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When does a minor outbreak become a major epidemic? Linking the risk from invading pathogens to practical definitions of a major epidemic

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Forecasting whether or not initial reports of disease will be followed by a major epidemic is an important component of disease management. Most estimates of the probability of a major epidemic involve assuming that infections occur according to a branching process. Surprisingly, however, these calculations can be carried out without the factors that differentiate a major epidemic from a minor outbreak ever being defined precisely. We assess potential implications of this lack of explicitness by considering the differences between three practically relevant possible definitions of a major epidemic. Specifically, we consider a major epidemic as an outbreak in which: i) a large number of hosts are infected simultaneously; ii) a large number of infections occur in total; and iii) disease persists in the population for a long time. We calculate the probability of a major epidemic under each of these definitions, initially considering the commonly used stochastic Susceptible-Infected-Susceptible model as a tractable case study allowing analytical progress. We show how the probability of a major epidemic can either be similar to, or very different from, the branching process estimate, depending on which definition of a major epidemic is used. We also show, using other models, that this ambiguity continues to hold in different epidemiological settings. In particular, we consider two additional models: the stochastic Susceptible-Infected-Removed model, and a more complex host-vector model parameterised for Zika virus. Our work highlights how the precise definition of a major epidemic must be considered carefully when estimating the risk posed by a new outbreak. It also demonstrates that the precise definition of a major epidemic must be tailored to the epidemiology of the host-pathogen system under consideration.
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When does a minor outbreak become a major epidemic? Linking the risk from
1
invading pathogens to practical definitions of a major epidemic
2
3
AUTHORS
4
R.N. Thompson1,2,3,4,*, C.A. Gilligan4, N.J. Cunniffe4
5
*Correspondence to: robin.thompson@chch.ox.ac.uk
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7
AFFILIATIONS
8
1Mathematical Institute, University of Oxford, Oxford, UK
9
2Department of Zoology, University of Oxford, Oxford, UK
10
3Christ Church, University of Oxford, Oxford, UK
11
4Department of Plant Sciences, University of Cambridge, Cambridge, UK
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13
ABSTRACT
14
Forecasting whether or not initial reports of disease will be followed by a major epidemic
15
is an important component of disease management. Most estimates of the probability of
16
a major epidemic involve assuming that infections occur according to a branching
17
process. Surprisingly, however, these calculations can be carried out without the factors
18
that differentiate a major epidemic from a minor outbreak ever being defined precisely.
19
We assess potential implications of this lack of explicitness by considering the differences
20
between three practically relevant possible definitions of a major epidemic. Specifically,
21
we consider a major epidemic as an outbreak in which: i) a large number of hosts are
22
infected simultaneously; ii) a large number of infections occur in total; and iii) disease
23
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(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
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persists in the population for a long time. We calculate the probability of a major epidemic
24
under each of these definitions, initially considering the commonly used stochastic
25
Susceptible-Infected-Susceptible model as a tractable case study allowing analytical
26
progress. We show how the probability of a major epidemic can either be similar to, or
27
very different from, the branching process estimate, depending on which definition of a
28
major epidemic is used. We also show, using other models, that this ambiguity continues
29
to hold in different epidemiological settings. In particular, we consider two additional
30
models: the stochastic Susceptible-Infected-Removed model, and a more complex host-
31
vector model parameterised for Zika virus. Our work highlights how the precise definition
32
of a major epidemic must be considered carefully when estimating the risk posed by a
33
new outbreak. It also demonstrates that the precise definition of a major epidemic must
34
be tailored to the epidemiology of the host-pathogen system under consideration.
35
36
KEYWORDS
37
mathematical modelling; infectious disease epidemiology; major epidemic; forecasting
38
39
SHORT TITLE
40
Assessing the practically relevant risk of a major epidemic
41
42
AUTHOR SUMMARY
43
When cases of disease are first reported in a new region or country, policy-makers must
44
decide if a disease control programme should be rolled out, as well as how to deploy
45
limited resources. Central to this assessment is understanding the risk of initial cases
46
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generating a major epidemic as opposed to fading out as a minor outbreak in which only
47
few individuals ever become infected. We show how this risk can be evaluated with
48
practical relevance, by considering consequences for disease control. For example, a
49
“major epidemic” might be defined as an outbreak likely to require more treatments than
50
are stockpiled. Once a major epidemic is defined precisely, a practically relevant estimate
51
for the probability of a major epidemic can be inferred using stochastic epidemiological
52
models. Estimating the risk posed by invading pathogens in this way contrasts with using
53
standard out-of-the-box estimates for the probability of a major epidemic, which do not
54
require a major epidemic to be defined formally and are instead based on restrictive
55
simplifying assumptions. At the start of any outbreak, careful consideration of precisely
56
what constitutes a “major epidemic” in that particular setting is vital for accurate
57
quantification of risk.
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1. INTRODUCTION
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61
Infectious disease epidemics in populations of humans, animals and plants represent a
62
recurring risk worldwide [17]. An important question for policy-makers towards the start
63
of an outbreak is whether initial cases will lead on to a major epidemic, or whether the
64
pathogen will rapidly die out instead [8,9]. An important practical consequence is that, if
65
an outbreak is likely to simply fade out, then costly interventions such as vaccination
66
[10,11], culling/felling [1218] and workplace or school closure [19] may well be
67
unnecessary [20].
68
69
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There is a well-known estimate for the probability of a major epidemic when a pathogen
70
is newly arrived in a host population, which in its simplest form is given by
71
72
!"#$
%
&'(#")*+,-*&,.
/
0)12
3
1
45
6
7%5/8)))))))))))))%1/
73
74
in which R0 is the basic reproduction number of the pathogen and I(0) is the number of
75
individuals that are currently infected. The estimate in equation (1) applies to a wide range
76
of models, including commonly-used models such as the Susceptible-Infected-
77
Susceptible (SIS) and Susceptible-Infected-Removed (SIR) models [21]. It can be derived
78
by assuming that infections occur according to a branching process (see Methods). In
79
models in which the infected class is sub-divided into different compartments, the value
80
of I(0) must be interpreted as the total number of individuals infected at t = 0. For example,
81
for the Susceptible-Exposed-Infectious-Removed model, the exponent in equation (1)
82
would in fact become E(0) + I(0) [9]. More sophisticated estimates that are based on the
83
branching process approximation can be derived for models including additional
84
epidemiological detail, such as more complex population structure [2224] and/or
85
infectious periods that are not exponentially distributed [25,26].
86
87
The quantity in equation (1), and particularly the version in which I(0) = 1, is used
88
extensively in the epidemiological modelling literature [8,9,21,2635]. It is also used in
89
real-time during emerging outbreaks. For example, it was used during the 2013-16
90
epidemic of Ebola virus disease in West Africa to estimate the chance that, if the virus
91
arrived in Nigeria, sustained transmission would follow in that country [30], and it has
92
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been considered in the context of flare-ups in new locations for the ongoing Ebola
93
epidemic in the Democratic Republic of the Congo [26].
