Supplemental information
When should a trophically and vertically transmitted parasite manipulate its intermediate hosts?
The case of Toxoplasma gondii
Appendix A: Computation of the basic reproductive ratio, R0
The following system describes the epidemiological dynamics of the parasite T. gondii in DHs,
IHs, and the environment:
𝐼1̇ = 𝑔𝑎𝑁2 𝜁𝐼2 (𝐾1 − 𝐼1 − 𝑅1 )/(𝑆2 + 𝜁𝐼2 ) − (𝑚1 + 𝑘1 𝑁1 + 𝛾)𝐼1
𝑅1̇ = 𝛾𝐼1 − 𝑏1 𝑅1
𝐸̇ = 𝜆𝐼1 − 𝑑𝐸
(A.1)
𝑆2̇ = 𝑏2 𝑆2 + (1 − 𝜋2 )𝑏2 𝐼2 − (𝑚2 + 𝑘2 𝑁2 + 𝛽2 𝐸 + 𝑎𝑁2 𝐾1 ⁄(𝑆2 + 𝜁𝐼2 ))𝑆2
𝐼2̇ = 𝜋2 𝑏2 𝐼2 + 𝛽2 𝐸𝑆2 − (𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁⁄(𝑆2 + 𝜁𝐼2 ) + 𝛼2 )𝐼2
with 𝑁1 = 𝐾1 , 𝐾1 − 𝐼1 − 𝑅1 = 𝑆1 and 𝑏1 = 𝑚1 + 𝑘1 𝐾1 .
From system (A.1), we computed 𝑅0 by using the methodology of Diekmann et al. (1990) and
van den Driessche and Watmough (2002), similarly to Lélu et al. (2010). F and V are
respectively the transmission and the transition matrices,
0
0
0
𝐹 = [𝜆
0 𝛽2 𝐾2∗
𝑔𝑎𝜁𝐾1
𝑏1 + 𝛾
0 ],𝑉 = [ 0
𝑏2 𝜋2
0
0
𝑑
0
0
0
].
𝑏2 + 𝑎𝐾1 (𝜁 − 1) + 𝛼2
with 𝐾2∗ = (𝑏2 − 𝑚2 − 𝑎𝐾1 )/𝑘2 , representing the number of prey at disease free equilibrium.
The next generation matrix, denoted NGM, was computed from these two matrices,
0
𝑁𝐺𝑀 = 𝐹𝑉
−1
0
0
= [𝜆/(𝑏1 + 𝛾)
0
𝛽2 𝐾2∗ /𝑑
𝑔𝑎𝜁𝐾1 /(𝑏2 + 𝑎𝐾1 (𝜁 − 1) + 𝛼2 )
0
].
𝑏2 𝜋2 /(𝑏2 + 𝑎𝐾1 (𝜁 − 1) + 𝛼2 )
1
The basic reproductive rate is the spectral radius of the matrix NGM and the maximum real
solution of the following cubic equation:
𝑔(𝑥) = 𝑥 3 −
𝑏2 𝜋2
𝜆𝛽2 𝐾2∗ 𝑔𝑎𝜁𝐾1
𝑥2 −
𝑏2 + 𝑎𝐾1 (𝜁 − 1) + 𝛼2
𝑑(𝑏1 + 𝛾)(𝑏2 + 𝑎𝐾1 (𝜁 − 1) + 𝛼2 )
(A.2)
Similarly to Lélu et al. (2010) and Turner et al. (submitted), the expressions 𝑏2 𝜋2 /(𝑏2 +
𝑎𝐾1 (𝜁 − 1) + 𝛼2 ) and 𝜆𝛽2 𝐾2∗ 𝑔𝑎𝜁𝐾1 ⁄(𝑑(𝑏1 + 𝛾)(𝑏2 + 𝑎𝐾1 (𝜁 − 1) + 𝛼2 )) can be assimilated to
partial 𝑅0 resulting respectively from vertical transmission 𝑅0𝑉 , and trophic transmission 𝑅0𝑇 .
Thus from (A.2), 𝑅0 satisfies the following equation:
𝑅0 3 − 𝑅0𝑉 𝑅0 2 − 𝑅0𝑇 = 0
(A.3)
Next we demonstrate that 𝑅0 is an increasing function of 𝑅0𝑉 and 𝑅0𝑇 .
First let’s prove that R0  R0V. Elementary calculus shows that the cubic function g is increasing
in the range [2 R0V /3, +) and negative in the range [0, 2 R0V /3]. Both relations 𝑔(𝑅0𝑉 ) =
−𝑅0𝑇 < 0 and 𝑔(𝑅0 ) = 0 imply 𝑅0 ≥ 𝑅0𝑉 .
Implicit differentiation of (A.3) with respect to 𝑅0𝑉 supplies
𝑑( 𝑅0 3 − 𝑅0𝑉 𝑅0 2 − 𝑅0𝑇 )
𝑑𝑅0
𝑑𝑅0
= 3𝑅0 2
− 𝑅0 2 − 2𝑅0𝑉 𝑅0
=0
𝑑𝑅0𝑉
𝑑𝑅0𝑉
𝑑𝑅0𝑉
𝑑𝑅
which yields 𝑑𝑅 0 = 3𝑅
𝑅0
0 −2𝑅0𝑉
0𝑉
≥ 0 because 𝑅0 ≥ 𝑅0𝑉 .
