1
Supplementary Material: Basic reproductive numbers
2
Here, differential equation analogs of recursion equations are presented (analogs are presented
3
for eq.’s 1-4, 5-7, 8-9, 14-16, and 17-18; analogs for eq.’s 19-21, 22-23, 24-26, and 27-28 are
4
available on request). The differential equations were used to calculate the basic reproductive
5
number (R0) for each model. The basic reproductive number is a threshold value: with values of
6
R0 < 1 the pathogen does not spread, whereas with R0 > 1 pathogen spread occurs. Derivation of
7
basic reproductive numbers was based on the approach described by van den Driessche and
8
Watmough (2002).
9
10
Models with Circulative Persistent Transmission. To calculate the basic reproductive number
11
(R0), the recursion equations must be rewritten as differential equations. With logistic vector
12
population growth and circulative persistent transmission (eq.’s 1-4 and eq.’s 5-7), the
13
differential equation analogs were:
14
𝑑𝑆⁄ = (𝐿 + 𝐼 + 𝑅)𝑟 − (𝑍 ) 𝜃𝑣𝑆
𝑑𝑡
𝑃
eq. S1
15
𝑑𝐿⁄ = (𝑍 ) 𝜃𝑣𝑆 − (𝜏 + 𝑟)𝐿
𝑑𝑡
𝑃
eq. S2
16
𝑑𝐼⁄ = 𝐿𝜏 − 𝐼(𝑟 + 𝑞)
𝑑𝑡
eq. S3
17
𝑑𝑅⁄ = 𝐼𝑞 − 𝑅𝑟
𝑑𝑡
eq. S4
18
𝑑𝑋⁄ = 𝑁𝑏 (1 − 𝑁) − 𝑋𝑚 − ( 𝐼 ) 𝛼𝑣𝑋
𝑑𝑡
𝐾
𝑃
eq. S5
19
𝑑𝑌⁄ = ( 𝐼 ) 𝛼𝑣𝑋 − 𝑌𝑚 − 𝑌𝜔
𝑑𝑡
𝑃
eq. S6
20
𝑑𝑍⁄ = 𝑌𝜔 − 𝑍𝑚
𝑑𝑡
eq. S7
21
Similarly, with fixed vector population size and circulative persistent transmission (eq.’s 1-4 and
22
eq.’s 14-16), the differential equation analogs were:
23
𝑑𝑆⁄ = (𝐿 + 𝐼 + 𝑅)𝑟 − (𝑍 ) 𝜃𝑣𝑆
𝑑𝑡
𝑃
eq. S8
24
𝑑𝐿⁄ = (𝑍 ) 𝜃𝑣𝑆 − (𝜏 + 𝑟)𝐿
𝑑𝑡
𝑃
eq. S9
25
𝑑𝐼⁄ = 𝐿𝜏 − 𝐼(𝑟 + 𝑞)
𝑑𝑡
eq. S10
26
𝑑𝑅⁄ = 𝐼𝑞 − 𝑅𝑟
𝑑𝑡
eq. S11
27
𝑑𝑋⁄ = (𝑌 + 𝑍)𝑚 − ( 𝐼 ) 𝛼𝑣𝑋
𝑑𝑡
𝑃
eq. S12
28
𝑑𝑌⁄ = ( 𝐼 ) 𝛼𝑣𝑋 − 𝑌𝑚 − 𝑌𝜔
𝑑𝑡
𝑃
eq. S13
29
𝑑𝑍⁄ = 𝑌𝜔 − 𝑍𝑚
𝑑𝑡
eq. S14
30
The method for calculating R0 described by van den Driessche and Watmough (2002) requires
31
assuming that plant and vector populations were at the disease free equilibrium. Accordingly, all
32
plants were assumed to be uninfected (i.e., [S, L, I, R] = [P, 0, 0, 0], where P is the total number
33
of plants) and the vector population was assumed to be at equilibrium abundance and consist of
34
only pathogen-free vectors (i.e., [X, Y, Z] = [N, 0, 0], where N = V in models with fixed vector
35
population size and 𝑁 =
36
populations were assumed to be of stable size, the equation for noninoculative vectors (X; S5 and
37
S12) could be omitted from both models (S1-S7 and S8-S14). With stable vector population
38
size, the number of noninoculative vectors in both models was N – Y – Z, where the two models
39
differed solely in their interpretation of N (see above). Eliminating the equation for
40