94
95
However, the probability of a major epidemic in equation (1) is derived without specifying
96
a precise definition of a major epidemic. Over many outbreaks under identical conditions,
97
if the population size is large and R0 is much greater than one, then the distribution of
98
possible epidemic sizes is bimodal according to simple epidemic models such as the
99
stochastic SIR model (Fig 1d see also [3638]): in other words, the final size of any
100
single outbreak can be in one of two possible ranges. For example, in Fig 1d, most
101
outbreaks either lead to 1-20 hosts ever infected or to 60-95 hosts ever infected. The
102
estimate for the chance of a major epidemic in equation (1) corresponds approximately
103
to the proportion of outbreaks that have a final size in the higher of these ranges. This
104
provides a natural definition for “minor outbreaks” and “major epidemics”, motivated
105
largely by the elegance of the mathematical modelling analysis. However, for practical
106
application it would often be more appropriate for the notion of a major epidemic to instead
107
be grounded in consequences for control of disease, depending on the specific system
108
and outbreak under consideration.
109
110
Here we consider three potential definitions of a major epidemic that might be practically
111
relevant in different outbreak scenarios. Specifically, these are:
112
113
Concurrent size. An outbreak in which the number of individuals simultaneously
114
infected exceeds the capacity for treatment.
115
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Final size. An outbreak in which the number ever infected exceeds the total number
116
of available treatments.
117
Duration. An outbreak that is not contained quickly and therefore persists for an
118
unacceptably long period.
119
120
We investigate under which circumstances the probability of a major epidemic under each
121
of these definitions matches the branching process estimate from equation (1). In our
122
analyses, we consider three stochastic epidemiological models that are representative of
123
different host responses to infection and capture different routes of transmission:
124
specifically, the SIS model, the SIR model, and a host-vector model parameterised for
125
Zika virus transmission. For the SIS and SIR models, the branching process estimate
126
corresponds to equation (1), and in the case of Zika virus the branching process estimate
127
is an adapted version of equation (1) that accounts for host-vector-host transmission (see
128
Methods).
129
130
The definitions of a major epidemic above might each be applicable in different scenarios.
131
For example, it might be natural to assume that, if the number of individuals infected
132
simultaneously remains below the capacity for treatment, then the outbreak can be
133
controlled and is therefore minor. The threshold capacity might derive from the number
134
of available hospital beds [39] or the availability of care workers [40]. This motivates the
135
“Concurrent size” definition above.
136
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However, such a definition cannot be applied ubiquitously. Policy-makers often have to
138
make decisions concerning how much treatment to stockpile. For example, in response
139
to growing awareness of the threat of an influenza pandemic, between 2006 and 2013
140
policy-makers in the UK stockpiled around 40 million units of antivirals at a cost of £424
141
million, leading to severe criticism when only 2.4 million units were needed; the majority
142
of which were used during the 2009 H1N1 influenza pandemic [41]. Another possible
143
definition of a major epidemic is therefore an outbreak in which the total number of
144
individuals ever infected and requiring treatment exceeds a critical value (the “Final size”
145
definition, above). This critical value might be set by the stock of available treatments for
146
use during the outbreak.
147
148
Finally, we consider a third definition of a major epidemic (the “Duration” definition). Under
149
this definition, a major epidemic is an outbreak that persists for an unacceptably long time.
150
An outbreak that fades out quickly may escape public attention. Even if an outbreak leads
151
to a significant number of hosts infected, if it ends relatively quickly then it might be
152
considered minor. For example, the first Ebola outbreak in the Democratic Republic of
153
the Congo in 2018 resulted in 53 cases, but was not considered a major epidemic due to
154
its fast containment [42], leading to commendation of the success of public health
155
measures. Consequently, an outbreak might only be classified as a major epidemic if it
156
persists for a threshold length of time.
157
158
We note that these definitions do not always coincide. In 1665-66, plague affected the
159
village of Eyam in the UK. The epidemic in the village was long-running, and a large
160
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number of individuals were killed (267 out of 350 in the village), yet model fits suggest
161
that a maximum of only 30 people were ever infected simultaneously [43,44]. As a result,
162
this epidemic might have been classified as a major epidemic according to the “Final size”
163
and “Duration” definitions, yet not under the “Concurrent size” definition, depending on
164
the precise values of the thresholds set in each case. This highlights the need to define
165
a major epidemic carefully, since an individual outbreak may or may not qualify as a major
166
epidemic, depending on the definition used.
167
168
We will show that the probability of a major epidemic depends on precisely how a major
169
epidemic is defined. The probability of a major epidemic under practical definitions may
170
or may not match the branching process estimate. The definition to use should therefore
171
be considered carefully before the risk of a major epidemic is assessed at the beginning
172
of each future outbreak. Only once the notion of a major epidemic has been formally
173
defined – based on criteria of practical relevance – can this risk be properly assessed.
174
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176
Figure 1. Schematic diagrams illustrating the population structures for the different models considered, and
177
an example distribution of final sizes for the stochastic SIR model. (a) The SIS model. (b) The SIR model.
178
(c) The model of Zika virus transmission. (d) Distribution of final sizes in the stochastic SIR model, with
179
population size N = 100, R0 = 1.8, I(0) = 1 and the rest of the population susceptible initially. The y-axis has
180
been truncatedthe height of the leftmost bar is 0.33.
181
182
2. METHODS
183
184
We conduct five main analyses. In the first analysis, we calculate the probability of a major
185
epidemic under the Concurrent size definition for the stochastic SIS model. We then
186
conduct two further analyses in which we calculate the probability of a major epidemic
187
under that definition for two other epidemiological models (the SIR model and Zika host-
188
vector model). We then focus on the stochastic SIS model, and perform two more
189
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analyses in which we calculate the probability of a major epidemic under different
190
definitions of a major epidemic (the Final size and Duration definitions). In each of these
191
five analyses, we compare the probability of a major epidemic for the particular
192
model/major epidemic definition pairing considered to the commonly used branching
193
process approximation for the probability of a major epidemic which does not require a
194
major epidemic to be defined formally.
195
196
Here, we describe the epidemiological models that we use, calculation of the branching
197
process approximation to the probability of a major epidemic for each model, and how the
198
probability of a major epidemic can be calculated under the Concurrent size definition for
199
each of the models considered. We then explain how the probability of a major epidemic
200
under each of the practically relevant definitions of a major epidemic can be obtained for
201
the SIS model.
202
203
Epidemiological models
204
205
Susceptible-Infected-Susceptible (SIS) model
206
According to the SIS model, at any time each individual in the population is classified to
207
be either (S)usceptible to or (I)nfected by the disease. The deterministic SIS model is
208
given by
209
210
-9
-:02;<9=><8
211
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-<
-:0;<92><?)))))%@/
212
213
We use the analogous stochastic model in most of our analyses, in which the net rate at
214
which any epidemiological event occurs is
;<9=><
. The probability that this next event is
215
an infection is
A7B
A7BCD7
and the probability that the next event is a recovery is
D7
A7BCD7
.
216
217
In this model, the basic reproduction number is given by
218
450;E
>?
219
220
Susceptible-Infected-Removed (SIR) model
221
Under the SIR model, at any time each individual in the population is classified according
222
to whether they are (S)usceptible to infection, (I)nfected, or (R)emoved and no longer
223
spreading the pathogen or available for infection. The deterministic SIR model is given by
224
225
-9
-:02;<98
226
-<
-:0;<92><?
227
-4
-:0><?))))))%F/
228
229
In the analogous stochastic model, the net rate at which any epidemiological event occurs
230
is still
;<9=><
, and the probability that the next event is an infection event is similarly
231
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unchanged at
A7B
A7BCD7
. However, the other possible next event is a removal, which occurs
232
with probability
D7
A7BCD7
. The basic reproduction number is again
233
450;E
>?
