key: cord-102269-lfdvl78a authors: Singer, B. J.; Thompson, R. N.; Bonsall, M. B. title: The effect of the definition of 'pandemic' on quantitative assessments of infectious disease outbreak risk date: 2020-10-05 journal: nan DOI: 10.1101/2020.10.02.20205682 sha: doc_id: 102269 cord_uid: lfdvl78a In the early stages of an outbreak, the term 'pandemic' can be used to communicate about infectious disease risk, particularly by those who wish to encourage a large-scale public health response. However, the term lacks a widely accepted quantitative definition. We show that, under alternative quantitative definitions of 'pandemic', an epidemiological metapopulation model produces different estimates of the probability of a pandemic. Critically, we show that using different definitions alters the projected effects on the pandemic risk of key parameters such as inter-regional travel rates, degree of pre-existing immunity, and heterogeneity of transmission rates between regions. Our analysis provides a foundation for understanding the scientific importance of precise language when discussing pandemic risk, illustrating how alternate definitions affect the conclusions of modelling studies. This serves to highlight that those working on pandemic preparedness must remain alert to the variability in the use of the term 'pandemic', and provide specific quantitative of analysis that we show to be sensitive to the pandemic definition. In the early stages of an infectious disease outbreak, it is important to determine whether the pathogen 18 responsible may go on to cause an epidemic or a pandemic [1] [2] [3] [4] [5] . There is extensive literature on determining 19 the probability of a major epidemic given a small population of initial infected hosts [6] [7] [8] [9] . This research has 20 produced a natural mathematical definition of an epidemic, based on the bimodal distribution of outbreak 21 sizes given by simple epidemiological models when R0 is larger but not close to one [10] . The term 'pandemic' 22 has no corresponding theoretical definition, and there is no consensus mathematical approach to determining 23 the probability of a pandemic. In this study, we examine how alternative definitions of 'pandemic' affect 24 quantitative estimates of pandemic risk assessed early in an infectious disease outbreak. 25 The term 'pandemic' is used extensively, appearing in phrases such as 'pandemic preparedness' [11] [12] [13] , 26 'pandemic influenza' [14] [15] [16] , and 'pandemic potential' [17] [18] [19] . A Google Scholar search returns 25,800 27 results using the term 'pandemic' for 2019 alone. 28 The International Epidemiology Association's Dictionary of Epidemiology defines a pandemic as "an 29 epidemic occurring worldwide, or over a very wide area, crossing international boundaries and usually 30 affecting a large number of people" [20] . Notably this definition makes an explicit reference to national there is little in common between all disease outbreaks that have been referred to as pandemics, except that 36 they have a wide geographical extension [24] . 37 These kinds of differences between pandemic definitions can often go unnoticed, but in certain circum-38 stances they can cause confusion between different stakeholders (e.g. between scientists and governments, or 39 governments and the public), who may not have a shared background understanding of the term. In 2009, 40 the World Health Organisation (WHO) declared a pandemic of H1N1 influenza, using criteria related to the 41 incidence and spread of the virus in different WHO regions [25] . The criteria did not include reference to of novel viruses [36] . Others treat the spread of a pathogen at a pandemic level as a context in which to 59 study transmission dynamics, without paying special attention to how those dynamics lead to a pandemic 60 as distinct from an epidemic or a more limited outbreak [37] [38] [39] . In this paper, we examine the effects 61 of alternate pandemic definitions on the analysis of key epidemiological questions. The results provide a 62 foundation for deciding the appropriate quantitative definition of 'pandemic' in a given context. 63 We make use of a metapopulation model to investigate the effects of pandemic definition on the results of 64 a quantitative assessment of the probability of a pandemic. Metapopulation models are commonly applied to 65 pathogens that spread between regions of the world, and so are appropriate for modelling pandemics [40] [41] [42] [43] [44] [45] . 