noninoculative vectors allowed both models (S1-S7 and S8-S14) to be represented by a single
41
system of equations:
𝐾(𝑏−𝑚)
𝑏
in models with logistic vector population growth). As vector
42
𝑑𝑆⁄ = (𝐿 + 𝐼 + 𝑅)𝑟 − (𝑍 ) 𝜃𝑣𝑆
𝑑𝑡
𝑃
eq. S15
43
𝑑𝐿⁄ = (𝑍 ) 𝜃𝑣𝑆 − (𝜏 + 𝑟)𝐿
𝑑𝑡
𝑃
eq. S16
44
𝑑𝐼⁄ = 𝐿𝜏 − 𝐼(𝑟 + 𝑞)
𝑑𝑡
eq. S17
45
𝑑𝑅⁄ = 𝐼𝑞 − 𝑅𝑟
𝑑𝑡
eq. S18
46
𝑑𝑌⁄ = ( 𝐼 ) 𝛼𝑣𝑋 − 𝑌𝑚 − 𝑌𝜔
𝑑𝑡
𝑃
eq. S19
47
𝑑𝑍⁄ = 𝑌𝜔 − 𝑍𝑚
𝑑𝑡
eq. S20
48
The infected compartments in eq.’s S15-20 were L, I, Y, and Z. Following Driessche and
49
Watmough (2002) and Wonham et al. (2004), the equations for the infected compartments were
50
written in vector format, separating terms describing appearance of new infections (a) from those
51
resulting in loss of infections or transition of infections between compartments (d).
52
53
54
55
56
57
(𝑍⁄𝑃)𝜃𝑣𝑆
(𝜏 + 𝑟)𝐿
𝐿
𝐼(𝑟 + 𝑞) − 𝐿𝜏
0
𝑑⁄ [ 𝐼 ] = 𝑎 − 𝑑 =
−[
]
𝑑𝑡 𝑌
𝐼
𝑌(𝑚 + 𝑤)
( ⁄𝑃 )𝛼𝑣𝑋
𝑍
𝑍𝑚 − 𝑌𝑤
[
]
0
eq. S21
Eq. S21 provides the corresponding Jacobian matrices A and D:
0
0
𝐴=
0
(0
0
0
𝛼𝑣𝑁
𝑃
0
0 𝜃𝑣
𝜏+𝑟
0
0
0
0 0
−𝜏 𝑟 + 𝑞
0
0
,𝐷=(
)
0 0
0
0
𝑚+𝑤 0
0
0
−𝑤
𝑚
0 0)
eq. S22, S23
Finally, the basic reproductive number (R0) was obtained by determining the dominant
eigenvalue of AD-1:
𝑅𝑜 = √(
𝛼𝑣(
𝑤
)
𝑤+𝑚
𝑞+𝑟
)(
𝜃𝑣(
𝑡
)
𝑟+𝑡
𝑚
𝑁
) (𝑃 )
eq. S24
58
The expression for R0 (eq. S24) consisted of three terms and was similar to R0 values reported for
59
other vector models (Madden et al. 2000, Wonham et al. 2004, Zeilinger and Daugherty 2014 ).
60
The first term represents movement of the pathogen from an infected plant to the vector. Thus,
61
the first term multiplies probability of a pathogen-free vector acquiring the pathogen (αv) to
62
probability of a vector surviving to become inoculative (𝑤+𝑚) and to average time an infected
𝑤
1
63
plant remains infectious (𝑞+𝑟). The second term represents movement of the pathogen from an
64
inoculative vector to a healthy plant. Thus, the second term multiplies probability of an
65
inoculative vector transmitting the pathogen (θv) to probability of a latently infected plant
66
surviving to become infectious (𝑟+𝑡) and to average lifespan of a vector (𝑚). The final term
67
( 𝑃 ) represents the number of insects per plant.
68
𝑡
1
𝑁
The procedure described above also was used to determine the basic reproductive number
69
for models that assessed mortality at the start of a time step (eq.’s 19-21 and eq.’s 24-26;
70
differential equation analogs available on request), providing:
71
𝑅𝑜 = √(
𝛼𝑣(1−𝑚)(
𝑤(1−𝑚)
)
𝑤(1−𝑚)+𝑚
𝑞+𝑟
)(
𝜃𝑣(
𝑡
)
𝑟+𝑡
𝑚
𝑁
) (𝑃 )
eq. S25
72
With fixed vector population size (eq.’s 24-26), N represented the fixed vector population size V.
