### The likelihood for the proportion of pregnant females by month

The females caught for sterilisation on a given day constitute a random sample of the females roaming the area designated for catching on that day. The number of pregnant females the sample is likely to contain depends on the proportion of recruited females that become pregnant in any given year, the timing of the breeding season in relation to the timing of catching and the spread of the breeding season over time. For example, the number is likely to be increased shortly before the peak of the breeding season. In this case, it will be further increased if the breeding season is of short duration and reduced if the season is more spread out. Thus the observed monthly variation in the proportion of females found to be pregnant (based on inspection of over 25,000 females caught for spaying from 1995 to 2006) provides information on fecundity. To estimate the proportion of females becoming pregnant in any given year we chose that proportion which, in conjunction with appropriate values for the timing and spread of the breeding season, mimics most closely the observed variation.

If *n*
_{
i
}females are collected in the *i*
^{th} sample the number found to be pregnant, *k*
_{
i
}, has a binomial distribution

\begin{array}{c}{n}_{i}\end{array}{C}_{{k}_{i}}{p}_{i}^{{k}_{i}}{\left(1-{p}_{i}\right)}^{{n}_{i}-{k}_{i}}

where *p*
_{
i
}is the probability that a female collected in the *i*
^{th} sample will be found to be pregnant. *p*
_{
i
}depends on *f*, the probability a recruited female becomes pregnant in any given year, which is the parameter of interest. It also depends on *μ* and *σ*, the mean and standard deviation whelping date, and on two further parameters: *D*
_{
p
}, the number of days prior to the whelping date for which a female can be seen, during the spaying operation, to be pregnant; and *D*
_{
l
}, the number of days following the whelping date for which a female can be seen to be lactating and will therefore not be collected for spaying. *D*
_{
p
}was set at 56 days and *D*
_{
l
}at 42 days. The probability a female collected in the *i*
^{th} sample is found to be pregnant is the probability her next litter is due within *D*
_{
p
}days given that her last litter was not born less than *D*
_{
l
}days ago, thus

{p}_{i}=\frac{f\cdot {\displaystyle \underset{{t}_{i}}{\overset{{t}_{i}+{D}_{p}}{\int}}N(t|\mu ,\sigma )dt}}{1-f\cdot {\displaystyle \underset{{t}_{i}-{D}_{l}}{\overset{{t}_{i}}{\int}}N(t|\mu ,\sigma )dt}}

where *t*
_{
i
}is the date of the *i*
^{th} sample expressed as days from the start of the year. To allow for the fact that, as a function of date, the variation in the proportion pregnant is cyclic, each normal density was replaced by the sum of the densities over the current, previous and following years and that sum scaled to integrate to one over a single year:

{p}_{i}=\frac{f\cdot {\displaystyle \sum _{j=-1}^{j=1}{\displaystyle \underset{{t}_{i}}{\overset{{t}_{i}+{D}_{p}}{\int}}N(t|\mu +j\cdot 365,\sigma )dt}}/{\displaystyle \sum _{j=-1}^{j=1}{\displaystyle \underset{1}{\overset{365}{\int}}N(t|\mu +j\cdot 365,\sigma )dt}}}{1-f\cdot {\displaystyle \sum _{j=-1}^{j=1}{\displaystyle \underset{{t}_{i}-{D}_{l}}{\overset{{t}_{i}}{\int}}N(t|\mu +j\cdot 365,\sigma )dt}}/{\displaystyle \sum _{j=-1}^{j=1}{\displaystyle \underset{1}{\overset{365}{\int}}N(t|\mu +j\cdot 365,\sigma )dt}}}

This expression for *p*
_{
i
}was used in the binomial distribution to calculate the probability of *k*
_{
i
}and those probabilities logged and summed over *i* to give the log likelihood of the observed variation in the proportion pregnant.

The asymptotic chi-squared distribution of a logged likelihood ratio was used to calculate confidence limits for the estimate of *f*. Figure 2 shows the change with *f* of the log likelihood maximised with respect to *μ* and *σ*. The horizontal line is drawn at 1.92 (i.e. *χ*
^{2}
_{1,0.05}/2) below the maximum of the log likelihood maximised with respect to all three parameters and the values of *f* at which it cuts the log likelihood curve provide lower and upper 95% confidence limits of 45.5% and 49.5% for the percentage of females that become pregnant in any given year. However, this calculation assumes that *D*
_{
p
}and *D*
_{
l
}are known without error whereas the period for which females can be seen to be lactating and are therefore not collected from the street is not known accurately. Allowing that period to vary from four to eight weeks reduces the lower confidence limit to 44% and increases the upper limit to 51%.