66 We explore two different kinds of pandemic definition, following Morens et al. 2009 [24] , specifically: • the family of transregional definitions, where a pandemic is defined as an outbreak in which the 68 number of regions experiencing epidemics meets or exceeds some threshold number n. We refer to 69 specific transregional definitions as n-region transregional definitions, e.g. a 3-region transregional 70 definition. • the interregional definition, where a pandemic is defined as an outbreak in which two or more non-72 adjacent regions experience epidemics. Note that these definitions require a specific sense of 'region'. These regions could be countries, or they 74 could be larger or smaller than individual countries-from counties to health zones to WHO regions. Our 75 metapopulation model (detailed in the Methods section below) can be used to model regions of any size. 76 We have chosen not to include definitions with criteria relating to the number of people infected or killed, 77 instead of, or in addition to, geographical extension. Extension is the only universal factor in pandemic 78 definitions, and so is the focus of the current study [24] . Three questions that feed into public health policy at the beginning of an outbreak are: • Would interventions restricting travel reduce the risk of a pandemic? • Does a portion of the population have pre-existing immunity, and does this affect pandemic risk? whose connections and weighting can be set at fixed values representing the rates of travel between regions. 95 We choose the simplest networks that can illustrate the effects of our different pandemic definitions-namely, 96 the star network, in which one central region is connected to all others with equal weighting and the non-97 central regions lack any other connections, and the fully connected network, in which each region is connected 98 to every other with equal weighting. Figure 1 illustrates that the connectivity of the full network is much 99 higher than that of the star network. Using the star network allows us to make the distinction between 100 adjacent and non-adjacent regions, thus allowing us to distinguish between transregional and interregional 101 pandemic definitions. Unless otherwise stated, all figures in the current study are generated with a transmission rate of β = 0.28 103 per day, a recovery rate of µ = 0.14 per day, and an inter-regional travel rate of 2 × 10 −4 per day. This : Pandemic probability for a range of between-region transmission rates and a range of pandemic definitions in terms of number of regions experiencing epidemics. The "pandemic threshold number" refers to the minimum number of regions that must experience epidemics before a pandemic is declared. Pandemic probability is, in general, sensitive to the pandemic definition used, but the degree of sensitivity depends on network structure and travel rates. a) Pandemic probability for a star network. Pandemic probability is, in general, highly sensitive to the pandemic definition used. b) Pandemic probability for a fully connected network. The sensitivity of pandemic probability to the pandemic definition used is significantly reduced at high travel rates. We can also look at the difference in pandemic probability between the transregional and interregional is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted October 5, 2020. . per day. In the full network there is sudden transition from higher risk to lower risk, as cross-immunity 162 approaches one. However, in the star network there is less circulation of the initial pathogen, so the effect 163 of cross-immunity is less dramatic. Increased cross-immunity can also increase the difference in risk for 164 different pandemic definitions-for the fully connected network, once cross-immunity reaches around α = 0.6, 165 differences in probability between different thresholds become visible that are much smaller at lower values. This suggests that the declaration of a pandemic may be more sensitive to the exact pandemic definition for 167 outbreaks of pathogens that encounter pre-existing immunity than for pathogens which encounter only fully 168 susceptible populations. However, this effect is not seen for the star network, in which the low connectivity 169 of the network results in larger differences in probability between different thresholds even at low cross-170 immunity. A topic of great concern during a pandemic is heterogeneity in risk between different countries or regions 173 [50, 51]. Cross-immunity can create one kind of heterogeneity, since it is common for previous exposure to 174 a pathogen to differ between regions. Another kind of heterogeneity is that due to different public health is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted October 5, 2020. . The second question was "Does a portion of the population have pre-existing immunity, and does 221 this affect pandemic risk?" The presence of immunity can significantly alter the results discussed in the 222 paragraphs above. In figure 5b , the leftmost column is equivalent to the column from figure 2b in which 223 λ = 2.0 × 10 −4 per day. However, as cross-immunity increases, a marked difference in the pandemic pro-224 bability between different definitions becomes visible. This shows that the conclusion that precise pandem-225 ic definitions are of reduced importance in a highly connected network with high travel rates is context 226 sensitive-if the population has high immunity, differences between definitions re-emerge. The third question was "How is the risk of a pandemic affected by differences between regions?" In figure is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted October 5, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted October 5, 2020. To determine how the final probabilities of