Mathematical methods

We describe the mathematical models used in analyzing the mating-cross data andextrapolating from these data to the population genetics of Wolbachia in a populationof tsetse. A mathematical model have been developed and well studied for Wolbachia-induced cytoplasmic incompatibility in Drosophila [4]. This model assumes non-overlappinggenerations, which is particularly at odds with tsetse life history [5]. To overcome this, weconstructed a continuous-time model for the time evolution of the number of Wolbachia-infected and uninfected tsetse in a population using differential equations. We first discussthis new population-genetic model and its analysis, showing that it has the same thresholdphenomenon as the earlier discrete-time model. Then we describe the Bayesian Markovchain Monte Carlo method used to estimate model parameters and derived quantitiesfrom the data.

1 Wolbachia Population Genetics

Let Np(t) and Nn(t) be the numbers of Wolbachia-infected and Wolbachia-uninfectedtsetse, respectively, at time t and N(t) = Np(t) +Nn(t). Then we have

dNn

dt= rf

!Nn

N+ (1! sh)

Np

N

"Nn (1a)

+ (1 + sr)r(1 + sf )fµ

!Nn

N+ (1! sh)

Np

N

"Np ! dNn,

dNp

dt= (1 + sr)r(1 + sf )f(1! µ)

#Nn

N+

Np

N

$Np (1b)

! d(1 + sd)Np,

where r is the offspring-production rate (that is, frequency of gonotrophic cycles) of Wol-bachia-uninfected tsetse, f is the probability of an Wolbachia-uninfected tsetse success-fully producing an offspring during a given gonotrophic cycle (i.e. the probability of a fer-tilized egg maturing to a larva), d is the death rate of Wolbachia-uninfected tsetse, and µis the proportion of Wolbachia-uninfected eggs of Wolbachia-infected mothers. For Wol-bachia-infected tsetse, 1 + sr is the relative offspring-production rate, 1 + sf is the relativereproduction success, sh is the proportion of fertilization of Wolbachia-uninfected eggs by

1

Number of Offspring

!" ‚ }"

µf(1 + sf )(1! sh)(1! µ)f(1 + sf )

!" | }"

µf(1 + sf )(1! µ)f(1 + sf )

~" | } f~" ‚ } f(1! sh)

Table 1: Matings and offspring per offspring cycle. Open circles are Wolbachia uninfectedand closed are Wolbachia infected. For the offspring, the expression is the relative numberof offspring, given that such a mating occurs.

Wolbachia-infected sperm that are not viable (sh = 1 means no such fertilizations produceoffspring), and 1 + sd is the relative death rate (Table 1). In equation (1a), the first term isbirths of Wolbachia-uninfected offspring from Wolbachia-infected mothers, while the sec-ond term is births of Wolbachia-uninfected offspring from Wolbachia-uninfected mothers.Wolbachia-uninfected eggs can be fertilized by Wolbachia-uninfected males (proportionNn/N of the population) successfully or by Wolbachia-infected males (proportion Np/Nof the population) with probability of success 1 ! sh. The final term in equation (1a) rep-resents death. For equation (1b), the first term is births of Wolbachia-infected offspring,which requires that the mother be Wolbachia-infected (Np), that she produces eggs at rate(1 + sr)r, those eggs mature to larva with probability (1 + sf )f , that proportion (1! µ) ofthose eggs are Wolbachia infected, and that she mates with either a Wolbachia-infectedmale (proportion Np/N of the population) or a Wolbachia-uninfected male (proportionNn/N of the population). The second term in equation (1b) represents death.

Using N = Nn +Np simplifies the model to

dNn

dt= rf

#1! sh

Np

N

$[Nn + (1 + sr)(1 + sf )µNp]! dNn,

dNp

dt= (1 + sr)r(1 + sf )f(1! µ)Np ! d(1 + sd)Np.

(2)

Letting p(t) = Np(t)/N(t) gives a single differential equation for the time evolution of theproportion of the population infected with Wolbachia

dp

dt= ! ({rf [1! (1 + sr)(1 + sf )(1! µ)] + sdd}

! {rf [1! (1 + sr)(1 + sf ) + sh] + sdd} p+ rfsh [1! µ(1 + sr)(1 + sf )] p

2%p.