Implicit differentiation of (A.3) with respect to 𝑅0𝑇 supplies
𝑑( 𝑅0 3 − 𝑅0𝑉 𝑅0 2 − 𝑅0𝑇 )
𝑑𝑅0
𝑑𝑅0
= 3𝑅0 2
− 2𝑅0𝑉 𝑅0
−1=0
𝑑𝑅0𝑇
𝑑𝑅0𝑇
𝑑𝑅0𝑇
𝑑𝑅
1
(3𝑅
0
0 −2𝑅0𝑉 )
which yields 𝑑𝑅 0 = 𝑅
0𝑇
≥ 0 because 𝑅0 ≥ 𝑅0𝑉 .
2
Thus the overall 𝑅0 is a complex function increasing with both partial rates 𝑅0𝑉 and 𝑅0𝑇 .
The reproductive ratio resulting from vertical transmission, 𝑅0𝑉 , always increases as vertical
transmission 𝜋2 increases, and decreases as manipulation 𝜁 (with 𝜁 ≥ 1) or virulence 𝛼2
increased. The reproductive ratio from trophic transmission, 𝑅0𝑇 , increases with manipulation if
𝑑𝑅0𝑇
𝑑𝜁
𝑔𝑎𝐾1 𝜆𝛽2 𝐾2∗ (𝑏2 +𝛼2 −𝑎𝐾1 )
2
1 +𝛾)(𝑏2 +𝑎𝐾1 (𝜁−1)+𝛼2 )
= 𝑑(𝑏
> 0, that is if (𝑏2 + 𝛼2 − 𝑎𝐾1 ) >0, which is always the case if
the IH population persists.
With the chosen parameters, 𝑅0 always increases with vertical transmission and manipulation
(excepted for 𝜋2 = 1 and 𝛼2 = 0, Fig. 2, main text). However, it is not always the case, for
example if the prey birth rate is reduced to 5/52 (instead of 6/52) we observe a threshold of
vertical transmission below which 𝑅0 decreases as manipulation increases (Fig A.1).
Figure A.1: Values of proportion of vertical transmission 𝜋2 and virulence rate 𝛼2 for which 𝑅0
increases (grey area) or decreases (white area) as a function of the manipulation coefficient 𝜁.
Parameters are the same as in the main text, excepted for the prey birth rate, 𝑏2 =5/52 (instead of
6/52).
3
Appendix B: Linking epidemiological and evolutionary dynamics.
B1. Considering 2 strains, a wild type wt and a mutant m and evolution of the manipulation rate
𝜁
System (A.1) is modified in order to allow for 2 pathogen strains, a wild type wt and a mutant m,
which only differ in their ability to manipulate IH behaviour. The following system represents
the dynamics of the infected hosts or the environmental stage for strain i (with i = m or wt):
𝐼1𝑖̇ = 𝑔𝑎𝑁2 𝜁 𝑖 𝐼2𝑖 (𝐾1 − 𝐼1 − 𝑅1 )/(𝑆2 + 𝜁 𝐼̅ 2 ) − (𝑚1 + 𝑟1 + 𝛾)𝐼1𝑖
𝐸̇ 𝑖 = 𝜆𝐼1𝑖 − 𝑑𝐸 𝑖
(B.1)
𝐼2𝑖̇ = 𝜋2 𝑏2 𝐼2𝑖 + 𝛽2 𝐸 𝑖 𝑆2 − (𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁 𝑖 ⁄(𝑆2 + 𝜁 𝐼̅ 2 ) + 𝛼2 )𝐼2𝑖
𝐼𝑖
with 𝐼𝑗 = ∑𝑖 𝐼𝑗𝑖 , 𝑆1 = 𝐾1 − 𝐼1 − 𝑅1 and where 𝜁 ̅ = ∑𝑖 𝐼2 𝜁 𝑖 refers to the average manipulation trait
2
value.