epidemics depend on the pairwise probabilities qjm, we use 307 a Markov chain. The states of this Markov chain assign one of three states to each region-N (for neutral), 308 where it is not yet determined whether the region will experience an epidemic, E (for epidemic), where 309 it is determined that the region will experience an epidemic but it is not yet determined in which further 310 regions it will cause epidemics, and T (for terminal), where it is determined that the region will experience 311 an epidemic and in which further regions it will cause epidemics due to onward transmission. As our model The initial probability of each global state z1z2...zn (where zi ∈ {N, E, T }) is given by: where Qj = min((1/R0,j) I j (0) , 1) is the probability that the initial population of infective individuals does is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted October 5, 2020. . https://doi.org/10.1101/2020.10.02.20205682 doi: medRxiv preprint initial global state is given by the product of the probabilities of each region being in the corresponding 334 initial regional state. In this system, all states in which no region is epidemic are absorbing, and in each transition at least 336 one epidemic state must become terminal. This means that the system must reach an absorbing state in at 337 most n transitions, since at least one region becomes terminal in each transition, and a fully terminal state 338 is absorbing. So the final probability vector p final is given by with T as the transition matrix and p initial as the vector whose elements defined by equation (5) The model described above can incorporate certain epidemiological details, such as heterogeneity of popu-344 lation parameters, but is restricted to treating quite simple disease dynamics. In this section we expand the 345 model to treat pathogens that give those who overcome infection cross-protection against future strains of 346 that pathogen. This is necessary to be able to investigate how pre-existing immunity changes how pandemic 347 definitions affect the results of our model. 348 We first describe the spread of a pathogen strain X using the methods above, introducing a superscript 349 X to the relevant parameters to mark the strain, e.g. R X 0 , R X (∞), and p X final . We assume that infection 350 with pathogen X confers cross-immunity α to a second strain of the pathogen, which we call Y . In each 351 population Pj we can define an effective basic reproductive number for Y in the case that Pj has experienced 352 an epidemic of X, which we call R Y e,j . This expression simply multiplies the basic reproductive number by the effective number of susceptible 354 individuals given the prevalence of cross-immunity in the population. It is through this expression that 355 cross-immunity enters the model-the parameter α does not otherwise appear in what follows. 356 We can write down an equation for the expected total number of individuals in Pj infected in an epidemic 357 of Y in analogy to equation (1). In the case where there has been no previous epidemic of X in Pj, the 358 expected epidemic size is the solution R Y j,noX (∞) of In the case where there has been a previous epidemic of X in Pj, the expected epidemic size is the solution We assume that individuals infected with Y travel at the same rate as individuals infected with X. We 362 then define the pairwise probabilities of transmission of Y between populations in analogy to equation (2). That is, is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted October 5, 2020. . https://doi.org/10.1101/2020.10.02.20205682 doi: medRxiv preprint where R Y c,m = R Y 0,m when Pm has not experienced a previous epidemic of X, R Y c,m = R Y e,m when Pm has 365 experienced a previous epidemic of X, R Y b,j (∞) = R Y noX,j (∞) when Pj has not experienced a previous 366 epidemic of X, and R Y b,j (∞) = R Y X,j (∞) when Pj has experienced a previous epidemic of X. These expressions for q Y jm can be substituted for qjm in equation (3) to yield a transition matrix for 368 modelling the spread of Y , which we will call T Y (s1s2...sn), where sj is the final state (either N or T ) of 369 the X outbreak in Pj. We find the initial probabilities of each state with regards to Y , p Y initial , in analogy 370 to equation (5), given an initial number of individuals infected with Y in each population I Y j (∞). To find the overall probability of each combination of epidemics of Y in various populations given a prior 376 probability of each combination of epidemics of X (given by p X final (s1s2...sn) defined in equation (6) is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted October 5, 2020. . https://doi.org/10.1101/2020.10.02.20205682 doi: medRxiv preprint Characteristics of Microbes Most 391 Likely to Cause Pandemics and Global Catastrophes Novel Coronavirus Outbreak in Wuhan, China, 2020: Intense Surveillance Is Vital 394 for Preventing Sustained Transmission in New Locations Effects of population density on the spread of disease Spread of infectious disease through clustered populations How big is an outbreak likely to be? 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