equation for the survival probability of the approximating infinite-type branching process is determined; if R-*<= 1, this equation has no nonzero solution, while if R-*>1, it is shown to have precisely one nonzero solution. A law of large numbers for the size of such a large outbreak is proved by exploiting a single-type branching process that approximates the size of the susceptibility set of a typical individual.

  • 3. Ball, Frank
    et al.
    Pellis, Lorenzo
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Reproduction numbers for epidemic models with households and other social structures II: Comparisons and implications for vaccination2016In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 274, p. 108-139Article in journal (Refereed)
    Abstract [en]

    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 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 >= R-0 when R-0 >= 1 and R-r <= R-0 when R-0 <= 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.

  • 4.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    House, Thomas
    Lloyd, Alun L.
    Mollison, Denis
    Riley, Steven
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Five challenges for stochastic epidemic models involving global transmission2015In: Epidemics, ISSN 1755-4365, E-ISSN 1878-0067, Vol. 10, p. 54-57Article in journal (Refereed)
    Abstract [en]

    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?

  • 5.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Leung, Ka Yin
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Who is the infector? General multi-type epidemics and real-time susceptibility processes2019In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 51, no 2, p. 606-631Article in journal (Refereed)
    Abstract [en]

    We couple a multi-type stochastic epidemic process with a directed random graph, where edges have random weights (traversal times). 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 the 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.

  • 6.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Inferring global network properties from egocentric data with applications to epidemics2015In: Mathematical Medicine and Biology, ISSN 1477-8599, E-ISSN 1477-8602, Vol. 32, no 1, p. 99-112Article in journal (Refereed)
    Abstract [en]

    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.

  • 7.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Maximizing the size of the giant2012In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 49, no 4, p. 1156-1165Article in journal (Refereed)
    Abstract [en]

    Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring 'on' the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight X-i with E(X-i) = mu, and where there is an edge between i and j with probability 1 - e(-XiXj/(mu n)), independently of other edges. The second model is the thinned configuration model in which then vertices of the ground graph have i.i.d. ground degrees, distributed as D, with E(D) = mu. The graph of interest is obtained by deleting edges independently with probability 1 - p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1 - q, where q depends on X or D and p. We investigate for which distributions X and D with given mu and p, 1 - q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.

  • 8.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Stochastic Epidemics in Growing Populations2014In: Bulletin of Mathematical Biology, ISSN 0092-8240, E-ISSN 1522-9602, Vol. 76, no 5, p. 985-996Article in journal (Refereed)
    Abstract [en]

    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.

  • 9.
    Deijfen, Maria
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rosengren, Sebastian
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    The Tail does not Determine the Size of the Giant2018In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 173, no 3-4, p. 736-745Article in journal (Refereed)
    Abstract [en]

    The size of the giant component in the configuration model, measured by the asymptotic fraction of vertices in the component, is given by a well-known expression involving the generating function of the degree distribution. In this note, we argue that the distribution over small degrees is more important for the size of the giant component than the precise distribution over very large degrees. In particular, the tail behavior of the degree distribution does not play the same crucial role for the size of the giant as it does for many other properties of the graph. Upper and lower bounds for the component size are derived for an arbitrary given distribution over small degrees d <= L and given expected degree, and numerical implementations show that these bounds are close already for small values of L. On the other hand, examples illustrate that, for a fixed degree tail, the component size can vary substantially depending on the distribution over small degrees.

  • 10.
    Fransson, Carolina
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    SIR epidemics and vaccination on random graphs with clustering2019In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 78, no 7, p. 2369-2398Article in journal (Refereed)
    Abstract [en]

    In this paper we consider Susceptible Infectious 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 R0, 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, R0 equals the perfect vaccine-associated reproduction number. Generalizations to groups larger than three are discussed briefly.

  • 11. Koval, Vyacheslav
    et al.
    Meester, Ronald
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Long range percolation on the hierarchical lattice2012In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 17, p. 1-21Article in journal (Refereed)
    Abstract [en]

    We study long-range percolation on the hierarchical lattice of order N, where any edge of length k is present with probability p(k) = 1 - exp (-beta(-k)alpha), independently of all other edges. For fixed beta, we show that alpha(c)(beta) ( the infimum of those alpha for which an infinite cluster exists a.s.) is non-trivial if and only if N < beta < N-2. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of alpha(c)(beta) as a function of beta. This means that the phase diagram of this model is well understood.

  • 12.
    Lashari, Abid Ali
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Branching process approach for epidemics in dynamic partnership network2018In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 76, no 1-2, p. 265-294Article in journal (Refereed)
    Abstract [en]

    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 R0. 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.

  • 13.
    Lashari, Abid
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Serafimović, Ana
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    The duration of an SIR epidemic on a configuration modelManuscript (preprint) (Other academic)
    Abstract [en]

    We consider the spread of a stochastic SIR (Susceptible, Infec-tious, Recovered) epidemic on a configuration model random graph.We focus especially on the final stages of the outbreak and providelimit results for the duration of the entire epidemic, while we allowfor non-exponential distributions of the infectious period and for bothfinite and infinite variance of the asymptotic degree distribution in thegraph.

    Our analysis relies on the analysis of some subcritical continuoustime branching processes and on ideas from first-passage percolation.

  • 14.
    Lashari, Abid
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Effect of vaccination on the duration of an SIR epidemic in homogeneously mixing and structured populationsManuscript (preprint) (Other academic)
    Abstract [en]

    This paper is concerned with the effects of vaccination on the properly scaled dura-tion of the stochastic SIR (Susceptible → Infected → Recovered/Removed) epidemicboth in homogeneous mixing populations and in populations structured through config-uration model random graphs. We examine uniform vaccination and leaky vaccinationin both homogeneous and structured populations. Furthermore, we consider acquain-tance vaccination on the configuration model. For these vaccination schemes, we studythe asymptotic time until the end of the epidemic and study the effect of the vaccinationon this duration.