We compute the changes of the frequency of the mutant in the three different compartments of
the model, where 𝑝1𝑚 =
𝐼1𝑚
𝐼1
; 𝑝𝐸𝑚 =
𝐸𝑚
𝐸
and 𝑝2𝑚 =
𝐼2𝑚
𝐼2
refer to the frequency of the mutant in the
DH, E and IH compartments respectively. For instance, the change in frequency of the mutant in
the DH compartment is computed as follows: 𝑝̇1𝑚 =
𝐼1̇𝑚
𝐼1
𝐼̇
− 𝑝1𝑚 𝐼1 . A similar reasoning is applied to
1
each compartment which yields:
𝑆1
((𝜁 𝑚
+𝜁
2 ̅ 𝐼2
𝑝̇1𝑚 = 𝑔𝑎𝑁2 𝑆
𝐼
− 𝜁 )̅ 𝑝2𝑚 + 𝜁 (̅ 𝑝2𝑚 − 𝑝1𝑚 )) 𝐼2
1
𝐼
𝑝̇𝐸𝑚 = (𝑝1𝑚 − 𝑝𝐸𝑚 )𝜆 𝐸1
(3a)
(3b)
𝐸
𝑎𝑁2 𝐾1
(𝜁 𝑚
̅ 𝐼2
2 +𝜁
𝑝̇2𝑚 = (𝑝𝐸𝑚 − 𝑝2𝑚 )𝛽2 𝐼 𝑆2 − 𝑆
2
− 𝜁 )̅ 𝑝2𝑚
(3c)
4
B2. General case with n strains and evolution of the manipulation rate 𝜁, vertical transmission
proportion 𝜋2 , and virulence 𝛼2
The following system consider 𝑛 strains and variations in manipulation rate 𝜁, vertical
transmission proportion 𝜋2 , and virulence 𝛼2 , between strains. The methodology developed by
Day and Gandon (2006) is used. The system for the variations of the total numbers of individuals
in each state is as following,
𝐼1̇ = 𝑔𝑎𝑁2 𝜁 𝐼̅ 2 (𝐾1 − 𝐼1 − 𝑅1 )/(𝑆2 + 𝜁 𝐼̅ 2 ) − (𝑏1 + 𝛾)𝐼1
𝑅1̇ = 𝛾𝐼1 − 𝑏1 𝑅1
𝐸̇ = 𝜆𝐼1 − 𝑑𝐸
(B.1)
𝑆2̇ = 𝑏2 𝑆2 + (1 − 𝜋̅2 )𝑏2 𝐼2 − (𝑚2 + 𝑘2 𝑁2 + 𝛽2 𝐸 + 𝑎𝑁2 𝐾1 ⁄(𝑆2 + 𝜁 𝐼̅ 2 ))𝑆2
̅ (𝑆2 + 𝜁 𝐼̅ 2 ) + 𝛼̅2 )𝐼2
𝐼2̇ = 𝜋̅2 𝑏2 𝐼2 + 𝛽2 𝐸𝑆2 − (𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁⁄
𝐼𝑖
with 𝐾1 − 𝐼1 − 𝑅1 = 𝑆1 and 𝑥̅ = ∑𝑖 𝐼2 𝑥 𝑖 .
2
The system following the changes in the number of infected host and oocysts of strain i in the
system is:
𝐼1𝑖̇ = 𝑔𝑎𝑁2 𝜁 𝑖 𝐼2𝑖 (𝐾1 − 𝐼1 − 𝑅1 )⁄(𝑆2 + 𝜁 𝐼̅ 2 ) − (𝑏1 + 𝛾)𝐼1𝑖
𝐸̇ 𝑖 = 𝜆𝐼1𝑖 − 𝑑𝐸 𝑖
(B.2)
𝐼2𝑖̇ = 𝜋2𝑖 𝑏2 𝐼2𝑖 + 𝛽2 𝐸 𝑖 𝑆2 − (𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁 𝑖 ⁄(𝑆2 + 𝜁 𝐼̅ 2 ) + 𝛼2𝑖 )𝐼2𝑖
Frequencies of strain i in each infectious compartment are denoted:
𝐼𝑖
𝑝1𝑖 = 𝐼1 the frequency of the strain i in the DH compartment (𝐼1 = ∑𝑖 𝐼1𝑖 ),
1
𝑝𝐸𝑖 =
𝐸𝑖
𝐸
the frequency of the strain i in the E compartment (𝐸 = ∑𝑖 𝐸 𝑖 ),
𝐼𝑖
𝑝2𝑖 = 𝐼2 the frequency of the strain i in the IH compartment (𝐼2 = ∑𝑖 𝐼2𝑖 ).
2
5
The variation in 𝑝1𝑖 is computed as following,
𝐼̇𝑖
𝐼̇
𝑝̇1𝑖 = 𝐼1 − 𝑝1 𝐼1 .
1
1
Using (B.1) and (B.2) it yields,
𝑆1
𝑖
̅ 𝐼2 ((𝜁
2 +𝜁
𝑝̇1𝑖 = 𝑔𝑎𝑁2 𝑆
𝐼
− 𝜁 )̅ 𝑝2𝑖 + 𝜁 (̅ 𝑝2𝑖 − 𝑝1𝑖 )) 𝐼2
(B.3a)
1
𝐼
𝑝̇𝐸𝑖 = (𝑝1𝑖 − 𝑝𝐸𝑖 )𝜆 𝐸1
(B.3b)
𝐸
𝑎𝑁2 𝐾1
𝑖
̅ 𝐼2 (𝜁
2 +𝜁
𝑝̇2𝑖 = (𝑝𝐸𝑖 − 𝑝2𝑖 )𝛽2 𝐼 𝑆2 +𝑏2 (𝜋2𝑖 − 𝜋̅2 )𝑝2𝑖 − 𝑆
2
− 𝜁 )̅ 𝑝2𝑖 − (𝛼2𝑖 − 𝛼̅2 )𝑝2𝑖
(B.3c)
The changes in the average values of the manipulation rate 𝜁 ,̅ the vertical transmission
proportion 𝜋̅2, and the virulence 𝛼̅2 , in each compartment, can be derived from system (B.3). For
example, the variation in the average manipulation rate in the DH compartment is 𝜁1̇ ̅ = ∑𝑖 𝜁𝑖 𝑝̇1𝑖 .
