Mathematical model of mating probability and fertilized egg production in helminth parasites

The distribution of parasites among hosts is a crucial aspect of host-parasite interactions. Typically, parasites exhibit a clustered or aggregated distribution, where a small proportion of hosts harbor the majority of parasites, while the majority of hosts remain parasite-free (Shaw and Dobson, 1995; Shaw et al., 1998). This distribution follows the widely recognized 20-80 rule, where 20% of individuals contribute to 80% of the parasite burden (Woolhouse et al., 1997).

Aggregation is considered a fundamental characteristic of parasitism and is often referred to as the ”First Law of Parasitism” (Crofton, 1971; Poulin, 2007).

In statistical terms, a distribution is considered aggregated when the variance-to-mean ratio of the number of parasites per host is significantly greater than one. To describe the distribution of parasites among hosts, the negative binomial distribution is commonly employed. This discrete and flexible distribution, characterized by two parameters (Fisher et al., 1941), fits well with the observed aggregation in nature (Bliss and Fisher, 1953). Consequently, several theoretical models of host-parasite dynamics implicitly assume the negative binomial distribution (Anderson and May, 1978; Adler and Kretzschmar, 1992).

However, it is important to note that the negative binomial distribution provides only a phenomenological description of aggregation, and its parameters are not explicitly linked to underlying processes governing host exposure to parasites and parasite success in infecting hosts (Duerr and Dietz, 2000; Gourbière et al., 2015).

In parasitology, the sex ratio of a parasite population is often quantified by the proportion of female parasites ($\alpha$) and the proportion of male parasites ($\beta=1-\alpha$). The male-to-female sex ratio, expressed as $\alpha/\beta:1$, serves as a significant metric in the study of parasitic infections. Another important metric is the mean number of female parasites per host. If $m$ represents the mean parasite burden, the mean number of female and male parasites can be calculated as $\alpha m$ and $\beta m$, respectively. By understanding the sex ratio and the mean number of female parasites per host, researchers can gain valuable insights into the transmission and spread of parasitic infections.

In this section we use random variables to model the distribution of parasites in hosts. Let $W$ be the random variable representing the number of parasites in a host, also known as the parasite burden. The variable $W$ is distributed according to the distribution of parasites in the hosts. In this section we assume that male and female parasites are distributed together in hosts. Therefore, we consider the number of female and male parasites per host as two dependent random variables, denoted $F$ and $M$, respectively. The total number of parasites per host, $W$, is then the sum of $F$ and $M$. To model the distribution of female parasites per host, we can model the number of female parasites per host, $F$, as a random sum of other random variables, modeled by a stopped-sum model (Johnson et al., 2005), defined as $F=\sum_{i=1}^{W}X_{i}$ where the $X_{i}$ are independent identically distributed (iid) Bernoulli random variables with parameter $\alpha$ ($X_{i}\sim\mathrm{Ber}(\alpha)$). Its probability generating function (pgf) is the function composition $G\circ G_{B}$, where $G$ is the pgf of the random variable $W$ and $G_{B}$ is the pgf of the Bernoulli distribution given by $G_{B}(s)=\beta+\alpha s$. Therefore, its pgf $G_{F}$ is of the form

The first moments of $F$ are

where $\mu$ and $\sigma^{2}$ are the mean and variance of $W$. The index of dispersion or variance-to-mean ratio (Cox and Lewis, 1966), $D=\frac{\sigma_{F}^{2}}{\mu_{F}}$, is given by

where $\frac{\sigma^{2}}{\mu}$ is the variance-to-mean ratio of $W$. Therefore, if $W$ is over-dispersed, so will $F$.

Similarly, we define the random variable $M$, the number of male parasites per host, as a randomly stopped-sum definity by $M=\sum_{i=1}^{W}Y_{i}$ where the $Y_{i}=1-X_{i}$ are iid Bernoulli random variables with $Y_{i}\sim\mathrm{Ber}(1-\alpha)$. Its pgf is of the form $G_{M}(s)=G(\alpha+\beta s)$ and the mean and variance are $\mu_{M}=\beta\mu$ and $\sigma_{M}^{2}=\beta^{2}\sigma^{2}+\alpha\beta\mu$, respectively. Note that for definition $F$ and $M$ are dependent random variables.

In practice, the distribution of the variable $W$ is known. Therefore, by obtaining the pgf $G_{F}$ and $G_{M}$, we can determine the distributions of the variables $F$ and $M$. A general class of distributions exists in which the statistical model of $W$ is conserved. This class includes the Poisson, binomial and negative binomial distributions, among others, which are widely used for counting models (Johnson et al., 2005).

In the paper, we define $G(s;\theta)$ by $\sum_{n\geq 0}p(n;\theta)s^{n}$, where $p$ is the probability mass function of the parasite distribution and $\theta$ is a parameter vector in which $\theta$ can include the mean, $m$, of the parasite distribution.