73
With logistic vector population growth (eq.’s 19-21), N represented the equilibrium abundance of
74
vectors which was determined by
75
(eq. S25) was similar to the expression for R0 with late vector mortality (eq. S24). With early
76
vector mortality (eq. S25), the probability of a pathogen-free vector acquiring the pathogen (αv)
77
and rate latent vectors become inoculative (w) were adjusted for vector mortality.
𝐾(𝑏(1−𝑚)−𝑚)
𝑏(1−𝑚)
. The expression for R0 with early vector mortality
78
79
Models with Non-Persistent Transmission. To calculate the basic reproductive number (R0),
80
the recursion equations were rewritten as differential equations. With logistic vector population
81
growth and non-persistent transmission (eq.’s 1-4 and eq.’s 8-9), the differential equation
82
analogs were:
83
𝑑𝑆⁄ = (𝐿 + 𝐼 + 𝑅)𝑟 − (𝑍 ) 𝜃𝑣𝑆
𝑑𝑡
𝑃
eq. S26
84
𝑑𝐿⁄ = (𝑍 ) 𝜃𝑣𝑆 − (𝜏 + 𝑟)𝐿
𝑑𝑡
𝑃
eq. S27
85
𝑑𝐼⁄ = 𝐿𝜏 − 𝐼(𝑟 + 𝑞)
𝑑𝑡
eq. S28
86
𝑑𝑅⁄ = 𝐼𝑞 − 𝑅𝑟
𝑑𝑡
eq. S29
87
𝑑𝑋⁄ = 𝑍𝑓 + 𝑁𝑏 (1 − 𝑁) − 𝑋𝑚 − ( 𝐼 ) 𝛼𝑣𝑋
𝑑𝑡
𝐾
𝑃
eq. S30
88
𝑑𝑍⁄ = ( 𝐼 ) 𝛼𝑣𝑋 − 𝑍𝑚 − 𝑍𝑓
𝑑𝑡
𝑃
eq. S31
89
Similarly, with fixed vector population size and non-persistent transmission (eq.’s 1-4 and eq.’s
90
17-18), the differential equation analogs were:
91
𝑑𝑆⁄ = (𝐿 + 𝐼 + 𝑅)𝑟 − (𝑍 ) 𝜃𝑣𝑆
𝑑𝑡
𝑃
eq. S32
92
𝑑𝐿⁄ = (𝑍 ) 𝜃𝑣𝑆 − (𝜏 + 𝑟)𝐿
𝑑𝑡
𝑃
eq. S33
93
𝑑𝐼⁄ = 𝐿𝜏 − 𝐼(𝑟 + 𝑞)
𝑑𝑡
eq. S34
94
𝑑𝑅⁄ = 𝐼𝑞 − 𝑅𝑟
𝑑𝑡
eq. S35
95
𝑑𝑋⁄ = 𝑍𝑚 + 𝑍𝑓 − ( 𝐼 ) 𝛼𝑣𝑋
𝑑𝑡
𝑃
eq. S36
96
𝑑𝑍⁄ = ( 𝐼 ) 𝛼𝑣𝑋 − 𝑍𝑚 − 𝑍𝑓
𝑑𝑡
𝑃
eq. S37
97
Again, the plant and vector populations were assumed to be at the disease free equilibrium.
98
Accordingly, all plants were assumed to be uninfected (i.e., [S, L, I, R] = [P, 0, 0, 0], where P is
99
the total number of plants) and the vector population was assumed to be at equilibrium
100
abundance and consist of only pathogen-free vectors (i.e., [X, Z] = [N, 0]), where N = V in
101
models with fixed vector population size and 𝑁 =
102
population growth). As vector populations were assumed to be of stable size, the equation for
103
noninoculative vectors (X; S30 and S36) could be omitted from both models (S26-S31 and S32-
𝐾(𝑏−𝑚)
𝑏
in models with logistic vector
104
S37). With stable vector population size, the number of noninoculative vectors in both models
105
was N – Z, where the two models differed solely in their interpretation of N (see above).