234
235
Zika transmission model
236
We consider the transmission of Zika virus according to the host-vector model of
237
Kucharski et al. [45], which we chose so that we could demonstrate how the probability
238
of a major epidemic can be calculated in a relatively complex epidemiological setting. In
239
the model, the numbers of the N hosts that are (S)usceptible, (E)xposed, (I)nfectious and
240
(R)emoved are tracked, as well as the vectors that are (SV)usceptible, (EV)xposed and
241
(IV)nfectious. The deterministic version of this model is given by
242
243
-9
-:02;<G98
244
-H
-:0;<G92IJH8
245
-<
-:0IJH2><8
246
-4
-:0><8
247
-9G
-: 0KEG2;G9G<
E2K9G?
248
-HG
-: 0;G9G<
E2
%
K=IG
/
HG8
249
-<G
-: 0IGHG2K<G?)))))))))))))))))%L/
250
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251
In the analogous stochastic model, the number of infected human hosts arising from a
252
single infected human (accounting for human-vector-human transmission) in an
253
otherwise entirely susceptible population of humans and vectors is given by
254
255
45
JGMNOPQ)7PM45
GJ 0;GIG;EG
>
%
K=IG
/
K8
256
257
where
45
JG 0)R
DAPSP
S
is the expected number of vectors infected (and going on to enter
258
the exposed class) by a single infectious human,
NOPQ)7P0TP
UCTP
is the proportion of
259
exposed vectors that become infectious and
45
GJ 0R
U;E
is the expected number of
260
humans infected by a single infectious vector.
261
262
The basic reproduction number is given by
263
264
450
V
;GIG;EG
>
%
K=IG
/
K8
265
266
where the square root accounts for the fact that it takes two generations for infected
267
humans to generate new infections, since new infections require host-vector-host
268
transmission [46,47]. We note that in some studies, e.g. [45], the square root is omitted
269
from the definition of
45
. In contrast to the expression calculated by Kucharski et al. [45],
270
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to facilitate simulation of the stochastic model we also explicitly track the total number of
271
vectors,
EG
, rather than the density.
272
273
Probability of a major epidemic (branching process estimate)
274
275
Standard estimate (for stochastic SIS/SIR models)
276
The commonly used estimate for the probability of a major epidemic when a pathogen
277
first arrives in a host population [8,9,21,27,2935] can be derived by assuming that
278
infections occur according to a branching process, making the assumptions that the
279
susceptible population is large and that infection lineages arising from different infected
280
hosts are independent. When a single infected host arrives in an otherwise susceptible
281
population, the branching process estimate for the probability of a major epidemic is given
282
by
283
284
!"#$
%
&'(#")*+,-*&,.
/
W
X
Y))Z#")45[18
12 1
45)))Z#")45\1?
285
286
This expression is derived in Text S1.
287
288
If instead there are I(0) infected individuals initially rather than one, then for no major
289
epidemic to occur, it is necessary for each initial infection lineage to die out, leading to
290
the approximation given in equation (1) whenever R0 > 1.
291
292
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Extension to more complex models
293
Here we show how the branching process estimate for the probability of a major epidemic
294
can be extended to more complex systems by considering the stochastic version of the
295
model of Zika virus given by the system of equations (4).
296
297
We denote the probability of no major epidemic occurring starting from i exposed or
298
infectious human hosts, j exposed vectors and k infectious vectors by qijk. We must
299
consider exposed and infectious vectors separately to account for the possibility that
300
exposed vectors die before becoming infectious.
301
302
Starting from a single infectious host introduced into an entirely susceptible population of
303
hosts and vectors, and conditioning on the next event, gives
304
]R55
=
)D
DC^P_P
_]555=^P_P
_
DC^P_P
_]RR5
.
305
306
Similarly,
307
]5R5
=
)U
UCTP]555=TP
UCTP]55R
,
308
]55R
=
)U
UCAS]555=AS
UCAS]R5R
.
309
310
We again assume that infection lineages are independent, permitting us to approximate
311
terms with two exposed or infectious individuals by non-linear terms involving single
312
exposed or infectious individuals, e.g.
]RR5W]R55]5R5
. Noting that
]55501
, then the
313
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three equations above can be solved to give expressions for
]R55
,
]5R5
and
]55R
. In
314
particular, the probability of a major epidemic starting from a single infected host is then
315
12]R550
X
Y)))))Z#")45[18
%
45
/
`21
%
45
/
`=45
GJ)Z#")45\1?))))))))%a/
316
In this expression,
45
GJ
is the expected number of humans infected by a single infectious
317
vector in an otherwise entirely susceptible population of humans and vectors.
318
319
Probability of a major epidemic (“Concurrent size” definition)
320
321
As described in the introduction, we first define a major epidemic to be an outbreak in
322
which the number of simultaneously infected individuals ever exceeds a threshold value,
323
which we denote by M. The value of M of relevance in practical applications might be set
324
by the capacity for treatment.
325
326
Deterministic SIS and SIR models
327
In the deterministic SIS model, the maximum number of simultaneously infected
328
individuals over the course of the outbreak is given by
329
330
<bcd0
e
f
g
E
3
12 1
45
6
))Z#")<
%
Y
/
[E
3
12 1
45
6
8
<
%
Y
/
)Z#")<
%
Y
/
\E
3
12 1
45
6
8
331
332
as shown in Text S1.
333
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334
In the deterministic SIR model, on the other hand,
335
336
<bcd0
h
)<i)Z#")<
%
Y
/
[<i8
<
%
Y
/
)Z#")<
%
Y
/
\<i8
337
338
where
<i02E
4Y=E
4Yjk
3
E
9
%
Y
/
4Y
6
=9
%
Y
/
=<
%
Y
/, and again this is shown in Text S1.
339
340
The probability of a major epidemic according to the “Concurrent size” definition under
341
the deterministic SIS and SIR models is then
342
343
!"#$
%
&'(#")*+,-*&,.
/
0
h
1))Z#")l[<bcd8
Y)Z#")l\<bcd8
344
345
where Imax is given by the corresponding expressions above depending which model is
346
used.
347
348
Stochastic SIS model
349
Under the stochastic SIS model, however, calculating the probability of a major epidemic
350
is more challenging [48]. Nevertheless, we can calculate this value analytically, which is
351
advantageous since approximating this quantity using model simulations can be time
352
consuming given that outbreaks under the SIS model can persist for long periods.
353
354
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We denote the probability of no major epidemic starting with I infected individuals as
]7
,
355
and assume that the rest of the population is susceptible. Conditioning on whether the
356
first event is an infection or recovery event gives
357
358
]70;<9
;<9=><]7CR=><
;<9=><]7mR8
359
360
for I = 1,2,…,M-1. We solve this tridiagonal system of equations with boundary conditions
361
q0 = 1 and qM = 0, since a major epidemic certainly does not occur if there are initially no
362
infected individuals, and certainly does if there are initially M infected individuals. Noting
363
that
90E2<
, these equations can be put in the form
364
365
]7CR2]70n7
%
]72]7mR
/
8
366
367
where
368
n70><
;<%E2</?
369
370
Iterating this gives
371
]7CR2]70o7
%
]R2]5
/
8))))))))%p/
372
373
where
374
o70
q
nr
7
rsR 8
375
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0
t
E
45
u
7
<v
w
SmR
7
x
?
376
377
Adding equation (6) to itself for I = 1,2,…,M - 1 and rearranging gives
378
379
]R0
y
o7
zmR
7sR
1=
y
o7
zmR
7sR ?
380
381
Instead adding equation (6) to itself for I = 1,2,…,k – 1 gives
382
383
]{0
y
o7
zmR
7s{
1=
y
o7
zmR
7sR ?