(3)

2

1.1 Model Analysis

We will show that model (3) can have a threshold, above which Wolbachia prevalenceincreases in time and below which prevalence decreases in time, however this thresholdis not present for all parameter values, exactly as for the model with non-overlappinggenerations [4]. Importantly, the presence of such a threshold indicates that there is someminimum effort needed for successful population transformation using Wolbachia as adriver [6, 2, 7].

Letting ! = rft and " = sdd/r gives the non-dimensional model

dp

d!= ! {[1! (1 + sr)(1 + sf )(1! µ) + "]

! [1! (1 + sr)(1 + sf ) + sh + "] p

+ sh [1! µ(1 + sr)(1 + sf )] p2&p,

(4)

with the basic biological conditions on the parameters

r > 0, d > 0, 0 < f # 1, 0 # µ < 1,

0 < sh # 1, sr > !1, sd > !1, !1 < sf <1! f

f.

(5)

From model (4),

dp

d!

''''p=0

= 0 anddp

d!

''''p=1

= !(1 + sr)(1 + sf )µ(1! sh) # 0, (6)

show that the interval 0 # p # 1 is forward invariant, meaning that if initially 0 # p(0) # 1,then 0 # p(t) # 1 for all t $ 0. This, in addition to fact that the model is a cubic polynomial,greatly restricts the possible phase space. Obviously, p0 = 0 is an equilibrium and, whenµ = 0 (perfect Wolbachia transmission) or sh = 1 (perfect CI), p = 1 is also an equilibrium.We consider only the general case of imperfect transmission (µ > 0) and imperfect CI(sh < 1) in detail, ignoring the special cases with perfect CI or perfect transmission. Briefly,in these special cases, for some parameter values, p = 1 is an unstable equilibrium, sothat Wolbachia can persist in the population at 100% prevalence until the introduction ofany Wolbachia-negative tsetse, at which point the Wolbachia prevalence tends to somelower value, perhaps even p0 = 0. In these case, at p = 1, Wolbachia-negative tsetse havehigher fitness than Wolbachia-positive tsetse, but are not present in the population: if theyare introduced, they replace at least a portion of the Wolbachia-positive population. Thisphenomenon of an unstable equilibrium with high Wolbachia prevalence does not occurin the general case because with imperfect transmission and imperfect CI, some numberof Wolbachia-negative are always present, being produced as offspring of Wolbachia-positive mothers.

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In general, there is a state with no Wolbachia in the population(p0); a state whereWolbachia has reached fixation in the population (pF ); and there may be a thresholdstate (pT ), above which Wolbachia goes to fixation and below which Wolbachia goes toextinction. There are three possibilities for the general case of imperfect transmission andimperfect CI:

1. Wolbachia can never persist in the population. Wolbachia-positive tsetse have lowerfitness at all levels of prevalence. (There is one equilibrium, pF = p0 = 0, and it isstable.)

2. The presence of any amount of Wolbachia leads to fixation in the population (i.e. thereis no release threshold). Wolbachia-positive tsetse have higher fitness at all levelsof prevalence. The fixation prevalence is less than 100% due to some failure oftransmission. (There are two equilibria, pT = p0 = 0 and pF , with 0 = p0 < pF < 1,p0 unstable, and pF stable.)

3. There is a release threshold (pT ) so that if Wolbachia prevalence is above the thresh-old, it will go to fixation, and if Wolbachia prevalence is below the threshold, it will bedriven out of the population. Wolbachia-positive tsetse have lower fitness at levels ofprevalence below the threshold and higher fitness at levels of prevalence above thethreshold. The fixation prevalence is less than 100% due to some failure of trans-mission. (There are three equilibria, p0 = 0, pT , and pF , with 0 = p0 < pT < pF < 1,p0 stable, pT unstable, and pF stable.)

1.2 Time to fixation

A quantity of interest is the amount of time it takes for Wolbachia to go from an initialintroduction in a small number of tsetse to fixation in the tsetse population. To get amore accurate estimate of this time in the presence of the unique reproductive biology oftsetse, we extended model (2) to separate the female population into females before thedeposition of their first pupal offspring and females after this deposition because the timefrom a female being deposited as a pupa to the deposition of her first pupal offspring issignificantly longer than the time between her subsequent pupal depositions.