106
Eliminating the equation for noninoculative vectors allowed both models (S26-S31 and S32-S37)
107
to be represented by a single system of equations:
108
𝑑𝑆⁄ = (𝐿 + 𝐼 + 𝑅)𝑟 − (𝑍 ) 𝜃𝑣𝑆
𝑑𝑡
𝑃
eq. S38
109
𝑑𝐿⁄ = (𝑍 ) 𝜃𝑣𝑆 − (𝜏 + 𝑟)𝐿
𝑑𝑡
𝑃
eq. S39
110
𝑑𝐼⁄ = 𝐿𝜏 − 𝐼(𝑟 + 𝑞)
𝑑𝑡
eq. S40
111
𝑑𝑅⁄ = 𝐼𝑞 − 𝑅𝑟
𝑑𝑡
eq. S41
112
𝑑𝑍⁄ = ( 𝐼 ) 𝛼𝑣𝑋 − 𝑍𝑚 − 𝑍𝑓
𝑑𝑡
𝑃
eq. S42
113
The infected compartments were L, I, and Z. The equations for the infected compartments were
114
rewritten in vector format, separating the terms describing appearance of new infections (a) from
115
those resulting in loss of infections or transition of infections between compartments (d).
116
117
118
119
120
121
(𝜏 + 𝑟)𝐿
(𝑍 ⁄ 𝑃)𝜃𝑣𝑆
𝐿
𝑑⁄ [ 𝐼 ] = 𝑎 − 𝑑 = [
0
] − [𝐼 (𝑟 + 𝑞) − 𝐿𝜏]
𝑑𝑡
(𝐼 ⁄ 𝑃)𝛼𝑣𝑋)
(𝑚 + 𝑓)𝑍
𝑍
eq. S43
Eq. S43 provides the Jacobian matrices A and D:
0
𝐴 = [0
0
0
0
𝛼𝑣𝑁
𝑃
𝜃𝑣
𝜏+𝑟
0 ] , 𝐷 = [ −𝜏
0
0
0
𝑟+𝑞
0
0
0 ]
𝑚+𝑓
eq.’s S44, S45
Finally, the basic reproductive number (R0) was obtained by determining the dominant
eigenvalue of AD-1:
𝛼𝑣
𝑅𝑜 = √(𝑞+𝑟) (
𝜃𝑣(
𝑡
)
𝑟+𝑡
𝑓+𝑚
𝑁
) (𝑃 )
eq. S46
122
The expression of R0 for a model with non-persistent transmission was similar to the expression
123
of R0 for a model with circulative persistent transmission (compare eq. S24 to S46). While the
124
expressions were similar, there were some important differences. With a non-persistent
125
pathogen, vectors become inoculative immediately after acquiring the pathogen. Accordingly,
126
with a non-persistent pathogen, the first term was not multiplied by probability of a vector
127
surviving to become inoculative (𝑤+𝑚). Similarly, the second term was multiplied by the
128
average period a vector remains inoculative (𝑓+𝑚).
129
𝑤
1
Using the same procedure, the basic reproductive number (R0) for models that assessed
130
mortality at the start of a time step (eq.’s 22-23 and eq.’s 27-28; differential equation analogs
131
available on request) was:
𝑡
132
𝜃𝑣( )
𝛼𝑣(1−𝑚)
𝑁
𝑟+𝑡
) (𝑚+(1−𝑚)𝑓
) (𝑃 )
𝑞+𝑟
𝑅𝑜 = √(
eq. S47
133
With fixed vector population size (eq.’s 27-28), N represented the fixed vector population size V.
134
With logistic vector population growth (eq.’s 22-23), N represented the equilibrium abundance of
135
vectors which was determined by
136
(eq. S47) was similar to the expression for R0 with late vector mortality (S46). With early vector
137
mortality (eq. S47), the probability of a pathogen-free vector acquiring the pathogen (αv) and the
138
rate vectors lost inoculativity (f) was adjusted for vector mortality.
𝐾(𝑏(1−𝑚)−𝑚)
𝑏(1−𝑚)
. The expression for R0 with early vector mortality
139
140
141
References for Supplementary Material
Madden, L. V., M. J. Jeger, and F. van den Bosch. 2000. A theoretical assessment of the
142
effects of vector-virus transmission mechanism on plant virus disease epidemics.
143
Phytopathol. 90: 576-594.
144
Van den Driessche, P., and J. Watmough. 2002. Reproduction number and sub-threshold
145
endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180:
146
29-48.
147
148
149
150
151
Wonham, M. J., T. de-Comina-Beck, and M. A. Lewis. 2004. An epidemiological model for
West Nile virus: invasion analysis and control applications. P. Roy. Soc. B 271: 501-507.
Zeilinger, A. R., and M. P. Daugherty. 2014. Vector preference and host defense against
infection interact to determine disease dynamics. Oikos 123: 613-622.