384
385
The probability of a major epidemic is then given by
386
387
!"#$
%
&'(#")*+,-*&,.
/
012]{0
e
|
|
f
|
|
g
Y))Z#")}0Y8
1
1=
y
o7
zmR
7sR ))Z#")}018
1=
y
o7
{mR
7sR
1=
y
o7
zmR
7sR ))Z#")1~}~l8
1))Z#")}l? ))))%€/
388
389
390
Stochastic SIR model
391
Under the stochastic SIR model, the probability of a major epidemic according to the
392
“Concurrent size” definition starting from any state (I,R) is calculated using an iterative
393
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approach. Denoting the probability of a major epidemic starting from state (I,R) by
78‚
,
394
then conditioning on what happens next gives
395
396
78‚ 0;<%E2<24/
;<%E2<24/=><7CR8‚=><
;<%E2<24/=><7mR8‚
397
398
This system can be solved with boundary conditions
58‚ 0Y8)•78SmzCR 0Y
and
z8‚ 01
.
399
To do this, the probability of a major epidemic is deduced for the following states (in
400
order): (I,R) = (M-1,N-M), (M-2,N-M), …, (1,N-M), (M-1,N-M-1), …,(1,N-M-1),… (M-
401
1,0),…,(1,0). For a schematic showing the order in which these probabilities are deduced,
402
see Fig S1.
403
404
Zika transmission model
405
The probability of a major epidemic under the “Concurrent size” definition is approximated
406
by simulating the model 10,000 times using the Gillespie direct method [49], and then
407
calculating the proportion of simulations in which the number of infected human hosts
408
exceeds M at any point during the simulation.
409
410
Probability of a major epidemic (“Final size” and “Duration” definitions)
411
412
We also consider the probability of a major epidemic according to the deterministic and
413
stochastic SIS model for the “Final size” and “Duration” definitions of a major epidemic.
414
Under the “Final size” definition, a major epidemic is assumed to be an outbreak in which
415
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at least F infections occur over the course of the outbreak. Under the “Duration” definition,
416
a major epidemic is defined to be an outbreak that persists for at least T days.
417
418
In the deterministic SIS model, whenever
45\1
, the outbreak persists indefinitely with
419
an infinite number of infection events. As a result, any outbreak is a major epidemic under
420
the “Final size” and “Duration” definitions.
421
422
In contrast, in any simulation of the stochastic SIS model, the number of infected
423
individuals will always reach zero, even if this takes a long time. As a result, the probability
424
of a major epidemic under the “Final size” and “Duration” definitions is not simply one or
425
zero depending on the value of
45
. We approximate the probability of a major epidemic
426
under these definitions by simulating the model 10,000 times using the Gillespie direct
427
method [49], and recording the proportion of simulations in which there are at least F
428
infections or that have a duration of at least T days.
429
430
3. RESULTS
431
432
To begin to understand outbreak dynamics under the SIS, SIR and Zika transmission
433
models, we first numerically solved the deterministic models given by the systems of
434
equations (2), (3) and (4) with
4501?a
in each case (Fig 2). In a deterministic setting, the
435
SIS model predicts the largest number of individuals simultaneously infected as well as
436
the most infections in total. Epidemics persisted forever (i.e. I remained larger than zero)
437
under all three models, although the number of infected hosts tended to zero under the
438
SIR and Zika transmission models.
439
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440
However, our main focus is the probability of a major epidemic soon after the pathogen
441
enters the system. For any definition of “major epidemic”, according to a deterministic
442
model the corresponding probability is either zero or one depending on the values of
443
model parameters. We therefore considered the more realistic stochastic models, in
444
which demographic stochasticity is included. In the following sections, first we calculate
445
the probability of a major epidemic for the stochastic SIS model under the Concurrent
446
size definition. We then consider different epidemiological models, as well as different
447
definitions of a major epidemic. In each case, the probability of a major epidemic for the
448
particular epidemiological model-definition of a major epidemic pair under consideration
449
is compared to the branching process approximation to the probability of a major epidemic
450
for that model. Results are shown in Figs 3-5, as well as summarised in Tables 1 and 2.
451
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452
Figure 2. Numerical solutions of the deterministic SIS, SIR and Zika virus transmission models when the
453
basic reproduction number is
4501?a
. (a) Number of infected individuals through time according to the
454
deterministic SIS model. (b) Cumulative number of infections through time according to the deterministic
455
SIS model. (c)-(d) Equivalent to a-b but for the deterministic SIR model. (e)-(f) Equivalent to a-b but for the
456
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deterministic Zika virus transmission model. Parameters for deterministic SIS and SIR models:
E018YYY
,
457
;0Y?YYY1a
per day,
1ƒ>01Y
days. Parameters for deterministic Zika virus transmission model:
E0
458
18YYY
,
)EG01Y8YYY
, 1/
IG01Y?a
days, 1/
IJ0a?„
days, 1/
>0a
days, 1/
K0€?…
days,
;0Y?YYYYp1a
per
459
day,
;G0Y?@@)
per day.
460
461
462
The probability of a major epidemic
463
464
We calculated the probability of a major epidemic according to the stochastic SIS model
465
under the “Concurrent size” definition of a major epidemic – i.e. major epidemics defined
466
as outbreaks in which a threshold number of simultaneously infected individuals is
467
exceeded. In this case, as described in Methods, it is possible to calculate the probability
468
of a major epidemic analytically.
469
470
We show the probability of a major epidemic for a range of values of the major epidemic
471
threshold, M, in Fig 3a. For R0 much larger than one, we found that the probability of a
472
major epidemic was approximated closely by the standard branching process estimate
473
for many values of the major epidemic threshold, M. When, however, R0 was close to
474
unity, the standard estimate corresponded to a single choice of M (see e.g. blue lines in
475
Fig 3a, where the solid line is close to the corresponding dotted line in only one place, i.e.
476
for a single value of M). The parameter regime in which R0 is close to one is important in
477
many real epidemiological systems since the aim of eradicative control strategies is
478
usually to reduce the reproduction number below one [50,51].
479
480
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481
Figure 3. Probability of a major epidemic under the SIS model, where a major epidemic is defined as an
482
outbreak in which M simultaneously infected individuals is exceeded (“Concurrent size” definition). (a)
483
Dependence on R0. Solid lines represent the true probability of a major epidemic (system of equations (7)),
484
dotted lines represent the branching process estimate (equation (1)), and dots show the maximum number
485
simultaneously infected in the analogous deterministic models (for values of M below this, the probability
486
of a major epidemic in the relevant deterministic model is 1).
45
is varied by changing the value of
;
. (b)
487
Equivalent to a, but showing dependence on the population size, N. (c) Equivalent to a, but showing
488
dependence on the initial number of infected individuals, I(0). (d). Single simulation of the stochastic SIS
489
model (blue), and numerical solution of deterministic SIS model (red dotted). The value of I in the stochastic
490
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simulation will continue to fluctuate about the deterministic value until I reaches 0. Parameter values
491
(except where stated): N = 1,000, R0 = 1.5, I(0) = 1 and R(0) = 0 and the remainder of the population
492
susceptible. In panel d,
;0Y?YYYY1a
per day and
1ƒ>01Y
days.
