A Comparative Analysis of the Relative Efficacy of Vector-Control Strategies Against Dengue Fever

Abstract

Dengue is considered one of the most important vector-borne infection, affecting almost half of the world population with 50 to 100 million cases every year. In this paper, we present one of the simplest models that can encapsulate all the important variables related to vector control of dengue fever. The model considers the human population, the adult mosquito population and the population of immature stages, which includes eggs, larvae and pupae. The model also considers the vertical transmission of dengue in the mosquitoes and the seasonal variation in the mosquito population. From this basic model describing the dynamics of dengue infection, we deduce thresholds for avoiding the introduction of the disease and for the elimination of the disease. In particular, we deduce a Basic Reproduction Number for dengue that includes parameters related to the immature stages of the mosquito. By neglecting seasonal variation, we calculate the equilibrium values of the model’s variables. We also present a sensitivity analysis of the impact of four vector-control strategies on the Basic Reproduction Number, on the Force of Infection and on the human prevalence of dengue. Each of the strategies was studied separately from the others. The analysis presented allows us to conclude that of the available vector control strategies, adulticide application is the most effective, followed by the reduction of the exposure to mosquito bites, locating and destroying breeding places and, finally, larvicides. Current vector-control methods are concentrated on mechanical destruction of mosquitoes’ breeding places. Our results suggest that reducing the contact between vector and hosts (biting rates) is as efficient as the logistically difficult but very efficient adult mosquito’s control.

Appendix: Some Comments on the Meaning of the Model’s Equations

In this appendix, we show how to include spatial heterogeneities in the model and, by doing so, we clarify the meaning of the model’s equations.

First, we assume that mosquitoes have a limited range of flight, which implies that the probability of transmission of infection from one infected mosquito to one susceptible host varies according to the distance between them.

Consider the first equation of system (1),

$$ \frac{dS_{H}}{dt} = - abI_{M}\frac{S_{H}}{N_{H}} - \mu_{H}S_{H} + r_{H}N_{H} \biggl( 1 - \frac{N_{H}}{\kappa_{H}} \biggr). $$

(40)

All the variables are densities. This implies that we are considering a very large region where the populations of mosquitoes and hosts are constant, that is, do not vary from point to point. Then, one might think that in Eq. (40) a mosquito in a certain place can bite a host which can be very far from it. This is not reasonable and it is not true for Eq. (40). To see this, consider the parameter a, the mosquitoes’ biting rate. We can write this as a=a′A, where a′ is the biting rate per unit area and A is the area where the mosquitoes’ flight ranges. Therefore, only humans inside this area are bitten by this mosquito. But, since the humans and mosquitoes populations are assumed as homogeneously distributed, this does not appear in the equations because in parameter a this effect is hidden.

Let us now introduce spatial heterogeneity. For this we should specify the position $\vec {r}$, representing the spatial location of individuals. Thus, let $S_{H}(\vec {r} )\,ds$ be the number of human susceptibles in the small area ds around the position $\vec {r}$.

Let us now consider how $S_{H}(\vec {r} )\,ds$ varies with time. Let $I_{M}(\vec {r}' )\,ds'$ be the number of infected mosquitoes in the small area ds′ around the position $\vec {r}'$. The total number of bites the infected mosquitoes population inflicts in a time interval dt is $a'I_{M}(\vec {r}' )\,ds'\,dt$. A fraction of those bites $F(\vert \vec {r} - \vec {r} ' \vert )$ is inflicted on the hosts at position $\vec {r}$, that is, $S_{H}(\vec {r} )\,ds$. Of course, $F(\vert \vec {r} - \vec {r} ' \vert )$ is a decreasing function of the distance $\vert \vec {r} - \vec {r} ' \vert$ between infected mosquitoes and susceptible humans. Thus, Eq. (40) becomes

$$\begin{aligned} \frac{dS_{H}(\vec {r} )}{dt} =& - b\frac{S_{H}(\vec {r} )}{N_{H}(\vec {r} )}\int \,d\vec {s}'\,a'\bigl(\vec {r} '\bigr) F \bigl(\bigl\vert \vec {r} - \vec {r} ' \bigr\vert \bigr)I_{M}\bigl(\vec {r} '\bigr) - \mu_{H}S_{H}(\vec {r} ) \\ &{} + r_{H}N_{H}( \vec {r} ) \biggl( 1 - \frac{N_{H}(\vec {r} )}{\kappa_{H}(\vec {r} )} \biggr). \end{aligned}$$

(41)

All the other equations in system (1) should be similarly modified and, of course, the result is very difficult to integrate. When $a'(\vec {r} ')F(\vert \vec {r} - \vec {r} ' \vert )$ is equal to $a'A\theta (\vert \vec {r} - \vec {r} ' \vert )$, and the densities are homogeneously distributed in space, we have

$$ b\frac{S_{H}(\vec {r} )}{N_{H}(\vec {r} )} \int \,d\vec {s}'\,a'\bigl( \vec {r} '\bigr) F\bigl(\bigl\vert \vec {r} - \vec {r} ' \bigr\vert \bigr) = b\frac{S_{H}}{N_{H}}I_{M} \int \,d\vec {s}'\,a'\bigl(\vec {r} '\bigr)F\bigl(\bigl\vert \vec {r} - \vec {r} ' \bigr\vert \bigr) = b\frac{S_{H}}{N_{H}}I_{M}a, $$

(42)

and Eq. (41) reduces to (40).

The above formalism is necessary when we are dealing with large regions of space, where heterogeneities are significant. However, for small regions, where heterogeneities can be neglected, the system of Eqs. (1) of the main text is a good approximation. The relative sensitivity of the transmission variables to the studied parameters, however, is not expected to be significantly influenced by spatial heterogeneities. Of course, the value of the transmission variables may vary from place to place but the relative sensitivity, the main objective of the present analysis, of these variables to the parameters should be the same.