| 1. |
- Ball, Frank, et al.
(författare)
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AN EPIDEMIC IN A DYNAMIC POPULATION WITH IMPORTATION OF INFECTIVES
- 2017
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Ingår i: The Annals of Applied Probability. - 1050-5164 .- 2168-8737. ; 27:1, s. 242-274
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Ball, Frank G., et al.
(författare)
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EPIDEMICS ON RANDOM INTERSECTION GRAPHS
- 2014
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Ingår i: The Annals of Applied Probability. - 1050-5164 .- 2168-8737. ; 24:3, s. 1081-1128
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Ball, Frank, et al.
(författare)
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Reproduction numbers for epidemic models with households and other social structures II : Comparisons and implications for vaccination
- 2016
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Ingår i: Mathematical Biosciences. - : Elsevier BV. - 0025-5564 .- 1879-3134. ; 274, s. 108-139
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Britton, Tom, et al.
(författare)
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A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2
- 2020
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Ingår i: Science. - : American Association for the Advancement of Science (AAAS). - 0036-8075 .- 1095-9203. ; 369:6505, s. 846-849
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Tidskriftsartikel (refereegranskat)abstract
- Despite various levels of preventive measures, in 2020, many countries have suffered severely from the coronavirus 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 similar to 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.
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| 5. |
- Britton, Tom, et al.
(författare)
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Five challenges for stochastic epidemic models involving global transmission
- 2015
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Ingår i: Epidemics. - : Elsevier BV. - 1755-4365 .- 1878-0067. ; 10, s. 54-57
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Tidskriftsartikel (refereegranskat)abstract
- 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?
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| 6. |
- Britton, Tom, et al.
(författare)
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Inferring global network properties from egocentric data with applications to epidemics
- 2015
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Ingår i: Mathematical Medicine and Biology. - : Oxford University Press (OUP). - 1477-8599 .- 1477-8602. ; 32:1, s. 99-112
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Britton, Tom, et al.
(författare)
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Maximizing the size of the giant
- 2012
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Ingår i: Journal of Applied Probability. - : Cambridge University Press (CUP). - 0021-9002 .- 1475-6072. ; 49:4, s. 1156-1165
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- Britton, Tom, et al.
(författare)
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Stochastic Epidemics in Growing Populations
- 2014
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Ingår i: Bulletin of Mathematical Biology. - : Springer Science and Business Media LLC. - 0092-8240 .- 1522-9602. ; 76:5, s. 985-996
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Britton, Tom, et al.
(författare)
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The risk for a new COVID-19 wave and how it depends on R-0, the current immunity level and current restrictions
- 2021
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Ingår i: Royal Society Open Science. - : The Royal Society. - 2054-5703. ; 8:7
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Tidskriftsartikel (refereegranskat)abstract
- The COVID-19 pandemic has hit different regions differently. The current disease-induced immunity level i 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 i plays a more influential roll than when immunity is obtained from vaccination. Secondly, by comparing regions with different R-0 and i it is shown that regions with lower R-0 and low i may need higher preventive measures (p(Min)) compared with regions having higher R-0 but also higher i, 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.
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| 10. |
- Britton, Tom, et al.
(författare)
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Who is the infector? General multi-type epidemics and real-time susceptibility processes
- 2019
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 51:2, s. 606-631
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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