493
494
495
In large host populations, the probability of a major epidemic as a function of M took the
496
form of a step function (Fig 3b). This suggested that, if the pathogen successfully invaded
497
the population, then the number of infected individuals would definitely reach a specific
498
maximum value which is determined by R0. For example, for outbreaks with R0 = 1.5, the
499
pathogen will invade the population with probability 0.33, and, if this occurs, then 67% of
500
the population will be infected simultaneously at some time during the epidemic.
501
502
The maximum value of I that will be reached in the stochastic SIS model in the large N
503
case is approximately
504
505
&'†
%
<
/
0&'†)
t
@E
t
12R
u
8E
u.
506
507
The expression
@E
t
12R
u is twice the maximum value of I in the corresponding
508
deterministic model. This is because, if the pathogen invades in the stochastic SIS model,
509
then I will fluctuate approximately symmetrically around the deterministic equilibrium
510
value (Fig 3d). It is likely, then, that by the time the pathogen dies out by reaching I = 0,
511
the number of infected individuals will at some stage also have reached approximately
512
double the value it fluctuated around too.
513
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514
We also note that, if the expression above for max(I) is reformulated to give the maximum
515
proportion of the population that is simultaneously infected, the resulting expression is
516
independent of the population size N. This can also be seen graphically – in Fig 3b, if the
517
pathogen successfully invaded the population then the maximum proportion of hosts that
518
were simultaneously infected was approximately independent of the size of the host
519
population, so long as N was sufficiently large.
520
521
522
Different epidemiological models
523
524
We considered the probability of a major epidemic (“Concurrent size” defintion) under the
525
SIR and Zika virus transmission models. For the stochastic SIR model, we used an
526
iterative method to calculate the probability of a major epidemic as described in Methods.
527
528
For the stochastic Zika virus transmission model, we simulated the model in a population
529
of N = 1,000 human hosts and NV = 10,000 vectors using the Gillespie direct algorithm
530
[49], using parameter values from Kucharski et al., 2016 [45] – see caption of Fig 4. The
531
value of R0 was then varied in Fig 4b by altering the parameter
;
that governs the rate at
532
which infected vectors infect susceptible human hosts.
533
534
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535
Figure 4. Probability of a major epidemic under the “Concurrent size” definition of a major epidemic, for the
536
SIR and Zika virus transmission models. (a) SIR model. Solid lines represent the true probability of a major
537
epidemic calculated using the iterative method described in Methods, dotted lines represent the branching
538
process estimate (equation (1)), and dots show the maximum number simultaneously infected in the
539
analogous deterministic models (for values of M below this, the probability of a major epidemic in the
540
relevant deterministic model is 1).
45
is varied by changing the value of
;
. (b) Equivalent to a, but for the
541
Zika virus transmission model (where M refers to the number of simultaneously infected hosts). For the
542
Zika transmission model, the true probability of a major epidemic is calculated by simulation, and the
543
branching process estimate is given by equation (5). For both models,
E018YYY
. Other parameters for the
544
Zika virus transmission model:
EG01Y8YYY
, 1/
IG01Y?a
days, 1/
IJ0a?„
days, 1/
>0a
days, 1/
K0€?…
545
days,
;G0Y?@@)
per day. Initial conditions for both models comprise of a single infected host, with all other
546
individuals (for the Zika transmission model, hosts and vectors) susceptible.
547
548
549
Under the stochastic SIR and Zika models, for R0 larger than and not close to one, the
550
maximum number of simultaneously infected individuals whenever the pathogen invaded
551
the host population was typically smaller than under the SIS model (cf. Fig 2).
552
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Nonetheless, we found qualitatively similar behaviour in these cases – the probability of
553
a major epidemic approximated using a branching process corresponded to a wide range
554
of values of the major epidemic threshold when R0 was high (Fig 4). However, even if that
555
is the case, the practically relevant value of the major epidemic threshold (e.g. the number
556
of treatments that have been stockpiled) may not give a probability of a major epidemic
557
that matches the branching process estimate. For example, if R0 = 2 and 250 treatments
558
were to have been stockpiled, for the SIR model the probability of a major epidemic under
559
the “Final size” definition is 0 (solid grey line in Fig 4a), yet the branching process estimate
560
for the probability of a major epidemic is 0.5 (dotted grey line in Fig 4a).
561
562
Alternative definitions of a major epidemic
563
564
For the stochastic SIS model, we then calculated the probability of a major epidemic for
565
different definitions of a major epidemic specifically, outbreaks in which there are at
566
least F infection events (the “Final size” definition Fig 5a) or outbreaks that persist for
567
at least T days (the “Duration” definition – Fig 5b).
568
569
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570
Figure 5. Probability of a major epidemic under the SIS model, for alternative definitions of a major epidemic.
571
(a) A major epidemic is defined as an outbreak in which more than F infections occur (“Final size” definition).
572
(b) A major epidemic is defined as an outbreak that persists for more than T days (“Duration” definition).
573
Solid lines represent the true probability of a major epidemic assessed via simulation of the stochastic
574
model, and dotted lines represent the branching process estimate (equation (1)). Results of the
575
deterministic model are not included in the figure, since under the deterministic SIS model epidemics persist
576
indefinitely and generate an infinite number of infections whenever
45\1
. Parameter values: N = 1,000,
577
I(0) = 1 and R(0) = 0 and the remainder of the population susceptible. In both panels,
45
is varied by
578
changing the value of
;
. In panel b,
1ƒ>01Y
days.
579
580
581
In the stochastic SIS model, if the pathogen invaded the host population then it tended to
582
persist for long periods. Consequently, the branching process estimate corresponded to
583
a very wide range of major epidemic thresholds under the “Final size” or “Duration”
584
definitions (i.e. values of F or T) compared to under the “Concurrent size” definition. As a
585
result, in these specific cases (i.e. when the stochastic SIS model was used and a major
586
epidemic was defined according to the “Final size” or “Duration” definitions) it can be
587
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concluded that the branching process approximation often leads to sensible estimates of
588
the risk posed by invading pathogens. Nonetheless, even in these cases, for very large
589
values of the major epidemic thresholds the probability of a major epidemic does not
590
match the branching process estimate, particularly when
45
was larger than but close to
591
one (see e.g. red line in Fig 5b).
592
593
SIS model
SIR model
Zika host-
vector model
R0 = 1.1
M in range
[21,54]
(Fig 3a)
M in range
[18,21]
(Fig 4a)
M in range
[6,7]
(Fig 4b)
R0 = 1.6
M in range
[8,681]
(Fig 3a)
M in range
[8,55]
(Fig 4a)
M in range
[3,47]
(Fig 4b)
R0 = 2
M in range
[6,866]
(Fig 3a)
M in range
[7,117]
(Fig 4a)
M in range
[2,83]
(Fig 4b)
594
Table 1. Effect of underlying epidemiology. For which values of the threshold (number of hosts
595
simultaneously infected, M) in the Concurrent size definition of a major epidemic is the branching process
596
approximation to the probability of a major epidemic accurate? Threshold values here are those for which
597
the branching process approximation is within 0.01 of the probability of a major epidemic. Results are
598
summarised for the SIS, SIR and Zika virus models, for the parameter values shown in the relevant figure
599
captions.
