A prevalence-based transmission model for the study of the epidemiology and control of soil-transmitted helminthiasis

Abstract

Much effort has been devoted by the World Health Organization (WHO) to eliminate soil-transmitted helminth (STH) infections by 2030 using mass drug administration targeted at particular risk groups alongside the availability to access water, sanitation and hygiene services. The targets set by the WHO for the control of helminth infections are typically defined in terms of the prevalence of infection, whereas the standard formulation of STH transmission models typically describe dynamic changes in the mean-worm burden. We develop a prevalence-based deterministic model to investigate the transmission dynamics of soil-transmitted helminthiasis in humans, subject to continuous exposure to infection over time. We analytically determine local stability criteria for all equilibria and find bifurcation points. Our model predicts that STH infection will either be eliminated (if the initial prevalence value, y(0), is sufficiently small) or remain endemic (if y(0) is sufficiently large), with the two stable points of endemic infection and parasite eradication separated by a transmission breakpoint. Two special cases of the model are analysed: (1) the distribution of the STH parasites in the host population is highly aggregated following a negative binomial distribution, and (2) no density-dependent effects act on the parasite population. We find that disease extinction is always possible for Case (1), but it is not so for Case (2) if y(0) is sufficiently large. However, by introducing stochastic perturbation into the deterministic model, we discover that chance effects can lead to outcomes not predicted by the deterministic model alone, with outcomes highly dependent on the degree of worm clumping, k. Specifically, we show that if the reproduction number and clumping are sufficiently bounded, then stochasticity will cause the parasite to die out. It follows that control of soil-transmitted helminths will be more difficult if the worm distribution tends towards clumping.

Fig 1. Comparisons of prevalence values generated by the models (3) (red dashed line) and (5) (black solid curve).

Both models (prevalence-based or mean-worm-burden-based) produce well-matched results for arbitrary k and initial values. Both predict that the infection will either die off or reach an endemic state.

Fig 2. Numerical solutions of equilibrium y* as a function of R₀ with different k values, but fixed z = 0.96.

Fig 3. The relationship between the eigenvalue (6) and the endemic equilibrium of the model (5) is demonstrated by varying k values (corresponding to 0 ≤ R₀ ≤ 5).

Linearization is one of the key methods employed in assessing stability, and it can be applied to determine the local stability of a model governed by ordinary differential equations. By definition [37], an equilibrium point is locally asymptotically stable if all eigenvalues have negative real parts, whereas it is unstable if at least one eigenvalue has positive real part. A local bifurcation occurs whenever the real part of an eigenvalue passes through zero.

Fig 4. The dynamics of model (5) when varying k, R₀ and initial value y₀.

By varying k and R₀ values, all solutions of this model converge to zero if y₀ < y_*, whereas the solutions of this model approach the endemic equilibrium y* whenever y₀ > y_* as t → ∞.

Fig 5. The vector field (5) derived using numerical solutions of the model (5), where z = 0.96, k = 0.5 and R₀ = 4.

Fig 6. Heatmaps of the vector field (5) and the rate of change of the model (5)—i.e., Eq (7)—are as shown in subfigures (a) and (b), respectively.

k = 0.5 and z = 0.96 are chosen to generate these two plots.

Fig 7. The comparisons of the analytical (9) and numerical (with approximations) (5) solutions with arbitrary initial points around y = 0 and y₀ < y_*.

Fig 8. The comparisons of analytical (12) and numerical (5) solutions with arbitrary initial points around the stable endemic equilibrium, y*.

Fig 9. The comparisons of analytical (12) and numerical (5) solutions with arbitrary initial points around the unstable endemic equlibrium, y_*.

Fig 10. The relationship between F(y;k,z) and y by varying parameter k.

$\mathcal{F}(y;k,z)$ is zero almost everywhere as k → 0.

Fig 11. The comparison of the analytical solution (16) and the numerical solution of (5) around y = 0 as k → 0.

All solutions, both analytical and numerical, are eventually converging to zero whenever y₀ < y_*. Hence disease elimination is possible in this case.

Fig 12. The relationship between F(y;k) and prevalence, y.

Fig 13. The dynamics of the model (18) with arbitrary k, R₀ and initial values.

Parasite extinction is possible if the initial value of y is sufficiently low. Otherwise, the disease will remain endemic.

Fig 14. The comparisons of analytical (21) and numerical (18) solutions with arbitrary k, R₀ and initial values around y = 0.

Both analytical and numerical solutions are eventually converging to zero.

Fig 15. Comparisons of the stochastic (23) and deterministic (5) models with arbitrary k, ρ and y₀ values.

By choosing parameter values that satisfy the conditions in Theorem 3, all solutions of the stochastic model (23) with arbitrary k and y₀ eventually converge to zero. However, if y₀ is sufficiently large, the stochastic and deterministic models produce conflicting results. That is, solutions of the stochastic model approach zero, whereas solutions of the deterministic model remain endemic.

Fig 16. Comparisons of the stochastic (23) and deterministic (5) models, with varying k, ρ and y₀ values.

By choosing parameter values such that the sufficient conditions as in Theorem 3 are violated, not all solutions of the stochastic model (23) approach zero. For small k and sufficiently large y₀ values, the solution of the stochastic model (23) fluctuates around the endemic equilibrium state (see (b)).