Let Np0(t) be the number of Wolbachia-infected female tsetse before the deposition oftheir first pupal offspring at time t, and Np1(t) be the number of Wolbachia-infected femaletsetse after the deposition of their first pupal offspring at time t. Let Nn0(t) and Nn1(t)be similarly defined for Wolbachia-uninfected female tsetse. For Wolbachia-uninfected fe-males, let r0 be the production rate for the first offspring and r1 be the production rate forsubsequent offspring (i.e. 1/r0 and 1/r1 are the mean times to first offspring and to subse-quent offspring, the duration of the gonotrophic cycle). For Wolbachia-infected females, let(1+sr0)r0 be the production rate for the first offspring and (1+sr1)r1 be the production rate

4

for subsequent offspring females, so that sr0 and sr1 are the relative offspring-productionbenefits for Wolbachia infection. Then we have

dNn0

dt= f

#1! sh

Np

N

${r0Nn0 + r1Nn1 + (1 + sf )µ [(1 + sr0)r0Np0 + (1 + sr1)r1Np1]}

! r0Nn0 ! dNn0,

dNn1

dt= r0Nn0 ! dNn1,

dNp0

dt= (1 + sf )f(1! µ) [(1 + sr0)r0Np0 + (1 + sr1)r1Np1]

! (1 + sr0)r0Np0 ! d(1 + sd)Np0,

dNp1

dt= (1 + sr0)r0Np0 ! d(1 + sd)Np1,

(7)

with Nn = Nn0 + Nn1, Np = Np0 + Np1, and N = Nn + Np. The !r0Nn0 term in the firstequation and the r0Nn0 term in the second equation are the transition of a Wolbachia-uninfected mother’s status at her first deposition; likewise the terms %(1+ sr0)r0Np0 in thethird and fourth equations are the same transition for Wolbachia-infected mothers. Notethat in the event of the failure to produce a first offspring, the female still moves into thenext stage class.

In a population free of Wolbachia (Np0 = Np1 = 0), the model becomes linear, givingthe population growth rate

# =

(r20(1! f)2 + 4r0r1f ! r0(1! f)! 2d

2(8)

and the stable stage distribution

Nn0

N= N!

n0 =2fr1

2fr1 + (1! f)r0 +(r20(1! f)2 + 4r0r1f

,

Nn1

N= N!

n1 =(1! f)r0 +

(r20(1! f)2 + 4r0r1f

2fr1 + (1! f)r0 +(r20(1! f)2 + 4r0r1f

.

(9)

(See e.g. Caswell [1].)To facilitate numerical solution, this model was converted from numbers of tsetse in

each class to the proportions

nn0 =Nn0

N, nn1 =

Nn1

N, np0 =

Np0

N, np1 =

Np1

N. (10)

5

The resulting model equations are

dnn0

dt= f(1! shnp) {r0nn0 + r1nn1 + (1 + sf )µ [(1 + sr0)r0np0 + (1 + sr1)r1np1]}

! r0nn0 ! dnn0 ! nn01

N

dN

dt,

dnn1

dt= r0nn0 ! dnn1 ! nn1

1

N

dN

dt,

dnp0

dt= (1 + sf )f(1! µ) [(1 + sr0)r0np0 + (1 + sr1)r1np1]

! (1 + sr0)r0np0 ! d(1 + sd)np0 ! np01

N

dN

dt,

dnp1

dt= (1 + sr0)r0np0 ! d(1 + sd)np1 ! np1

1

N

dN

dt,

(11)

where

1

N

dN

dt= f {(1! shnp)(r0nn0 + r1nn1)

+ (1 + sf )(1! µshnp) [(1 + sr0)r0np0 + (1 + sr1)r1np1]}! d(1 + sdnp).

(12)

The death rate, d, was chosen so that a Wolbachia-free population has a stable size(i.e. the growth rate # = 0) and the relative mortality cost of Wolbachia infection wasassumed to be sd = 0. For Wolbachia-uninfected tsetse, we assumed a mean of 60days between the deposition of a female pupa and that female depositing her first pupa(r0 = 1/60 d"1) and then 10-day intervals between her subsequent pupal depositions(r1 = 1/10 d"1). For Wolbachia-infected tsetse, we assumed 50 days to first deposition(so (1 + sr0)r0 = 1/50 d"1 =& sr0 = 0.2) and 10 days between subsequent depositions(so (1 + sr1)r1 = 1/10 d"1 =& sr1 = 0). Sensitivity analysis was performed on therelative costs sr0 , sr1 , and sd (Figures 5–7).