600
601
602
603
Concurrent
size
Final size
Duration
R0 = 1.1
M in range
[21,54]
(Fig 3a)
F in range
[138,878]
(Fig 5a)
T in range
[183,458]
(Fig 5b)
R0 = 1.6
M in range
[8,681]
(Fig 3a)
F greater
than 18
(Fig 5a)
T greater
than 51
(Fig 5b)
R0 = 2
M in range
[6,866]
(Fig 3a)
F greater
than 10
(Fig 5a)
T greater
than 34
(Fig 5b)
604
Table 2. Effect of definition of a major epidemic. For which values of the thresholds in the more practically
605
relevant definitions of a major epidemic is the branching process approximation to the probability of a
606
major epidemic accurate? Threshold values here are those for which the branching process
607
approximation is within 0.01 of the probability of a major epidemic. Results are summarised for the SIS
608
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model, for the parameter values shown in the relevant figure captions. Final size values were tested up to
609
a maximum of F = 2,000 (Fig 5a) and duration values were tested up to T = 6,000 (Fig 5b).
610
611
4. DISCUSSION
612
613
Determining whether or not an emerging outbreak is likely to develop into a major
614
epidemic is useful for planning intervention and containment strategies. When a pathogen
615
arrives in a new location, the probability of a major epidemic can be derived by assuming
616
that infections occur according to a branching process. For simple models such as the
617
stochastic SIS and SIR models, this leads to the probability of a major epidemic in
618
equation (1). It is also possible to calculate the probability of a major epidemic according
619
to the branching process approximation using models with other epidemiological
620
complexity, as we showed by considering the case of host-vector transmission (see
621
equations (4) and (5)).
622
623
However, the branching process estimate for the probability of a major epidemic is not
624
necessarily accurate when a definition of a major epidemic is used that addresses
625
practical aspects of disease control (Fig 3). The branching process estimate corresponds
626
to a range of choices of the epidemic thresholds in our definitions when R0 is much greater
627
than one, or when the population size is extremely large (see e.g. different values of M in
628
Fig 3a,b). However, when R0 is close to one and the population is not large, the standard
629
estimate can correspond to a single choice of the epidemic threshold (see e.g. blue and
630
red lines in Fig 3a and Tables 1 and 2). For specific outbreaks, even when both R0 and
631
the population size are large, the branching process estimate may not be relevant – since
632
the range of choices of the major epidemic threshold that the standard estimate
633
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corresponds to may or may not include the specific threshold of practical importance in
634
the outbreak, for example the number of hospital beds available. Consequently, using the
635
standard branching process estimate for the probability of a major epidemic could lead to
636
the risk of a major epidemic being incorrectly assessed, potentially including scenarios in
637
which a major epidemic develops when previously deemed unlikely. Our main conclusion
638
that the branching process estimate for the probability of a major epidemic may or may
639
not match the true probability of a major epidemic when a practically relevant definition is
640
used holds for a range of epidemiological systems (Fig 4) as well as different definitions
641
of a major epidemic that apply in alternative settings (Fig 5). We note that exactly how
642
well the branching process estimate reflects a range of threshold values depends on the
643
underlying model as well as the relevant quantity differentiating major epidemics from
644
minor outbreaks (cf. Figs 3a and 5).
645
646
We considered practical definitions of a major epidemic that were based on thresholds
647
such as the availability of treatment. A previous study has concluded that epidemiological
648
modellers should report the precise cutoff used to define a major epidemic in model
649
simulations [52]. We support this conclusion, and indeed some authors have done this –
650
for example, Keeling et al. [53] define a major epidemic to be an outbreak in which at
651
least one-third of the population becomes infected. However, we contend furthermore that
652
the precise threshold used should be motivated by the value that is most practically
653
relevant in the particular system under consideration.
654
655
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
Under the first definition of a major epidemic that we considered (the “Concurrent size”
656
definition), the probability of a major epidemic was assessed in the context of the capacity
657
for treatment by estimating whether or not a threshold number of simultaneously infected
658
individuals was likely to be exceeded. This definition may be practically relevant in a range
659
of scenarios. For example, real-time analysis of a diphtheria epidemic in Cox Bazar’s in
660
Bangladesh involved assessing the number of hospital beds that were needed [39]. The
661
number of hospital beds that were already available might have provided a practically
662
relevant major epidemic threshold. Another example is citrus greening disease in Brazil,
663
for which a law was introduced that stated that a citrus grove must be destroyed if 28%
664
of trees in the grove were infected and symptomatic [54,55]. This fraction of trees could
665
correspond to a threshold in a representative model similar to the thresholds that we
666
considered here. Other examples for which interventions are introduced as soon as a
667
threshold in the number simultaneously infected is reached include the development of
668
the National Chlamydia Screening Programme in the United Kingdom in 2002 in response
669
to the large size of the infected population [56].
670
671
However, no single definition of a major epidemic will be relevant in all situations. We also
672
considered two other definitions of a major epidemic. In one of these (the “Final size”
673
definition), whether or not an outbreak was classified as a major epidemic referred to the
674
total number of infection events over the course of the outbreak, rather than the maximum
675
number simultaneously infected. This might correspond to the total number of individuals
676
ever requiring treatment, which may be an important threshold if treatments have been
677
stockpiled prior to the outbreak [41]. This definition might also be relevant if, for example,
678
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(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
a policy-maker has to choose how to deploy resources between two different epidemics.
679
If there are only sufficient resources to contain one outbreak, then it might be preferable
680
to choose the one that is likely to generate more infections. In other real-world scenarios,
681
other definitions might be appropriate. We also considered major epidemics defined as
682
outbreaks that persist for a threshold length of time (the “Duration” definition). Different
683
definitions of a major epidemic might appear contradictory – for example, treatment can
684
act to reduce the total number of infections yet increase the duration of the outbreak [57],
685
making a major epidemic less likely under the “Final size” definition of a major epidemic
686
but more likely under the “Duration” definition. In fact, integrating interventions into our
687
models to examine their effects on the probability of a major epidemic under different
688
practical definitions is a candidate for further study.
689
690
Our intention here was to use very simple models to demonstrate the principle that
691
different definitions of a major epidemic lead to different probabilities of a major epidemic
692
each of which might be practically relevant in different scenarios. Although simple models
693
are commonly used, accurate outbreak forecasts require a model carefully matched to
694
the epidemiology of the host-pathogen system, potentially including asymptomatic
695
transmission [9,58], spread between spatially distinct regions [29], or explicit modelling of
696
control interventions [12,20,5961]. For certain definitions, it may be necessary to include
697
convalescent hosts in the model explicitly. For example, if convalescing individuals take
698
up resources, such as beds in treatment rooms or hospitals, and the definition of a major
699
epidemic is linked to the availability of resources (as in the case of the “Concurrent size”
700
definition), then these individuals should be modelled, potentially by including them in a
701
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(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
new compartment following the infectious class. More complex definitions of major
702
epidemics could also be used, for example requiring multiple criteria to be satisfied for an
703
outbreak to be classified as a major epidemic. In these more complicated scenarios,
704
analytic calculations of the probability of a major epidemic might not be possible. Model
705
simulations can then be used to assess the probability of a major epidemic, as we showed
706
using a host-vector model of Zika virus disease transmission (Fig 4b).
707
708
In summary, we have shown that how precisely to define a major epidemic should be
709
considered carefully in future studies. The definition of a major epidemic should be
710
designed to match the questions of interest in the particular setting being considered.
711
Standard approximations based on branching processes should not always be used as
712
the default. Only once a “major epidemic” has been defined precisely can the probability
713
of a major epidemic occurring be properly assessed. Providing an explicit demonstration
714
of the consequences of not considering a practically relevant definition in evaluating the
715
risk of an epidemic is the key contribution of this paper.
716
717
718
SUPPLEMENTARY MATERIAL
719
720
S1 Text.