This yields to the following equations, that are grouped according to the different compartment
DH (1), Environment (E) and IH (2),
2
𝜎𝜁𝜁
𝜁1̇ ̅
𝜁2̅ − 𝜁1̅
𝑔𝑎𝑁 𝑆 𝐼
2
(𝜋̅̇2,1 ) = [(𝜎𝜁𝜋2 ) + 𝜁2̅ (𝜋̅2,2 − 𝜋̅2,1 )] (𝑆 +𝜁2̅ 𝐼 1)𝐼2
2
2 1
2
𝛼̅2,2 − 𝛼̅2,1
𝛼̅̇2,1
𝜎𝜁𝛼
2
(B.4a)
𝜁𝐸̇ ̅
𝜁1̅ − 𝜁𝐸̅
𝐼
(𝜋̅̇2,𝐸 ) = (𝜋̅2,1 − 𝜋̅2,𝐸 ) 𝜆 𝐸1
𝛼̅2,1 − 𝛼̅2,𝐸
𝛼̅̇2,𝐸
(B.4b)
2
𝜎𝜁𝜁
𝜎𝜋22 𝜁
𝜎𝛼22 𝜁
𝜁2̇ ̅
𝜁𝐸̅ − 𝜁2̅
𝛽 𝐸𝑆
𝑎𝑁 𝐾
2
(𝜋̅̇2,2 ) = (𝜋̅2,𝐸 − 𝜋̅2,2 ) 2 2 +𝑏2 (𝜎𝜋22 𝜋2 ) − 2̅ 1 (𝜎𝜁𝜋
) − (𝜎𝛼22 𝜋2 )
2
𝐼2
𝑆2 +𝜁 𝐼2
2
𝛼̅2,𝐸 − 𝛼̅2,2
𝜎𝜋22 𝛼2
𝜎𝛼22 𝛼2
𝛼̅̇2,2
𝜎𝜁𝛼
(B.4c)
2
𝑗
with 𝜎𝑥𝑦 , the covariance between traits x and y in the compartment j.
This system may be used to study the evolution of the average values of the different traits with
considering constraints between traits.
6
However if one allow traits to evolve independently from each other, then from (B.3.c) and
(B.4.c), one may expect vertical transmission to be maximized and virulence to be decreased to
its lower level. Similarly to the result presented in the main text, the average manipulation rate
may depend on the epidemiological dynamics which is impacted by the maximal and the lowest
values vertical transmission and virulence may respectively take.
7
Appendix C: Invasion analysis
We investigate the outcome of a mutant parasite introduced in a wild type population of parasite
at equilibrium. The two strains differ only in their ability to manipulate the IHs. System (C.1)
represents the dynamics of the mutant or the resident, depending on whether i = m or wt:
𝐼1𝑖̇ = 𝑔𝑎𝑁2 𝜁 𝑖 𝐼2𝑖 (𝐾1 − 𝐼1 − 𝑅1 )/(𝑆2 + 𝜁 𝐼̅ 2 ) − (𝑏1 + 𝛾)𝐼1𝑖
𝐸̇ 𝑖 = 𝜆𝐼1𝑖 − 𝑑𝐸 𝑖
(C.1)
𝐼2𝑖̇ = 𝜋2 𝑏2 𝐼2𝑖 + 𝛽2 𝐸𝑖 𝑆2 − (𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁 𝑖 ⁄(𝑆2 + 𝜁 𝐼̅ 2 ) + 𝛼2 )𝐼2𝑖
𝐼𝑖
with, 𝜁 ̅ = ∑𝑖 𝐼2 𝜁𝑖 and 𝐼2 = 𝐼2𝑤𝑡 + 𝐼2𝑚 .
2
The reproductive success of a mutant, 𝑅𝑚 , is usually computed similarly to 𝑅0 but with
considering that the wild type population is at its endemic equilibrium. However because 𝑅0 had
a complex expression we modified 𝑅𝑚 computation in order to obtain a tractable expression
which kept the following threshold property: 𝑅𝑚 > 1 yields the invasion by the mutant. The
computation was simplified by moving the term 𝜋2𝑚 𝑏2 from matrix F to matrix V and ensuring
that the eigenvalues of matrix -V are all negative,
0
𝐹 = [𝜆
0
𝑔𝑎𝜁 𝑚 (𝐾1 −𝐼1∗ −𝑅1∗ )𝑁2∗
0
𝑆2∗ +𝜁 𝑤𝑡 𝐼2∗
0
𝛽2 𝑆2∗
𝑏1 + 𝛾
𝑉=[ 0
0
0
𝑑
0
0
0
],
0
0
],
𝑎𝐾1 𝜁 𝑚 𝑁2∗
∗
−𝜋2 𝑏2 + 𝑚2 + 𝑘2 𝑁2 + 𝛼2 + 𝑆∗ +𝜁𝑤𝑡𝐼∗
2
2
𝑎𝐾 𝜁 𝑚 𝑁 ∗
1
2
𝑚
𝑉[3,3] > 0 if 𝜋2 𝑏2 < 𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 + 𝑆 ∗+𝜁
𝑤𝑡 𝐼 ∗ , which is a necessary condition for 𝐼2 > 0 at
2
2
endemic equilibrium (see third equation of system C.1). The stability of the equilibriums was
verified numerically for the values or range of values of the parameters.