2 Bayesian Markov Chain Monte Carlo Parameter Estimation

Following Gelman et al. [3], the Bayesian Markov chain Monte Carlo (MCMC) method wasimplemented using the Metropolis–Hastings algorithm, with a multi-dimensional normaldistribution for the jumps. Maximum-likelihood estimates of the parameters were foundfirst, using standard numerical minimization and the covariance approximated from theHessian matrix. From these maximum-likelihood estimates, importance resampling wasdone to get 10 sets of 1000 samples each to start the MCMC routine. The MCMC was runwith 10 parallel sequences to derive the stopping criterion based on the variances withinand between sequences.

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From the mating-cross data, for each cross let N be the number of females in theexperiment and let P be the number of pupae deposited, with subscript fm denoting thefemale and male’s type, respectively. The three types of tsetse are wild type (GmmWt

in the main text), denoted here by W , harboring Wolbachia, Wigglesworthia, & Sodalis;ampicillin treated (GmmWig-), denoted by A, cleared of Wigglesworthia, but harboring Wol-bachia & Sodalis; and tetracycline treated (GmmApo), denoted by T , cleared of Wolbachia,Wigglesworthia, & Sodalis.

Assuming that each number of offspring is a binomial random variable gives the simpleprobability models for the data

PWW ' B (NWW , qWW ) ,

PWT ' B (NWT , qWT ) ,

PTW ' B (NTW , qTW ) ,

PTT ' B (NTT , qTT ) ,

PAA ' B (NAA, qAA) ,

(13)

where B(N, p) is the standard binomial random variable for N Bernoulli trials with individ-ual success probability q. Then the likelihood function for each of the models is simplythat of the binomial distribution,

L(f, sf , sh, µ|P ) =

#N

P

$qP (1! q)N"P , (14)

with the appropriate N , P , and q. The probabilities of pupal deposition for each matingcross are

qWW = fW (1! µsh),

qWT = fW ,

qTW = fT (1! sh),

qTT = fT ,

qAA = fA(1! µsh).

(15)

The reproduction successes are

fA = fT (1 + sf,Wol),

fW = fT (1 + sf,Wol)(1 + sf,Wig),(16)

due to the relative reproduction-success benefits of the mother harboring Wolbachia (sf,Wol)and Wigglesworthia (sf,Wig). (Because ampicillin only cleared Wigglesworthia and tetra-cycline cleared Wolbachia, Wigglesworthia, & Sodalis, we were unable to estimate theimpact of each bacteria separately. Here we are assuming that the difference between

7

ampicillin-treated and tetracycline-treated tsetse is entirely due to the absence of Wol-bachia, and not to the absence Sodalis.)

Each of the parameters fW , fT , fA, sh, and µ are a proportion, so we use uninformativepriors of uniform distributions on (0, 1) for the parameters. The parameters were thentransformed using the logit function

logit(x) = log(x)! log(1! x), (17)

transforming them from the interval (0, 1) to (!(,+().In addition to estimating the parameters, quantities derived from those parameters

were also estimated for each sample set of parameters.

• Reproduction-success benefits were calculated as

sf,Wol = fA/fT ! 1 and sf,Wig = fW /fA ! 1. (18)

• Fixation prevalence (pF ) was calculated as an equilibrium of population-geneticsmodel (11). For each sample set of parameters, a nonlinear root finder was used tofind the model equilibrium for Wolbachia fixation. The fixation prevalence was thencalculated as pF = (Np0 +Np1)/N . For this comparison, Wolbachia-negative tsetsewere assumed to have reproduction success

f = fT (1 + sf,Wig) (19)

(i.e. the reproduction success with Wigglesworthia & Sodalis), while Wolbachia-positive tsetse had reproduction success

fW = f(1 + sf,Wol) (20)

(with Wolbachia, Wigglesworthia, & Sodalis).

• The release threshold (qT ), defined as the number of newly-emerged Wolbachia-positive tsetse (Np0) that must be released to achieve eventual fixation of Wolbachia.This release threshold is relative to the size of the population, so that, for example,qT = 0.01 indicates that release of the size of 1% of the population.