721
722
S1 Figure. Schematic showing how the probability of a major epidemic under the
723
“Concurrent size” definition can be deduced for the stochastic SIR model. The probability
724
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(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
corresponding to each state (I,R) is deduced iteratively in the order 1,2,3... (red). The
725
black arrows indicate which previous values are used to inform each new deduction.
726
727
COMPETING INTERESTS
728
We have no competing interests.
729
730
AUTHORS’ CONTRIBUTIONS
731
All authors designed the study. RNT undertook the research and wrote the first draft of
732
the manuscript. All authors revised the manuscript.
733
734
FUNDING
735
This research was funded by BBSRC via a G2O PhD studentship (RNT), and by Christ
736
Church, Oxford, via a Junior Research Fellowship (RNT).
737
738
REFERENCES
739
740
1. Daszak P, Cunningham AA, Hyatt AD. Emerging infectious diseases of wildlife —
741
threats to biodiversity and human health. Science (80- ). 2000;287: 443–450.
742
2. Anderson PK, Cunningham AA, Patel NG, Morales FJ, Epstein PR, Daszak P.
743
Emerging infectious diseases of plants: pathogen pollution, climate change and
744
agrotechnology drivers. Trends Ecol Evol. 2004;19: 535–544.
745
doi:10.1016/j.tree.2004.07.021
746
3. Taylor LH, Latham SM, Woolhouse MEJ. Risk factors for human disease
747
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
emergence. Philos Trans R Soc B. 2001;356: 983–989.
748
doi:10.1098/rstb.2001.0888
749
4. Morens DM, Folkers GK, Fauci AS. The challenge of emergin and re-emerging
750
infectious diseases. Nature. 2004;430: 242–249.
751
5. Jones KE, Patel NG, Levy MA, Storeygard A, Balk D, Gittleman JL, et al. Global
752
trends in emerging infectious diseases. Nature. 2008; 990–994.
753
doi:10.1038/nature06536
754
6. Fisher MC, Henk DA, Briggs CJ, Brownstein JS, Madoff LC, McCraw SL, et al.
755
Emerging fungal threats to animal, plant and ecosystem health. Nature. 2012;484:
756
186–194. doi:10.1038/nature10947
757
7. Thompson RN, Brooks-Pollock E. Detection, forecasting and control of infectious
758
disease epidemics: modelling outbreaks in humans, animals and plants. Philos
759
Trans R Soc B. 2019;374: 20190038.
760
8. Craft ME, Beyer HL, Haydon DT. Estimating the probability of a major outbreak
761
from the timing of early cases: an indeterminate problem ? PLoS One. 2013;8:
762
e57878. doi:10.1371/journal.pone.0057878
763
9. Thompson RN, Gilligan CA, Cunniffe NJ. Detecting presymptomatic infection is
764
necessary to forecast major epidemics in the earliest stages of infectious disease
765
outbreaks. PLoS Comput Biol. 2016;12: e1004836.
766
doi:10.1371/journal.pcbi.1004836
767
10. Ramsay ME. Measles: the legacy of low vaccine coverage. Arch Dis Child.
768
2013;98: 752–754. doi:10.1136/archdischild-2013-304292
769
11. Rohani P, Earn DJD, Grenfell BT. Impact of immunisation on pertussis
770
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
transmission in England and Wales. Lancet. 2000;355: 285–286.
771
12. Ferguson NM, Donnelly CA, Anderson RM. The foot-and-mouth epidemic in Great
772
Britain: Pattern of spread and impact of interventions. Science (80- ). 2001;292:
773
11551160.
774
13. Kao RR. The role of mathematical modelling in the control of the 2001 FMD
775
epidemic in the UK. Trends Microbiol. 2002;10: 279–286.
776
14. Keeling MJ. Models of foot-and-mouth disease. Proc R Soc B Biol Sci. 2005;272:
777
11951202.
778
15. Keeling MJ, Woolhouse MEJ, Shaw DJ, Matthews L, Chase-Topping M, Haydon
779
DT, et al. Dynamics of the 2001 UK foot and mouth epidemic: Stochastic
780
dispersal in a heterogeneous landscape. Science (80- ). 2001;294: 813–818.
781
16. Pautasso M, Aas G, Queloz V, Holdenrieder O. European ash (Fraxinus
782
excelsior) dieback – A conservation biology challenge. Biol Conserv. Elsevier Ltd;
783
2013;158: 37–49. doi:10.1016/j.biocon.2012.08.026
784
17. Cunniffe NJ, Laranjeira FF, Neri FM, DeSimone RE, Gilligan CA. Cost-effective
785
control of plant disease when epidemiological knowledge is incomplete: Modelling
786
Bahia bark scaling of citrus. PLoS Comput Biol. 2014;10: e1003753.
787
doi:10.1371/journal.pcbi.1003753
788
18. Chan M-S, Jeger MJ. An analytical model of plant virus disease dynamics with
789
roguing and replanting. J Appl Ecol. 1994;31: 413–427.
790
19. Wu JT, Cowling BJ, Lau EHY, Ip DKM, Ho L-M, Tsang T, et al. School closure
791
and mitigation of pandemic (H1N1) 2009, Hong Kong. Emerg Infect Dis. 2010;16:
792
538. doi:10.3201/eid1603.091216
793
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
20. Thompson RN, Gilligan CA, Cunniffe NJ. Control fast or control smart: When
794
should invading pathogens be controlled? PLoS Comput Biol. 2018;14.
795
doi:10.1371/journal.pcbi.1006014
796
21. Keeling MJ, Rohani P. Modeling infectious diseases in humans and animals.
797
Princeton University Press; 2008.
798
22. Lloyd AL, Zhang J, Root AM. Stochasticity and heterogeneity in host-vector
799
models. J R Soc Interface. 2007;4: 851–863. doi:10.1098/rsif.2007.1064
800
23. Lloyd-Smith JO, Schreiber SJ, Kopp PE, Getz WM. Superspreading and the effect
801
of individual variation on disease emergence. Nature. 2005;438: 355–359.
802
doi:10.1038/nature04153
803
24. Nishiura H, Cook AR, Cowling BJ. Assortativity and the probability of epidemic
804
extinction: a case study of pandemic influenza A (H1N1-2009). Interdiscip
805
Perspect Infect Dis. 2011; 194507.
806
25. Anderson D, Watson R. On the spread of a disease with gamma distributed latent
807
and infectious periods. Biometrika. 1980;67: 191–198.
808
26. Thompson RN, Jalava K, Obolski U. Sustained transmission of Ebola in new
809
locations: more likely than previously thought. Lancet Infect Dis. 2019;
810
doi:10.1016/S1473- 3099(19)30360-3
811
27. Swinton J. Extinction times and phase transitions for spatially structured closed
812
epidemics. Bull Math Biol. 1998;60: 215–230.
813
28. Merler S, Ajelli M, Fumanelli L, Parlamento S, Pastore y Piontti A, Dean NE, et al.
814
Containing Ebola at the Source with Ring Vaccination. PLoS Negl Trop Dis.
815
2016;10: 1–11. doi:10.1371/journal.pntd.0005093
816
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
29. Thompson RN, Thompson C, Pelerman O, Gupta S, Obolski U. Increased
817
frequency of travel in the presence of cross-immunity may act to decrease the
818
chance of a global pandemic. Philos Trans R Soc B. 2019;374: 20180274.
819
doi:10.1098/rstb.2018.0274
820
30. Althaus CL, Low N, Musa EO, Shuaib F, Gsteiger. Ebola virus disease outbreak
821
in Nigeria: Transmission dynamics and rapid control. Epidemics. 2015;11: 80–84.