In this paper we considered a polygamous mating system for parasites, i.e. a male parasite can fertilize all female parasites in the same host (May and Woolhouse, 1993). Since the expression $\sum_{j=0}^{n-1}p_{n}\binom{n}{j}\alpha^{j}\beta^{n-j}$ is the probability of having at least one male parasite in a parasite burden of size $n$, we obtained the following result

Then we can estimate the mating probability of a female parasite as the quotient of the mean number of fertilized female parasites per host, $\alpha m-\alpha G^{\prime}(\alpha)$, and the mean number of female parasites per host, $\alpha m$. Therefore, a expression for mating probability (denoted by $\Phi$) as a function of mean parasite burden $m$ is given by

In population ecology, density-dependent processes occur when population density affects growth rates. In the case of parasites, these processes can affect their fecundity, establishment and survival in the host. For example, in helminth parasites, their fecundity has been observed to decrease as the parasite burden in the host increases (Churcher et al., 2006; Walker et al., 2009). This phenomenon is known as density-dependent fecundity, and it is explained by a negative exponential function that relates per capita fecundity to parasite burden

where $\lambda(n)$ is the per capita female fecundity within a host with a parasite burden of size $n$, $\lambda_{0}$ is the intrinsic fecundity in the absence of density-dependent effects, and $\gamma$ is the density-dependent intensity. To simplify the notation in the rest of the text, we will express female fecundity by $\lambda(n)=\lambda_{0}z^{n-1}$ where $z=e^{-\gamma}$. In summary, density-dependent effects are an important factor in parasite ecology, and may have significant implications for parasite population dynamics. For more information on this subject, see the study by Hall and Holland (2000) on Ascaris lumbricoides.

In this section we will consider the most common statistical models used to describe the distribution of parasites among hosts.

In the previous section, we introduced the random variable $W$, which represents the number of parasites in a host. We also introduced two additional random variables, $F$ and $M$, representing the number of female and male parasites per host, respectively.

In this section we focus on the case where male and female parasites are distributed separately among hosts. Therefore, we assume that the random variables $F$ and $M$ are independent. This assumption allows us to investigate and verify the following properties:

where $G$, $G_{F}$ and $G_{M}$ are probability generating function of the variables $W$, $F$ and $M$, respectively. From the definition of these random variables, we obtain that the first moments of the variables $F$ and $M$ are respectively

where $\mu$ and $\sigma^{2}$ are the mean and variance of $W$. The variance-to-mean ratios of the variables $F$ and $M$ are equal to $\frac{\sigma^{2}}{\mu}$, which is the variance-to-mean ratio of $W$. Therefore, if $W$ is over-dispersed, $F$ and $M$ will also be over-dispersed. If we compare these variance-to-mean ratios with the case of the dependent variables (see (3)), we can see that the independent variables show a greater over-dispersion when $W$ is over-dispersed.

We now present an expression for each of the variables studied in the section 2. The proofs of these expressions can be found in the appendix LABEL:formulasind. Where $m$ is the mean of $W$ and $p_{M}$ is the probability mass function of $M$.

• •

  Mean number of fertilized female parasites

  $\displaystyle\alpha m\left[1-p_{M}(0)\right]$

  (26)

• •

  Mating probability

  $\displaystyle\Phi=1-p_{M}(0)$

  (27)

• •

  Mean egg production per host

  $$\lambda_{0}G_{M}(z)G^{\prime}_{F}(z)$$

  (28)

• •

  Mean fertilized egg production per host

  $$\lambda_{0}G_{M}(z)G^{\prime}_{F}(z)\left[1-\frac{p_{M}(0)}{G_{M}(z)}\right]$$

  (29)

• •

  Mating probability with density-dependent effects

  $$\phi=1-\frac{p_{M}(0)}{G_{M}(z)}$$

  (30)

• •

  Mean effective transmission contribution by female parasite

  $$\psi=\frac{G_{M}(z)G^{\prime}_{F}(z)}{\alpha m}$$

  (31)

• •

  Contribution of mean fertilized egg production for mean-based deterministic model of parasite burden

  $$\lambda_{0}\alpha m\psi(m)\phi(m)$$

  (32)

In this section, we show the results obtained for some of the statistical models used in the section 2.5.

Simulation algorithms presented in this section are based on the following assumptions and rules:

• •

  We considered a host population of size $N$.

• •

  The parasite burden of each host is a random variable $W$. Female and male burdens are random variables, $F$ and $M$ respectively. The values of the random variables will be denoted by $w$, $x$ and $y$ respectively.

• •

  Distributed together: The individual parasite burden per host is a value $w$ associated to the variable W with some assumed distribution (Poisson, negative binomial, etc.). The individual female parasite burden ($x$) is then obtained as the result of $w$ Bernoulli trials with the success parameter $\alpha$, where $\alpha$ is the sex ratio of the female parasites. The individual male parasite burden is then obtained as $y=w-x$.

• •

  Distributed separately: The values of the individual female and male parasite burdens ($x$ and $y$, respectively) are associated with the variables $F$ and $M$, respectively. These variables have assumed distributions (Poisson, negative binomial, etc.) that satisfy the equations (24)(25). The individual parasite burden per host, $w$, is then obtained as $w=x+y$.

• •

  The egg production per host is given by $xz^{w-1}$, where $z=e^{-\gamma}$ with $\gamma$ the density-dependence intensity.

• •

  The infective egg production per host is given by $I(y\neq 0)xz^{w-1}$, where $I$ is the indicator function of the set $y\neq 0$.

• •

  The fertilized female parasites are female parasites where the individual male parasite burden is non-zero ($y\neq 0$).

• •

  The mean effective transmission contribution per female parasite is obtained by quotient of the mean number egg production per host and the mean of female parasites per host.

• •

  The mating probability is obtained by quotient of the mean number of fertilized female parasites per host and the mean number of female parasites per host.

• •

  The mating probabillity with density-dependent effects is obtained by quotient of the mean number infective egg production and the mean number egg production.

All the simulations were carried out in RStudio (Version 2022.12.0+353).

In this section, we present the results obtained from Monte Carlo simulations, considering a distribution of parasites in the host population based on negative binomial and zero-inflated negative binomial models.