For a given value of q, the initial condition

nn0(0) = N!n0, nn1(0) = N!

n1, np0(0) = q, np1(0) = 0. (21)

was used, where N!n0 and N!

n1 are the stable stage distribution for a Wolbachia-freepopulation given in (9). The population-genetics model was solved numerically for10 years. At the end of 10 years, for large values of q the Wolbachia is near fixation,while for small values of q the Wolbachia is near extinction. A bisection algorithmwas used to find the critical value of q.

8

Figure 1: Time to fixation (! ) vs. release size (q). The red box and bars show the medianand 95% credible interval for the baseline parameter values, while the blue circles andbars show the median 95% credible interval as the parameter is varied.

Figure 2: Time to fixation (! ) vs. relative fixation threshold (T ). Note that here the x-axis is1 ! T . The red box and bars show the median and 95% credible interval for the baselineparameter values, while the blue circles and bars show the median 95% credible intervalas the parameter is varied.

• The time to fixation (! ) was calculated by numerically solving the population-geneticsmodel from a small initial release of Wolbachia-infected tsetse (q = 0.1) until thepopulation reached 95% (i.e. T = 0.95) of its fixation prevalence, pF . Figures 1 and2 show the sensitivity of time to fixation to changes in the initial-release size q andrelative fixation threshold T used. These were generated by calculating the time tofixation for all of the sample parameter sets generated by the MCMC routine overranges of q and T .

9

2.1 Sensitivity Analysis

Sensitivity analysis of fixation prevalence (pF ), release threshold (qT ), and time to fixation(! ) to various model parameters were also performed (Figures 3–12). Only one parameterwas varied at a time from the baseline parameter values of q = 0.1 for release size;T = 0.95 for the relative fixation threshold; r0 = 1/60 days"1 for the rate of first deposition;r1 = 1/10 days"1 for the rate of subsequent depositions; sr0 = 0.2 for the relative costto rate of first deposition; sr1 = 0 for the relative cost to rate of subsequent depositions;sd = 0 for the relative mortality cost; and sh, µ, f , sf,Wol, and sf,Wig from the MCMCsamples.

• Rate of first deposition (r0), rate of subsequent depositions (r1), relative benefit torate of first deposition (sr0), relative benefit to rate of subsequent depositions (sr1),and relative mortality cost (sd) were not estimated from the experimental data. Forranges of each of these parameters calculated, the model outputs were calculatedusing all of the sample parameter sets from the MCMC routine (Figures 3–7).

• Proportion incompatible crosses that fail (sh), transmission failure (µ), and reproduc-tion success of wild-type tsetse (fW ) were estimated from the data. To generatesensitivity to these parameters, the above model outputs were calculated with fixedvalues of one of these parameters and using the MCMC sample parameter sets forthe other estimated parameters (Figures 8–10).

• The relative benefits to reproduction success for Wolbachia (sf,Wol) and for Wig-glesworthia (sf,Wig) were estimated indirectly from the data by equations (18), wherefW , fT , and fA were directly estimated from data. To estimate the sensitivity,sf,Wol values were chosen in a range and for each of those values, fT was set tofA/(1 + sf,Wol) and the MCMC parameter sets were used for the other parameters.Likewise, sf,Wig values were chosen in a range and for each of those values, fAwas set to fW /(1 + sf,Wig) and the MCMC parameter sets were used for the otherparameters (Figures 11 & 12).

The model shows strong sensitivity to transmission failure (µ) and relative reproductive-success benefit to Wolbachia infection (sf,Wol), with some MCMC samples showing Wol-bachia is unsustainable altogether for µ $ 0.3 or sf,Wol # !0.1 (Figures 9 & 11). Themodel shows weaker sensitivity to large negative values of relative benefit to rate of sub-sequent depositions (sr1), large values of relative cost to mortality (sd), large values ofproportion of incompatible crosses that fail (sh), and large values of relative reproductive-success benefit to Wigglesworthia infection (Figures 6–8 & 12). There was weak sensi-tivity to rate of first deposition (r0), rate of subsequent depositions (r1), relative benefit torate of first deposition (sr0), and wild-type reproduction success (fW ) (Figures 3–5 & 10).

Local sensitivity and elasticity analysis of fixation prevalence and time to fixation atthe baseline parameter values was also performed by varying the non-zero parameters

10

Figure 3: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. rateof first deposition (r0). The red box and bars show the median and 95% credible intervalfor the baseline parameter values, while the blue circles and bars show the median 95%credible interval as the parameter is varied.