8
𝑅𝑚 is the maximum real eigenvalue of the matrix 𝐹. 𝑉 −1, which yields,
𝑅𝑚 =
𝜆𝛽2 𝑆2∗ 𝑔𝑎𝜁 𝑚 (𝐾1 − 𝐼1∗ − 𝑅1∗ )𝑁2∗
3
√
(𝑏1 + 𝛾)𝑑(𝑆2∗ + 𝜁 𝑤𝑡 𝐼2∗ ) (−𝜋2 𝑏2 + 𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 +
𝑎𝜁 𝑚 𝐾1 𝑁2∗
)
𝑆2∗ + 𝜁 𝑤𝑡 𝐼2∗
with 𝑋 ∗ representing the equilibrium value of a variable 𝑋.
For the following computation, we used (𝑅𝑚 )3 which has the same threshold property of 𝑅𝑚 .
We are interested in the derivative of (𝑅𝑚 )3 according to the manipulation coefficient of the
mutant, 𝜁𝑚 :
𝑑(𝑅𝑚 )3
=
𝑑𝜁𝑚
𝜆𝛽2 𝑆2∗ 𝑔𝑎(𝐾1 − 𝐼1∗ − 𝑅1∗ )𝑁2∗ (−𝜋2 𝑏2 + 𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 )
(𝑏1 +
𝛾)𝑑(𝑆2∗
+
𝜁 𝑤𝑡 𝐼2∗ ) (−𝜋2 𝑏2
+ 𝑚2 +
𝑘2 𝑁2∗
𝑎𝜁 𝑚 𝐾1 𝑁2∗ 2
+ 𝛼2 + ∗
)
𝑆2 + 𝜁 𝑤𝑡 𝐼2∗
Equation C.2 cancel out if −𝜋2 𝑏2 + 𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 = 0, i.e., if 𝜋2 =
Equation C.2 is always positive if 𝜋2 <
selected, and conversely when 𝜋2 >
𝑚2 +𝑘2 𝑁2∗ +𝛼2
𝑏2
𝑚2 +𝑘2 𝑁2∗ +𝛼2
𝑏2
.
9
𝑚2 +𝑘2 𝑁2∗ +𝛼2
𝑏2
(C.2)
.
, which means that increasing manipulation is
Appendix D: Invasion analysis for the model that differentiates male and female IHs.
Both the mutant and the wild type parasite strains can manipulate differentially male and female
IHs. We investigate the outcome of a mutant parasite introduced in a wild type population of
parasite at equilibrium. The two strains differ only in their ability to manipulate male and female
IHs. System (D.1) represents the dynamics of the mutant or the resident, depending on whether i
= m or wt, respectively:
𝑖
𝑖 𝑖
̅̅𝐼2𝑀 )
𝐼1𝑖̇ = 𝑔𝑎𝑁2 (𝐾1 − 𝐼1 − 𝑅1 ) (𝜁𝐹𝑖 𝐼2𝐹
+ 𝜁𝑀
𝐼2𝑀 )⁄(𝑆2𝐹 + 𝑆2𝑀 + 𝜁̅𝐹 𝐼2𝐹 + ̅𝜁̅𝑀
−(𝑏1 + 𝛾)𝐼1𝑖
𝐸̇ 𝑖 = 𝜆𝐼1𝑖 − 𝑑𝐸 𝑖
𝑖̇
𝐼2𝐹
(D.1)
=
𝑖
𝜋2 𝑏2 𝐼2𝐹
𝑖
+ 𝛽2 𝐸 𝑆2𝐹
𝑖
−(𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁𝐹𝑖 ⁄(𝑆2𝐹 + 𝑆2𝑀 + 𝜁̅𝐹 𝐼2𝐹 + ̅𝜁̅̅𝑀̅𝐼2𝑀 ) + 𝛼2 )𝐼2𝐹
𝑖̇
𝑖
𝐼2𝑀
= 𝜋2 𝑏2 𝐼2𝐹
+ 𝛽2 𝐸 𝑖 𝑆2𝑀
𝑖 ⁄(𝑆
𝑖
̅
̅̅̅̅
−(𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁𝑀
2𝐹 + 𝑆2𝑀 + 𝜁𝐹 𝐼2𝐹 + 𝜁𝑀 𝐼2𝑀 ) + 𝛼2 )𝐼2𝑀
𝐼𝑖
𝑖
with 𝑆1 = 𝐾1 − 𝐼1 − 𝑅1 , 𝐼2𝑗 = ∑𝑖 𝐼2𝑗
and 𝜁̅𝑗 = ∑𝑖 𝐼 2𝑗 𝜁𝑗𝑖 , ( j= F or M).
