under these definitions often coincides with the standard branching process estimate for 24 the major epidemic probability. However, these practically relevant risk assessments can 25 also be different from the major epidemic probability, as well as from each other. This outbreak is to assess the risk posed by the invading pathogen, including whether initial 42 cases will lead on to a major epidemic or whether the pathogen will die out rapidly instead 43 [8, 9] . An important practical consequence is that, if an outbreak is likely simply to fade 44 out, then costly interventions such as vaccination [10, 11] , culling/felling/roguing of plants 45 48 There is a well-known estimate for the probability of a major epidemic when a pathogen 49 is newly arrived in a host population, which in its simplest form is given by or not an outbreak is likely to persist beyond the initial stochastic phase, becoming a major 76 epidemic does not guarantee that the outbreak will overwhelm available control 77 resources. Over many outbreaks under identical conditions, if the population size is large 78 and R0 is much greater than one, then the distribution of possible epidemic sizes is 79 bimodal according to simple epidemic models such as the stochastic SIR model (Fig 1d  80 -see also [38] [39] [40] [41] [42] ). In other words, the final size of any single outbreak is almost always that have a final size in the higher of these ranges. The outbreaks within the higher range, 86 however, do not necessarily represent outbreaks in which available control resources are 87 exceeded. For practical assessments of the threat from an invading pathogen, it would 88 often therefore be appropriate for the notion of a severe epidemic to be grounded in 89 consequences for disease control, depending on the specific system and outbreak under 
The x-axis has been split into bars of width 20 (so that, for example, the first bar corresponds to the 202 probability that between 1 and 20 individuals are ever infected).
We conduct five analyses. In the first three analyses, we consider the stochastic SIS 207 model, the SIR model and the Zika host-vector model, and assess the risk that a single 208 initial case will lead to an outbreak with a large "concurrent size". Our final two analyses 209 focus on the stochastic SIS model. We calculate the probability of an outbreak going on 210 to exceed a pre-specified total number of infections (the "total infections" metric) or time 211
(the "duration" metric). 212
Here, we describe the epidemiological models that we use, calculation of the branching 214 process estimate of the major epidemic probability for each model, and how the 215 probability of a severe epidemic can be calculated under the "concurrent size" metric for 216 each of the models considered. We then explain how the probability of a severe epidemic 217 under the other practically relevant metrics can be obtained for the SIS model, although 218 our methodology generalises immediately to any model for which a method of stochastic 219 simulation is available. where represents the infection rate between each susceptible-infected pair and is the 231 rate at which each infected host recovers and becomes susceptible again. We use the 232 analogous stochastic model in most of our analyses, in which the net rate at which any 233 epidemiological event occurs is + . At any time prior to the end of the outbreak, the 234 probability that this next event is an infection is In the analogous stochastic model, the net rate at which any epidemiological event occurs is still + , and the probability that the next event is an infection event is 252 similarly unchanged at hosts are sufficiently abundant that the number of hosts that each vector bites is 266 independent of the total number of hosts). The deterministic version of this model is then 267
(4) 275
The parameters and ) govern the rates at which infectious vectors infect susceptible 277 hosts and infectious hosts infect susceptible vectors, respectively. The mean latent 278 periods of infections in hosts and vectors are given by 1/ * and 1/ ) . The parameter 279 is the rate of removal of infectious hosts, and describes the death rate of every vector. 280
In the analogous stochastic model, the number of infected human hosts arising from a 281 single infected human (accounting for human-vector-human transmission) in an 282 otherwise entirely susceptible population of humans and vectors is given by 283
is the expected number of vectors infected (and going on to enter 287 the exposed class) by a single infectious human,
is the proportion of 288 
The commonly used estimate for the major epidemic probability when a pathogen first 306 arrives in a host population [8, 9, 35, 36, 21, 27, [29] [30] [31] [32] [33] [34] can be derived by assuming that 307
infections occur according to a branching process, making the assumptions that the 308 susceptible population is large and that infection lineages arising from different infected 309 hosts are independent. When a single infected host arrives in an otherwise susceptible 310 population, the branching process estimate for the major epidemic probability is given by 311
This expression is derived in Text S1. 315
If instead there are I(0) infected individuals initially rather than one, then for no major 317 epidemic to occur, it is necessary for each initial infection lineage to die out, leading to 318 the approximation given in equation (1) whenever R0 > 1. 319 320