    We show that, depending on the degree distribution, uniform vaccination witha perfect vaccine may lead to both an increase and decrease in the duration of anepidemic in structured populations, whereas in homogeneously mixing populations,vaccination with a perfect vaccine either prevents or prolongs an epidemic in the largepopulation limit. In homogeneously mixing populations, the leaky vaccine has a similareffect as uniform vaccination on the duration of the epidemic, whereas in structuredpopulations, we conjecture that the leaky vaccine always increases the duration of anepidemic. For the acquaintance vaccination scheme we give, through the derivationof the effective degree distribution of unvaccinated individuals, a recipe to obtain theasymptotic duration of an epidemic and show that acquaintance vaccination may bothdecrease and increase the duration of an epidemic.

  • 15.
    Lashari, Abid
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ball, Frank
    Sirl, David
    Modeling the spread of two successive SIR epidemics on a configuration model networkManuscript (preprint) (Other academic)
    Abstract [en]

    We present a stochastic model for two successive SIR (Susceptible, Infectious, Recov-ered) epidemics in the same network structured population. Individuals infected duringthe first epidemic might have (partial) immunity for the second one. The first epidemic isanalysed through a bond percolation model, while the second epidemic is approximated bya three-type branching process in which the types of individuals depend on their position inthe percolation clusters used for the first epidemic. This branching process approximationenables us to calculate a threshold parameter and the probability of a large outbreak for thesecond epidemic.

    We illustrate our results through two examples. In the first example individuals infectedby the first epidemic are independently either completely susceptible or completely immuneto the second epidemic. The probability of being completely immune is the same for allindividuals infected in the first epidemic. In the second example the recovered individual inthe first epidemic have reduced susceptibility and infectivity for the second epidemic.

  • 16.
    Leung, Ka Yin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Britton, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Who is the infector? Epidemic models with symptomatic and asymptomatic cases2018In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 301, p. 190-198Article in journal (Refereed)
    Abstract [en]

    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 p 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,R- is caused by symptomatic individuals? The latter fraction only depends on the type-specific reproduction numbers, while the former fraction p 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 p 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.

  • 17. Ouboter, Tanneke
    et al.
    Meester, Ronald
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Stochastic SIR epidemics in a population with households and schools2016In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 72, no 5, p. 1177-1193Article in journal (Refereed)
    Abstract [en]

    We study the spread of stochastic SIR (Susceptible Infectious 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.

  • 18. Pellis, Lorenzo
    et al.
    Ball, Frank
    Bansal, Shweta
    Eames, Ken
    House, Thomas
    Isham, Valerie
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Eight challenges for network epidemic models2015In: Epidemics, ISSN 1755-4365, E-ISSN 1878-0067, Vol. 10, p. 58-62Article in journal (Refereed)
    Abstract [en]

    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.

  • 19. Pellis, Lorenzo
    et al.
    Ball, Frank
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Reproduction numbers for epidemic models with households and other social structures. I. Definition and calculation of R-02012In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 235, no 1, p. 85-97Article in journal (Refereed)
    Abstract [en]

    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.

  • 20. Riley, Steven
    et al.
    Eames, Ken
    Isham, Valerie
    Mollison, Denis
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Five challenges for spatial epidemic models2015In: Epidemics, ISSN 1755-4365, E-ISSN 1878-0067, Vol. 10, p. 68-71Article in journal (Refereed)
    Abstract [en]

    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.

  • 21.
    Spricer, Kristoffer
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Characterizing the Initial Phase of Epidemic Growth on some Empirical Networks2018In: Stochastic Processes and Applications / [ed] Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić, Springer, 2018, p. 315-334Conference paper (Refereed)
    Abstract [en]

    A key parameter in models for the spread of infectious diseases is the basic reproduction number R0">R 0  R0 , which is the expected number of secondary cases a typical infected primary case infects during its infectious period in a large mostly susceptible population. In order for this quantity to be meaningful, the initial expected growth of the number of infectious individuals in the large-population limit should be exponential. We investigate to what extent this assumption is valid by simulating epidemics on empirical networks and by fitting the initial phase of each epidemic to a generalised growth model, allowing for estimating the shape of the growth. For reference, this is repeated on some elementary graphs, for which the early epidemic behaviour is known. We find that for the empirical networks tested in this paper, exponential growth characterizes the early stages of the epidemic, except when the network is restricted by a strong low-dimensional spacial constraint.

  • 22.
    Trapman, Pieter
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ball, Frank
    Dhersin, Jean-Stephane
    Viet, Chi
    Wallinga, Jacco
    Britton, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Inferring R-0 in emerging epidemics-the effect of common population structure is small2016In: Journal of the Royal Society Interface, ISSN 1742-5689, E-ISSN 1742-5662, Vol. 13, no 121, article id 20160288Article in journal (Refereed)
    Abstract [en]

    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.

  • 23.
    Trapman, Pieter
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lambert, Amaury
    UPMC Univ Paris 06.
    Splitting trees stopped when the first clock rings and Vervaat's transformation2013In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 50, no 1, p. 208-227Article in journal (Refereed)
    Abstract [en]

    We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let Nt denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (Nt, T, X(T)), where X(T)is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y'M) conditional on {M≠0}. HereM+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; IM is the infimum of the path YM, which in turn is defined as Y killed at its last visit of 0 before e; and Y'M is the Vervaat transform of YM. This identity yields an explanation for the geometric distribution of NT (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {NT=n}, and also on {NT=n,T<a}, the ages and residual lifetimes of the n alive individuals at timeT are i.i.d. and independent of n. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital

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