2𝑗
The reproductive success of a mutant is computed similarly to Annex C:
0
𝑔𝑎𝑁2∗ (𝐾1 −𝐼1∗ −𝑅1∗ )𝜁𝐹𝑚
∗
∗ +𝜁 𝑤𝑡 𝐼 ∗ +𝜁 𝑤𝑡 𝐼 ∗
𝑆2𝐹 +𝑆2𝑀
𝐹 2𝐹
𝑀 2𝑀
𝑚
𝑔𝑎𝑁2∗ (𝐾1 −𝐼1∗ −𝑅1∗ )𝜁𝑀
∗
∗
𝑤𝑡
∗
𝑤𝑡
∗
𝑆2𝐹 +𝑆2𝑀 +𝜁𝐹 𝐼2𝐹 +𝜁𝑀 𝐼2𝑀
0
∗
𝛽2 𝑆2𝐹
∗
𝛽2 𝑆2𝑀
0
0
0
0
0
0
0
𝐹= 𝜆
0
[0
𝑏1 + 𝛾
0
𝑉=
0
[ 0
0
𝑑
0
0
0
0
𝑉3,3
−𝜋2 𝑏2
0
0
0 ,
𝑉4,4 ]
10
,
]
𝑎𝐾1 𝑁2∗ 𝜁𝐹𝑚
∗
𝑤𝑡 ∗
𝑤𝑡 ∗
2𝐹 +𝑆2𝑀 +𝜁𝐹 𝐼2𝐹 +𝜁𝑀 𝐼2𝑀
with 𝑉3,3 = −𝜋2 𝑏2 + 𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 + 𝑆 ∗
and 𝑉4,4 = 𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 + 𝑆 ∗
,
𝑚
𝑎𝐾1 𝑁2∗ 𝜁𝑀
∗
𝑤𝑡 ∗
𝑤𝑡 ∗
2𝐹 +𝑆2𝑀 +𝜁𝐹 𝐼2𝐹 +𝜁𝑀 𝐼2𝑀
𝑅𝑚 is the maximum real eigenvalue of the matrix 𝐹. 𝑉 −1, which yields,
𝑚 ∗
𝑚
∗
𝜆𝛽2 𝑔𝑎𝑁2∗ (𝐾1 − 𝐼1∗ − 𝑅1∗ )
𝜁𝑀
𝑆2𝑀 𝑆2𝐹
𝜁𝑀
𝜋2 𝑏2
𝑚
𝑅𝑚 = √
(
+
(𝜁
+
))
𝐹
𝑤𝑡 ∗ )
∗
∗
∗
(𝑏1 + 𝛾)𝑑(𝑆2𝐹
𝑉4,4
𝑉3,3
𝑉4,4
+ 𝑆2𝑀
+ 𝜁𝐹𝑤𝑡 𝐼2𝐹
+ 𝜁𝑀
𝐼2𝑀
3
𝑚
Then we study the derivative of (𝑅𝑚 )3 according to 𝜁𝑀
and 𝜁𝐹𝑚 :
-
Manipulation in males:
∗
𝑑(𝑅𝑚 )3
𝜆𝛽2 𝑔𝑎𝑁2∗ (𝐾1 − 𝐼1∗ − 𝑅1∗ )(𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 )
𝜋2 𝑏2 𝑆2𝐹
∗
=
(𝑆
+
)
2𝑀
𝑚
2
𝑤𝑡 ∗ )(𝑉
∗
∗
∗
𝑑𝜁𝑀
𝑉3,3
(𝑏1 + 𝛾)𝑑(𝑆2𝐹
+ 𝑆2𝑀
+ 𝜁𝐹𝑤𝑡 𝐼2𝐹
+ 𝜁𝑀
𝐼2𝑀 4,4 )
(D.2)
All the terms of equation (D.2) are always positive or equal to 0, thus (𝑅𝑚 )3 always increases
𝑚
with 𝜁𝑀
.
-
Manipulation in females:
∗
𝑑(𝑅𝑚 )3
𝜆𝛽2 𝑔𝑎𝑁2∗ (𝐾1 − 𝐼1∗ − 𝑅1∗ )𝑆2𝐹
=
2
𝑤𝑡 ∗ )(𝑉
∗
∗
∗
𝑑𝜁𝐹𝑚
(𝑏1 + 𝛾)𝑑(𝑆2𝐹
+ 𝑆2𝑀
+ 𝜁𝐹𝑤𝑡 𝐼2𝐹
+ 𝜁𝑀
𝐼2𝑀 3,3 )
𝑚
𝑎𝐾1 𝑁2∗ 𝜁𝑀
∗
× (𝜋2 𝑏2 (1 + ∗
𝑤𝑡 ∗ )𝑉 ) − (𝑚2 + 𝑘2 𝑁2 + 𝛼2 ))
∗
∗
(𝑆2𝐹 + 𝑆2𝑀
+ 𝜁𝐹𝑤𝑡 𝐼2𝐹
+ 𝜁𝑀
𝐼2𝑀 4,4
Equation (D.3) cancels out when
(𝑚2 + 𝑘2 𝑁2∗ + 𝛼2 )
𝜋2 =
𝑏2 (1 +
𝑚
𝑎𝐾1 𝑁2∗ 𝜁𝑀
𝑤𝑡 ∗ )𝑉 )
∗
∗
∗
(𝑆2𝐹
+ 𝑆2𝑀
+ 𝜁𝐹𝑤𝑡 𝐼2𝐹
+ 𝜁𝑀
𝐼2𝑀 4,4
11
(D.3)
When 𝜋2 <
(𝑚2 +𝑘2 𝑁2∗ +𝛼2 )
, (D.3) is always positive and (𝑅𝑚 )3 always
∗ 𝜁𝑚
𝑎𝐾1 𝑁2
𝑀
𝑏2 (1+ ∗
)
𝑤𝑡
∗
∗
∗
(𝑆2𝐹 +𝑆2𝑀 +𝜁𝐹 𝐼2𝐹 +𝜁𝑤𝑡
𝑀 𝐼2𝑀 )𝑉4,4
increases with 𝜁𝐹𝑚 . When 𝜋2 >
(𝑚2 +𝑘2 𝑁2∗ +𝛼2 )
𝑎𝐾1 𝑁∗ 𝜁𝑚
, (D.3) is always negative and
2 𝑀
𝑏2 (1+ ∗
)
𝑤𝑡 ∗
∗
(𝑆2𝐹 +𝑆∗2𝑀 +𝜁𝑤𝑡
𝐹 𝐼2𝐹 +𝜁𝑀 𝐼2𝑀 )𝑉4,4
(𝑅𝑚 )3 always decreases with 𝜁𝐹𝑚 .
