The Dynamics of Ascaris lumbricoides Infections

Abstract

The Anderson–May model of human parasite infections and specifically that for the intestinal worm Ascaris lumbricoides is reconsidered, with a view to deriving the observed characteristic negative binomial distribution which is frequently found in human communities. The means to obtaining this result lies in reformulating the continuous Anderson–May model as a stochastic process involving two essential populations, the density of mature worms in the gut, and the density of mature eggs in the environment. The resulting partial differential equation for the generating function of the joint probability distribution of eggs and worms can be partially solved in the appropriate limit where the worm lifetime is much greater than that of the mature eggs in the environment. Allowing for a mean field nonlinearity, and for egg immigration from neighbouring communities, a negative binomial worm distribution can be predicted, whose parameters are determined by those in the continuous Anderson–May model; this result assumes no variability in predisposition to the infection.

Appendix

1.1 Solution of (1)

We write (1) in characteristic form:

$$\begin{aligned} \dot{m}=1,&\dot{E}=-\mu _\mathrm{e}E,\dfrac{}{}\nonumber \\\dot{m}=1,&\dot{H}=-\mu _\mathrm{h}H,\dfrac{}{}\nonumber \\\dot{m}=1,&\dot{I}=-\mu _iI, \end{aligned}$$

(56)

with the boundary conditions (2) taking the parametric form

$$\begin{aligned} t=\tau ,\quad E=E_0(\tau ),\quad H=\beta 'L(\tau ),\quad I=H(\tau ,\tau _3)\quad \mathrm{at}\quad m=0 \end{aligned}$$

(57)

(these give the solutions for $t>m$; for $t<m$ we would use an initial condition at $t=0$, but since m is finite, this just produces a transient which washes through the system). The solution of (56) and (57) is given parametrically by

$$\begin{aligned}&m=t-\tau ,\quad E=E_0(\tau )\hbox {e}^{-\mu _\mathrm{e}(t-\tau )},\quad H=\beta 'L(\tau )\hbox {e}^{-\mu _h(t-\tau )},\nonumber \\&I=H(\tau ,\tau _3)\hbox {e}^{-\mu _i(t-\tau )}, \end{aligned}$$

(58)

whence we obtain (3).