Here we show how the branching process estimate for the major epidemic probability can 322 be derived for more complex systems by considering the stochastic version of the host- We denote the probability of no major epidemic occurring starting from i exposed or 331 infectious human hosts, j exposed vectors and k infectious vectors by qijk. We must 332 consider exposed and infectious vectors separately to account for the possibility that 333 exposed vectors die before becoming infectious. 334
Starting from a single infectious host introduced into an entirely susceptible population of 336 hosts and vectors, and conditioning on the next event, gives 337
because the next event can then either be recovery of the infectious host (with 339
starting from a single exposed or infectious vector gives 341
We again assume that infection lineages are independent, permitting us to approximate 344 terms with two exposed or infectious individuals by non-linear terms involving single 345 exposed or infectious individuals, e.g. ..! ≈ .!! !.! . Noting that !!! = 1, the three 346 equations above can be solved to give expressions for .!! , !.! and !!. . In particular, 347 the probability of a major epidemic starting from a single infected host is then 348
In this expression, ! )* is the expected number of humans infected by a single infectious 350 vector in an otherwise entirely susceptible population of humans and vectors. 351
Probability of a severe epidemic ("concurrent size" metric) 353
As described in the introduction, we first define a major epidemic to be an outbreak in 355 which the maximum number of individuals infected simultaneously is above a threshold , 389 . 398
The probability of a severe epidemic according to the "concurrent size" metric is then 400
for 1 < (0) < , 1 for (0) ≥ . We also consider the probability of a severe epidemic according to the stochastic SIS 429 model for the "total infections" and "duration" metrics. Specifically, we estimate the 430 probability that at least F infections occur over the course of the outbreak (prior to 431 outbreak extinction), and the probability that the outbreak persists for at least T days. However, our main focus is assessing the risk from an invading pathogen, according to 452 the "concurrent size", "total infections" and "duration" metrics for defining severe 453 epidemics. We therefore considered the more realistic stochastic models, in which 454 demographic stochasticity is included. In the following sections, first we calculate the 455 In large host populations, the probability of a severe epidemic as a function of M took the 510 form of a step function (Fig 3b) . This suggested that, if the pathogen successfully invaded 511 the population, then the number of infected individuals would definitely reach a specific 512 maximum value which is determined by R0. For example, for outbreaks with R0 = 1.5, the 513 pathogen will invade the population with probability 0.33 (i.e. the probability of a major 514 epidemic), and, if this occurs, then around two-thirds of the population will be infected 515 simultaneously at some time during the epidemic. invades in the stochastic SIS model (so that a major epidemic occurs), I will fluctuate 525 approximately symmetrically around the deterministic equilibrium value (Fig 3d) . This is 526 consistent with the previous finding that, in the stochastic SIS model, the quasi-stationary 527 distribution can be approximated by the (symmetric) normal distribution when ! is larger We considered the probability that the maximum number of individuals infected 545 simultaneously is at least a pre-specified threshold (i.e. a severe epidemic occurs, using 546 the "concurrent size" metric) under the SIR and Zika virus transmission models. For the 547 stochastic SIR model, we used an iterative method to calculate this probability as 548 Under the stochastic SIR and Zika virus transmission models, for R0 larger than and not 575 close to one, the maximum number of simultaneously infected individuals whenever the 576 pathogen invaded the host population was typically smaller than under the SIS model (cf. 577 Fig 2) . Nonetheless, we found qualitatively similar behaviour in these cases -the 578 probability of a severe epidemic was similar to the major epidemic probability 579 approximated using a branching process for a wide range of values of the severe 580 epidemic threshold when R0 was high (Fig 4) . However, even if that is the case, the 581 practically relevant value of the severe epidemic threshold (e.g. the number of available 582 hospital beds) may mean that the severe epidemic probability does not match the major 583 epidemic probability. For example, if R0 = 2 and 250 beds are available, for the SIR model 584 the probability of a severe epidemic under the "concurrent size" definition is 0 (solid grey 585 line in Fig 4a) , yet the branching process estimate for the probability of a major epidemic 586 is 0.5 (dotted grey line in Fig 4a) . For the stochastic SIS model, we then calculated the probability of a severe epidemic 591 using different metrics to define a severe epidemic -specifically, outbreaks in which there 592 are at least F infection events (the "total infections" metric - Fig 5a) or outbreaks that 593 persist for at least T days (the "duration" metric - Fig 5b) . 