822
31. Abdullah N, Kelly JT, Graham SC, Birch J, Gonçalves-Carneiro D, Mitchell T, et
823
al. Structure-guided identification of a nonhuman morbillivirus with zoonotic
824
potential. J Virol. 2018;92: e01248-18.
825
32. Keeling MJ, Danon L. Mathematical modelling of infectious diseases. Br Med Bull.
826
2009;92: 33–42. doi:10.1093/bmb/ldp038
827
33. Keeling MJ, Ross J V. On methods for studying stochastic disease dynamics. J R
828
Soc Inferface. 2008;5: 171–181. doi:10.1098/rsif.2007.1106
829
34. Kessler DA, Shnerb NM. Solution of an infection model near threshold. Phys Rev
830
E. 2007;76: 010901. doi:10.1103/PhysRevE.76.010901
831
35. Park AW, Gubbins S, Gilligan CA. Invasion and persistence of plant parasites in a
832
spatially structured host population. Oikos. 2001;94: 162–174.
833
36. Kendall DG. Deterministic and stochastic epidemics in closed populations. Proc
834
3rd Berkeley Symp Math Stat Prob. 1956;4: 149–165.
835
37. de Jong MCM. Mathematical modelling in veterinary epidemiology: why model
836
building is important. Prev Vet Med. 1995;25: 183–193.
837
38. Nasell I. 'The threshold concept in stochastic epidemic and endemic models’ in D.
838
Mollison, Epidemic models: their structure and relation to data. Cambridge
839
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
University Press; 1995.
840
39. Finger F, Funk S, White K, Siddiqui MR, Edmunds WJ, Kucharski AJ. Real-time
841
analysis of the diphtheria outbreak in forcibly displaced Myanmar nationals in
842
Bangladesh. BMC Med. 2019;17: 58. doi:10.1038/d41586-018-05810-w
843
40. Lamontagne F, Clément C, Fletcher T, Jacob ST, Fischer II WA, Fowler RA.
844
Doing today’s work superbly well - treating Ebola with current tools. N Engl J Med.
845
2014;371: 15651566.
846
41. O’Dowd A. Government says it would stockpile Tamiflu again. Br Med J.
847
2014;349: g6386. doi:10.1136/bmj.g6386
848
42. Butler D. Africa’s latest Ebola outbreak comes to swift end. Nature. 2018; 2018
849
2020.
850
43. Brauer F. “Compartmental models in epidemiology” in F. Brauer, P. van den
851
Driessche, J. Wu - Mathematical Epidemiology. Springer; 2014. doi:10.1007/978-
852
3-540-78911-6
853
44. Raggett GF. A stochastic model of the Eyam plague. J Appl Stat. 1982;9: 212
854
225. doi:10.1080/02664768200000021
855
45. Kucharski AJ, Funk S, Eggo RM, Mallet H-P, Edmunds WJ, Nilles EJ.
856
Transmission Dynamics of Zika Virus in Island Populations : A Modelling Analysis
857
of the 2013 – 14 French Polynesia Outbreak. PLoS Negl Trop Dis. 2016;10:
858
e0004726. doi:10.1371/journal.pntd.0004726
859
46. van den Driessche P, Watmough J. Further notes on the basic reproduction
860
number. In “Mathematical Epidemiology.” Springer; 2008.
861
47. Heffernan JM, Smith RJ, Wahl LM. Perspectives on the basic reproductive ratio. J
862
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
R Soc Inferface. 2005;2: 281–293. doi:10.1098/rsif.2005.0042
863
48. Artalejo JR, Economou A, Lopez-Herrero MJ. The maximum number of infected
864
individuals in SIS epidemic models: Computational techniques and quasi-
865
stationary distributions. J Comput Appl Math. 2010;233: 2563–2574.
866
doi:10.1016/j.cam.2009.11.003
867
49. Gillespie DT. Exact stochastic simulation of coupled chemical reactions. J Phys
868
Chem. 1977;8: 2340–2361.
869
50. Cori A, Ferguson NM, Fraser C, Cauchemez S. A new framework and software to
870
estimate time-varying reproduction numbers during epidemics. Am J Epidemiol.
871
2013;178: 1505–1512. doi:10.1093/aje/kwt133
872
51. Thompson RN, Stockwin JE, Gaalen RD Van, Polonsky JA, Kamvar ZN, Demarsh
873
PA, et al. Improved inference of time-varying reproduction numbers during
874
infectious disease outbreaks. Epidemics. 2019;19: 100356.
875
doi:10.1016/j.epidem.2019.100356
876
52. Orbann C, Sattenspiel L, Miller E, Dimka J. Defining epidemics in computer
877
simulation models: How do definitions influence conclusions? Epidemics. Elsevier
878
B.V.; 2017;19: 24–32. doi:10.1016/j.epidem.2016.12.001
879
53. Keeling MJ. The implications of network structure for epidemic dynamics. Theor
880
Popul Biol. 2005;67: 1–8. doi:10.1016/j.tpb.2004.08.002
881
54. Craig AP, Cunniffe NJ, Parry M, Laranjeira FF, Gilligan CA. Grower and regulator
882
conflict in management of the citrus disease Huanglongbing in Brazil: A modelling
883
study. J Appl Ecol. 2017;55: 1956–1965. doi:10.1111/1365-2664.13122
884
55. Belasque Jr. J, Bassanezi RB, Yamamoto PT, Ayres AJ, Tachibana A, Violante
885
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
AR, et al. Letter to the editor: Lessons from Huanglongbing management in Sao
886
Paulo state, Brazil. J Plant Pathol. 2010;92: 285–302.
887
56. LaMontagne DS, Fenton KA, Randall S, Anderson S, Carter P. Establishing the
888
National Chlamydia Screening Programme in England: results from the first full
889
year of screening. Sex Transm Infect. 2004;80: 335–341.
890
57. Hollingsworth TD, Klinkenberg D, Heesterbeek H, Anderson RM. Mitigation
891
strategies for pandemic influenza A: Balancing conflicting policy objectives. PLoS
892
Comput Biol. 2011;7: e1001076. doi:10.1371/journal.pcbi.1001076
893
58. Hyatt-Twynam SR, Parnell S, Stutt ROJH, Gottwald TR, Gilligan CA, Cunniffe NJ.
894
Risk-based management of invading plant disease. New Phytol. 2017;214: 1317–
895
1329.
896
59. Thompson RN, Hart WS. Effect of confusing symptoms and infectiousness on
897
forecasting and control of Ebola outbreaks. Clin Infect Dis. 2018;67.
898
doi:10.1093/cid/ciy248
899
60. Thompson RN, Cobb RC, Gilligan CA, Cunniffe NJ. Management of invading
900
pathogens should be informed by epidemiology rather than administrative
901
boundaries. Ecol Modell. 2016;324. doi:10.1016/j.ecolmodel.2015.12.014
902
61. Cunniffe NJ, Cobb RC, Meentemeyer RK, Rizzo DM, Gilligan CA. Modeling when,
903
where, and how to manage a forest epidemic, motivated by sudden oak death in
904
California. Proc Natl Acad Sci USA. 2016;113: 5640–5645.
905
doi:10.1073/pnas.1602153113
906
907
.CC-BY 4.0 International licenseIt is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint. http://dx.doi.org/10.1101/768853doi: bioRxiv preprint first posted online Sep. 13, 2019;
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