### The likelihood for the distribution of intervals preceding a second catching event

After being released back into the urban environment a small number of spayed females are caught by the catching teams a second time because the ear notch that identifies the female as having been spayed was missed. An even smaller number are recorded as having been caught a second time because they are terminally ill or injured and are therefore returned to the clinic for euthanasia.

The spayed females caught a second time provide a random sample of minimum times for which females persist in the population following being spayed and hence information on their survival rate. For example, if their survival was extremely low then almost all second catching events would occur soon after the operation. Data records were available from 2002 to 2006 inclusive. Over this period tattoo markings were recorded on about 0.6% of the adult female dogs caught, giving a sample size of 62 intervals between the first and second catching events. We used the proportion of females recorded as having been caught a second time and the distribution of time intervals to estimate the annual survival of recruited females.

Those statistics depend on the probability of survival in combination with the probabilities of the two types of catching in each successive month. Thus the likelihood of the observed proportion and intervals was maximised with respect to three parameters: the probability of survival, *S*; the probability per month of being caught a second time because of the ear notch having been missed, *R*
_{
h
}; and the probability of being caught and recorded a second time because of terminal illness or injury *R*
_{
s
}(only a proportion of dogs that were euthanised were checked for the presence of a tattoo mark).

The release of each sterilised female resulted in one of three types of event: released in a certain month and not caught a second time; first released in a certain month and caught a second time, after a certain interval, for euthanasia; and first released in a certain month and caught a second time after a certain interval because the ear notch was missed but not caught again. No females were recorded as caught more than twice.

The probability of the first type of event is the sum, over every month following the release, of the probability that the female survives until that month without being caught a second time and then dies in that month, plus the probability the female survives until the last month without being caught a second time. Let *n*
_{1i
}represent the number of females released in month *i* that were not caught subsequently. For each of those females the probability that they were not caught subsequently was:

{P}_{1i}={\displaystyle \sum _{m=0}^{t-i-1}{(S(1-{R}_{h}-{R}_{s}))}^{m}(1-S)+{(S(1-{R}_{h}-{R}_{s}))}^{(t-i)}}

where *t* is the final month in which dogs were caught.

The *n*
_{2ij
}females released in month *i* that were caught for euthanasia in month *j* must have survived and avoided capture for *j-i* months before being caught, hence for each of them the probability of that event was:

*P*
_{2ij
}= (*S*(1 - *R*
_{
h
}- *R*
_{
s
}))^{(j-i)}
*R*
_{
s
}

(6)

The *n*
_{3ij
}females released in month *i* that were caught because the ear notch was missed in month *j* must have survived and avoided capture for *j-i* months before being caught and then have avoided capture subsequently, hence for each of them the probability of that event was:

*P*
_{3ij
}= (*S*(1 - *R*
_{
h
}- *R*
_{
s
}))^{(j-i) }
*R*
_{
h
}
*P*
_{1j
}

(7)

Thus, assuming the probability of each event is independent, the log likelihood equals:

{\displaystyle \sum _{i=1}^{f-1}{n}_{1i}\mathrm{log}({P}_{1i})}+{\displaystyle \sum _{i=1}^{f-i}{\displaystyle \sum _{j=i+1}^{f}{n}_{2ij}\mathrm{log}({P}_{2ij})+}}{\displaystyle \sum _{i=1}^{f-i}{\displaystyle \sum _{j=i+1}^{f}{n}_{3ij}\mathrm{log}({P}_{3ij})}}

*R*
_{
h
}and *R*
_{
s
}are taken as constants, so to avoid bias in the estimate of *S* it is necessary that, given a spayed female has survived to a certain month, the probability she is caught again in that month is independent of the period since spaying. That assumption could be violated if some spayed females had an increased mortality rate when first released and hence an increased probability of being recaptured early as dogs suffering from terminal illness or injury. In that case we would expect an excess of short intervals in the histogram of intervals relating to dogs caught for euthanasia. However figure 3a shows that the distributions of intervals prior to the two types of catching are very similar.

There is also a potential for changes in *R*
_{
h
}and *R*
_{
s
}over time to bias the survival estimate. The number of tattoo markings recorded as a percentage of the total number of females collected has been fairly constant since 2002 but there was some reduction in the effort to look for markings on dogs collected in 2004, when only half the usual number of marks were recorded. However, by that time the age distribution of spayed females in the population had stabilised and reducing the values of *R*
_{
h
}and *R*
_{
s
}to reflect the observed variation changed the survival estimate by less that 0.5%.