12
Appendix E: Invasion analysis for the model that differentiates male and female IHs considering
sexual transmission from male to female IHs.
Both the mutant and the wild type parasite strains can manipulate differentially male and female
IHs. We investigate the outcome of a mutant parasite introduced in a wild type population of
parasite at equilibrium. The infection term writes 𝛽3 𝑆
𝑐3 𝐼2𝑀
2𝑀 +𝑐3 𝐼2𝑀
𝑆2𝐹 , with 𝛽3being the rate of
copulation (we assumed 5 events per year) and 𝑐3 is a coefficient accounting for the higher
success of mating for infected males than susceptible ones (1.5, derived from Dass et al. 2011).
The two strains differ only in their ability to manipulate male and female IHs, respectively 𝜁𝑀
and 𝜁𝐹 . System (E) represents the dynamics of the mutant or the resident, depending on whether i
= m or wt, respectively:
𝑖
𝑖 𝑖
̅̅𝐼2𝑀 )
𝐼1𝑖̇ = 𝑔𝑎𝑁2 (𝐾1 − 𝐼1 − 𝑅1 ) (𝜁𝐹𝑖 𝐼2𝐹
+ 𝜁𝑀
𝐼2𝑀 )⁄(𝑆2𝐹 + 𝑆2𝑀 + 𝜁̅𝐹 𝐼2𝐹 + ̅𝜁̅𝑀
−(𝑏1 − 𝛾)𝐼1𝑖
𝐸̇ 𝑖 = 𝜆𝐼1𝑖 − 𝑑𝐸 𝑖
𝑖̇
𝑖
𝐼2𝐹
= 𝜋2 𝑏2 𝐼2𝐹
+ 𝛽2 𝐸 𝑖 𝑆2𝐹 +𝛽3 𝑆
𝑐3 𝐼2𝑀
2𝑀 +𝑐3 𝐼2𝑀
(E.1)
𝑆2𝐹
𝑖
−(𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁𝐹𝑖 ⁄(𝑆2𝐹 + 𝑆2𝑀 + 𝜁̅𝐹 𝐼2𝐹 + ̅𝜁̅̅𝑀̅𝐼2𝑀 ) + 𝛼2 )𝐼2𝐹
𝑖̇
𝑖
𝐼2𝑀
= 𝜋2 𝑏2 𝐼2𝐹
+ 𝛽2 𝐸 𝑖 𝑆2𝑀
𝑖 ⁄(𝑆
𝑖
̅
̅̅̅̅
−(𝑚2 + 𝑘2 𝑁2 + 𝑎𝑁2 𝐾1 𝜁𝑀
2𝐹 + 𝑆2𝑀 + 𝜁𝐹 𝐼2𝐹 + 𝜁𝑀 𝐼2𝑀 ) + 𝛼2 )𝐼2𝑀
𝐼𝑖
𝑖
with 𝑆1 = 𝐾1 − 𝐼1 − 𝑅1 , 𝐼2𝑗 = ∑𝑖 𝐼2𝑗
and 𝜁̅𝑗 = ∑𝑖 𝐼 2𝑗 𝜁𝑗𝑖 , ( j= F or M).
2𝑗
The reproductive success of a mutant is computed similarly to Annex C:
13
0
𝜆
𝐹= 0
0
0
∗
𝛽2 𝑆2𝐹
𝐴𝑝𝑟𝑒𝑑 𝜁𝐹𝑚
0
0
[0
∗
𝛽2 𝑆2𝑀
0
𝑏1 + 𝛾
0
𝑉=
0
[ 0
0
𝑑
0
0
With 𝐴𝑝𝑟𝑒𝑑 = 𝑆∗
𝑚
𝐴𝑝𝑟𝑒𝑑 𝜁𝑀
0
,
𝛽3 𝑐3 𝑆2𝐹
∗ +𝑐 𝐼∗
𝑆2𝑀
3 2𝑀
0
]
0
0
−𝜋2 𝑏2 + 𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝐹𝑚 + 𝛼2
−𝜋2 𝑏2
𝑔𝑎𝑁2∗ (𝐾1 −𝐼1∗ −𝑅1∗ )
∗
𝑤𝑡 ∗
𝑤𝑡 ∗
2𝐹 +𝑆2𝑀 +𝜁𝐹 𝐼2𝐹 +𝜁𝑀 𝐼2𝑀
0
0
0
,
𝑚
𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝑀
+ 𝛼2 ]
𝑎𝑁2∗ 𝐾1
∗
𝑤𝑡 ∗
𝑤𝑡 ∗
2𝐹 +𝑆2𝑀 +𝜁𝐹 𝐼2𝐹 +𝜁𝑀 𝐼2𝑀
, 𝑀2 = 𝑚2 + 𝑘2 𝑁2∗ and 𝑀𝑝𝑟𝑒𝑑 = 𝑆∗
.