(a) A severe epidemic is defined as an outbreak in which at least F infections occur ("total infections" metric).
(b) A severe epidemic is defined as an outbreak that persists for at least T days ("duration" metric). Solid In the stochastic SIS model, if the pathogen invaded the host population then it tended to 610 persist for long periods. Consequently, the probability of a severe epidemic using the 611 "total infections" or "duration" metrics is approximately equal to the major epidemic 612 probability for a wide range of values of the severe epidemic thresholds (i.e. values of F 613 or T) compared with under the "concurrent size" definition. However, even in these cases, 614
for small or very large values of the severe epidemic thresholds, the probability of a severe 615 epidemic does not match the major epidemic probability, particularly when ! is larger 616 than but close to one (see e.g. red line in Fig 5b) . 617
Zika host-vector model R0 = 1.1 21 ≤ ≤ 54 (Fig 3a) 18 ≤ ≤ 21 (Fig 4a) 6 ≤ ≤ 7 (Fig 4b) R0 = 1.6 8 ≤ ≤ 681 (Fig 3a) 8 ≤ ≤ 55 (Fig 4a) 3 ≤ ≤ 47 (Fig 4b) R0 = 2 6 ≤ ≤ 866 (Fig 3a) 7 ≤ ≤ 117 (Fig 4a) 2 ≤ ≤ 83 (Fig 4b) 619 (Fig 3a) 138 ≤ ≤ 878 (Fig 5a) 183 ≤ ≤ 458 (Fig 5b) R0 = 1.6 8 ≤ ≤ 681 (Fig 3a) ≥ 18 (Fig 5a) ≥ 51 (Fig 5b) R0 = 2 6 ≤ ≤ 866 (Fig 3a) ≥ 10 (Fig 5a) ≥ 34 (Fig 5b) 629 
"duration" threshold values were tested up to T = 6,000 (Fig 5b) .
Evaluating the risk from an invading pathogen early in a potential severe epidemic is vital 640 for planning interventions and determining whether or not current control resources are 641 sufficient. When a pathogen arrives in a new location, the probability that initial cases will 642 lead to a major epidemic as opposed to fading out as a minor outbreak can be 643 approximated by assuming that infections occur according to a branching process. This 644 probability represents the risk that the outbreak will persist beyond the initial stochastic 645 phase in which case numbers are low. For simple models such as the stochastic SIS and 646 SIR models, this corresponds to the major epidemic probability in equation (1). This can 647 be extended to estimate the major epidemic probability using models with additional complexity, as we demonstrated by considering the case of host-vector transmission (see 649 equations (4) and (5)). 650 651 However, the threat from an invading pathogen can also be assessed by estimating the 652 probability that the outbreak will become "severe" and overwhelm available control 653 resources. For example, the probability that the peak prevalence exceeds the capacity of 654 treatment facilities (e.g. the number of hospital beds) can be estimated (Figs 3 and 4) , as 655 well as the probability that the outbreak exceeds a threshold in the total number of 656 infections or has a long duration (Fig 5) . In each case, the probability of a severe epidemic 657 depends on the value of the threshold set (e.g. the exact number of hospital beds) to 658 differentiate severe epidemics from other outbreaks, as well as the metric used to define 659 a severe epidemic. This highlights the need to consider the precise definition of a "severe 660 epidemic" carefully when assessing the risk from an invading pathogen. 661 662 When R0 is much greater than one or when the population size is extremely large, 663 however, the probability of a severe epidemic is constant for a range of values of the 664 threshold differentiating severe epidemics from other outbreaks (see e.g. different values 665 of M in Fig 3a,b) . In these cases, the probability of a severe epidemic will often match 666 between definitions (Table 2) . This is perhaps unsurprising since, for example, an 667 outbreak with a large total number of infections may well also have a large peak 668 prevalence. The probability of a severe epidemic is then approximately equal to the major 669 epidemic probability calculated in the standard way (Table 2) while these values were reported clearly, we emphasise that the precise type of threshold 692 and the value used should be chosen according to practical relevance in the particular 693 system under consideration. 694
In summary, we have shown that the precise definition of a severe epidemic should be 808 considered carefully in future studies that aim to evaluate the risk when a pathogen first 809 arrives in a host population. Only once a "severe epidemic" has been defined precisely 