11

Figure 4: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. rateof subsequent depositions (r1). The red box and bars show the median and 95% credibleinterval for the baseline parameter values, while the blue circles and bars show the median95% credible interval as the parameter is varied.

12

Figure 5: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. rel-ative benefit to rate of first deposition of Wolbachia infection (sr0). The red box and barsshow the median and 95% credible interval for the baseline parameter values, while theblue circles and bars show the median 95% credible interval as the parameter is varied.

13

Figure 6: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. rel-ative benefit to rate of subsequent depositions of Wolbachia infection (sr1). The red boxand bars show the median and 95% credible interval for the baseline parameter values,while the blue circles and bars show the median 95% credible interval as the parameter isvaried.

14

Figure 7: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. rela-tive mortality cost of Wolbachia infection (sd). The red box and bars show the median and95% credible interval for the baseline parameter values, while the blue circles and barsshow the median 95% credible interval as the parameter is varied.

15

Figure 8: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. pro-portion of crosses between a Wolbachia-positive sperm and Wolbachia-negative eggswhere the egg fails to fertilize (sh). The red box and bars show the median and 95%credible interval for the baseline parameter values, with the horizontal bars showing thecredible interval of the parameter from the MCMC estimation and the vertical bars showingthe credible interval for time to fixation, while the blue circles and bars show the median95% credible interval as the parameter is varied.

16

Figure 9: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. prob-ability of failure for Wolbachia transmission from mother to offspring (µ). The red box andbars show the median and 95% credible interval for the baseline parameter values, withthe horizontal bars showing the credible interval of the parameter from the MCMC esti-mation and the vertical bars showing the credible interval for fixation prevalence, releasethreshold, or time to fixation, while the blue circles and bars show the median 95% credibleinterval as the parameter is varied.

17

Figure 10: Fixation prevalence (pF ), release threshold (qT ) and time to fixation (! ) vs. re-productive success for wild-type tsetse (fW ). The red box and bars show the median and95% credible interval for the baseline parameter values, with the horizontal bars showingthe credible interval of the parameter from the MCMC estimation and the vertical barsshowing the credible interval for release threshold or time to fixation, while the blue circlesand bars show the median 95% credible interval as the parameter is varied.

18

Figure 11: Fixation prevalence (pF ), release threshold (qT ), and time to fixation (! ) vs. rel-ative fecundity benefit of Wolbachia infection (sf,Wol). The red box and bars show themedian and 95% credible interval for the baseline parameter values, with the horizontalbars showing the credible interval of the parameter from the MCMC estimation and thevertical bars showing the credible interval for fixation prevalence, release threshold, ortime to fixation, while the blue circles and bars show the median 95% credible interval asthe parameter is varied.

19

Figure 12: Fixation prevalence (pF ), release threshold (qT ), and time to fixation (! ) vs. rel-ative fecundity benefit of Wigglesworthia infection (sf,Wig). The red box and bars show themedian and 95% credible interval for the baseline parameter values, with the horizontalbars showing the credible interval of the parameter from the MCMC estimation and thevertical bars showing the credible interval for fixation prevalence, release threshold, ortime to fixation, while the blue circles and bars show the median 95% credible interval asthe parameter is varied.

20

Figure 13: Sensitivity and elasticity of fixation prevalence.

by a relative increment of 1% of their baseline values, while the parameters with zeroas the baseline value were varied by the absolute increment of 0.01. Each parameterwas both increased and decreased by its increment and the median fixation prevalenceand median time to fixation were then calculated over the sample parameter set fromthe MCMC estimation. Fixation prevalence showed strongest sensitivity and elasticityto proportion of incompatible crosses that fail (sh) and to transmission failure (µ) andrelatively little sensitivity and elasticity to the other parameters (Figure 13). Time to fixationshowed substantial sensitivity and elasticity to a broader range of parameters (Figure 14).

21

Figure 14: Sensitivity and elasticity of time to fixation.

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[3] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis. Chap-man & Hall, New York, 1995. ISBN 0-412-03991-5.

[4] A. Hoffmann, M. Turelli, and L. Harshman. Factors affecting the distribution of cyto-plasmic incompatibility in Drosophila simulans. Genetics, 126(4):933–948, 1990.

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