𝑅𝑚 is the maximum real eigenvalue of the matrix 𝐹. 𝑉 −1, and thus the maximal real solution of
the following polynomial,
4
𝜌 −𝜌
−𝜌
−
𝑟
𝑏2 𝛽3 𝑐3 𝜋2 𝑆2𝐹
3
𝑚
𝑟
𝑟 )(−
(𝑐3 𝐼2𝑀
+ 𝑆2𝑀
𝜋2 𝑏2 + 𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝐹𝑚 + 𝛼2 )( 𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝑀
+ 𝛼2 )
𝑟 𝑚
𝑟
𝑟 )
𝑟
𝑟
𝑟 )𝜁 𝑚 𝑚
𝐴𝑝𝑟𝑒𝑑 𝛽2 𝜆 ((𝛼2 + 𝑀2 )𝑆2𝐹
𝜁𝐹 + (𝑏2 𝜋2 (𝑆2𝐹
− 𝑆2𝑀
+ (𝛼2 + 𝑀2 )𝑆2𝑀
+ 𝑀𝑝𝑟𝑒𝑑 (𝑆2𝐹
+ 𝑆2𝑀
𝐹 )𝜁𝑀 )
𝑚
𝑑(𝑏1 + 𝛾)(− 𝜋2 𝑏2 + 𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝐹𝑚 + 𝛼2 )(𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝑀
+ 𝛼2 )
𝑟 𝑟
𝐴𝑝𝑟𝑒𝑑 𝛽2 𝛽3 𝑐3 𝜆𝑆2𝐹
𝑆2𝑀 𝜁𝐹𝑚
𝑚
𝑟
𝑟 )(−
𝑑(𝑏1 + 𝛾)(𝑐3 𝐼2𝑀
+ 𝑆2𝑀
𝜋2 𝑏2 + 𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝐹𝑚 + 𝛼2 )( 𝑀2 + 𝑀𝑝𝑟𝑒𝑑 𝜁𝑀
+ 𝛼2 )
The threshold of proportion of vertical transmission represent situations where 𝑅𝑚 = 1, thus
replacing 𝑅𝑚 by unity in the polynomial allows us to work with a tractable expression. The
derivation of this expression with respect to the manipulative coefficients for females and males
yields the following thresholds of vertical transmission for females,
𝜋2𝐹 =
𝑚
∗
∗
∗
∗
∗ )+𝑀
∗
∗
𝛽2 𝜆(𝛼2 +𝑀2 )𝐴𝑝𝑟𝑒𝑑 (𝑐3 𝐼2𝑀
𝑀2 +𝛽3 𝑐3 𝑆2𝑀
+𝑀2 𝑆2𝑀
+𝛼2 (𝑐3 𝐼2𝑀
+𝑆2𝑀
𝑝𝑟𝑒𝑑 (𝑐3 𝐼2𝑀 +𝑆2𝑀 )𝜁𝑀 )
∗ +(𝑐 𝐼∗ +𝑆 ∗ )(𝛼 +𝑀 +2 𝑀
𝑚
𝑏2 ((𝑏1 +𝛾)𝛽3 𝑐3 𝑑𝑀𝑝𝑟𝑒𝑑 +𝛽2 𝜆𝐴𝑝𝑟𝑒𝑑 (𝛽3 𝑐3 𝑆2𝑀
3 2𝑀
2
2
𝑝𝑟𝑒𝑑 𝜁𝑀 )))
2𝑀
and for males,
14
,
(E.2)
𝜋2𝑀 =
𝑚
∗ ((𝛼
∗
∗
∗
∗
∗
2
𝛽2 𝜆𝐴𝑝𝑟𝑒𝑑 𝑆2𝑀
2 +𝑀2 ) (𝑐3 𝐼2𝑀 +𝑆2𝑀 )+𝑀𝑝𝑟𝑒𝑑 (𝑐3 𝐼2𝑀 (𝛼2 +𝑀2 )−𝛽3 𝑐3 𝑆2𝐹 +(𝛼2 +𝑀2 )𝑆2𝑀 )𝜁𝐹 )
∗ −𝛽 𝜆(𝛼 +𝑀 )𝐴
∗
∗
∗
∗
𝑏2 ((𝑏1 +𝛾)𝛽3 𝑐3 𝑑𝑀𝑝𝑟𝑒𝑑 𝑆2𝐹
2
2
2 𝑝𝑟𝑒𝑑 (𝑆2𝐹 −𝑆2𝑀 ) (𝑐3 𝐼2𝑀 +𝑆2𝑀 ))
.
(E.3)
Below these thresholds, manipulation is selected in females and males respectively and
conversely above.
The following figure represents these thresholds for varying mating rates.
Figure E.1: Threshold values of vertical transmission and parasite virulence allowing the
evolution of parasite manipulation in the presence of sexual transmission between males and
females for 3 different transmission rates through mating β3, (a) β3 = 3/52; (b) β3 = 5/52 (same as
figure 4b in the main text); (c) β3 = 7/52. Shaded and hatched areas represent values of vertical
transmission and virulence rates favouring manipulation in males and females respectively. Note
𝑤𝑡
that the curves are obtained assuming no manipulation of the IH behaviour, i.e. 𝜁𝑀
= 𝜁𝐹𝑤𝑡 =
𝑚
𝜁𝑀
= 𝜁𝐹𝑚 = 1.
References
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