Comparing the Effectiveness of Malaria Vector-Control Interventions Through a Mathematical Model

Although some malaria-control programs are beginning to combine insecticide-treated nets (ITNs) and indoor residual spraying (IRS), little is known about the effectiveness of such combinations. We use a mathematical model to compare the effectiveness of ITNs and IRS with dichlorodiphenyltrichloroethane (DDT) or bendiocarb, applied singly and in combination, in an epidemiological setting based in Namawala, Tanzania, with Anopheles gambiae as the primary vector. Our model indicates that although both IRS (with DDT) and ITNs provide personal protection, humans with only ITNs are better protected than those with only IRS, and suggests that high coverage of IRS with bendiocarb may interrupt transmission, as can simultaneous high coverage of ITNs and IRS with DDT. When adding a second vector-control intervention, it is more effective to cover the unprotected population first. Although our model includes some assumptions and approximations that remain to be addressed, these findings should be useful for prioritizing and designing future field research.

where ë x û is the floor function, that is, the greatest integer less than x ; and the binomial coefficient is The derived parameters are, 1 The system (1) has a unique fixed point [1, (6)], with k + and k l + defined in (2). This fixed point is globally asymptotically stable if all the eigenvalues of the evolution matrix (¡ in ref. 1) of the system of equations (1) are inside the unit circle. While we could not show this analytically for all parameter values, 1 we numerically verified that the magnitudes of all eigenvalues of ¡ for each of our simulations here were less than 1. For this globally asymptotically stable fixed point, we calculate the field-measurable quantities, 1 Parous rate: Host-biting rate: Vectorial capacity:

DETAILS OF THE HUMAN INFECTIVITY TO MOSQUITOES AS A FUNCTION OF EIR
We let x represent the EIR (measured per person per year) and f ( x ) represent the human infectivity to mosquitoes as a function of EIR. We require a realistic f ( x ) that captures the biology of malaria transmission through humans and the simulation results in Figure 3 to have the following properties: There exists an f < f * such that lim ( ) .
x f x f →∞ = We define f ( x ) using piecewise rational functions as, We fit the rest of the parameters in (5)  . , . , . , so that f ( x ) (5) satisfies all seven properties.

PARAMETER VALUES AND THE EFFECTS OF INTERVENTIONS
We use baseline parameter values in the absence of interventions (except for P D i and P E i which we discuss in the section on IRS with DDT) from ref. 1, which are based largely on data from Killeen and Smith 27 for Anopheles gambiae in Namawala, Tanzania. We assume that each intervention affects a certain number of mosquito survival and infection parameters (that we describe below) while other parameters remain unchanged by the intervention. Thus some parameter values change depending on whether a type of host has or does not have an intervention. We note again that N i is the number of hosts of type i and varies depending on the coverage level of the intervention, and not on the biological properties of the intervention itself.
Except for parameters that are defined as natural numbers, we use two significant figures for their values. The parameter values used for each individual intervention are given in Table 4 . Parameter values for simulations of combined interventions are calculated as described below.
Insecticide-treated nets. The effects of ITNs were modeled in ref. 1 and we use the same parameter values here. The availability rate of ITN users is reduced by 44% and the survival probability of a mosquito biting a human is reduced by 46%, divided equally between before and after biting. Using the notation, i = 1 representing ITN users and i = 2 unprotected humans, we model the effects of ITNs as, Indoor residual spraying with DDT. We assume that IRS with DDT deters mosquitoes from entering houses, thus reducing the human availability rate, a i , and kills resting mosquitoes after they rest on walls, thus reducing the mosquito's probability of surviving the resting phase, P D i , but does not affect any of the other parameters in the model. We use data from Smith and Webley 12 to determine the numerical values for these reductions. In this subsection, we use the notation of i = 1 to denote humans in houses sprayed with DDT and i = 2 to denote unprotected humans. Table IX 12 shows the deterrent effect of DDT on Anopheles gambiae in the Magugu area of Tanzania over each month since treatment of the hut. Since the model contains only constant parameters, we average the deterrent effect over all months to give a value of 56%. We assume this corresponds to a 56% decrease in the probability that a mosquito encounters a human protected by IRS-DDT, P A i , .
Assuming N = 2 and N 1 = N 2 from 12 (the number of inhabitants in the treated and untreated experimental huts were the same) in (3), some algebraic manipulations show that, . Table VI 12 shows the number of An. gambiae counted and collected in treated and untreated verandah-trap huts in Magugu for every month since treatment. The table divides the mosquitoes by feeding status and whether they were dead or alive. We assume that the proportion of dead fed mosquitoes out of all fed mosquitoes is the probability of a mosquito dying while resting. Averaged over all months, we find that for huts treated with IRS, P D 1 m = 0.25 and for untreated huts, P D 1 m = 0.0064 . Thus, and P E i did not matter as long as the product matched the probability of surviving a feeding cycle in the absence of intervention, P f . We now use this data from Smith and Webley 12 to set P D i in the absence of IRS and choose a corresponding value of P E i that matches the original P f , We thus use P D i = 0.99 and P E i = 0.88 for the baseline values in the absence of an intervention.
Indoor residual spraying with bendiocarb. Laboratory studies by Evans (1993) 28 on Anopheles gambiae showed that the irritancy effect of bendiocarb was statistically similar to that of distilled water. We assume in our simulations that IRS with bendiocarb does not repel mosquitoes and thus does not reduce the availability rate of humans in houses sprayed with bendiocarb. Using the notation in this subsection of i = 1 for humans in houses sprayed with bendiocarb and i = 2 for unprotected humans, we thus have, We use data from Sharp et al. (2007) 13 to determine the effects of bendiocarb ( Ficam ™ ) on mosquito mortality. They collected data on An. gambiae s.s. on Bioko island on the number of mosquitoes and their sporozoite rates in sentinel huts, before and after treatment. They list the number of mosquitoes collected per trap per 100 nights in Table 1 as 23.9 before treatment and 1.9 after treatment. In the absence of more detailed information on the mosquitoes, we assume that all mosquitoes caught have survived the resting phase.
Since we are considering two separate time points, we assume there is only one type of host, and label the time before spraying as ( j ) = 2 and the time after spraying as ( j ) = 1. The number of mosquitoes caught per night 1 is, x N P P P P P P P P P P We substitute (6) into the ratio of x (1) to x (2) to solve for P D 1 1 ( ) , P P x P P P P P P P x P P .

Parameter values for two concurrent interventions.
When modeling humans with multiple interventions, we assume that the effects of the interventions are independent and cumulative. For example, humans that are protected by both ITNs and IRS with DDT, the availability rate of the protected human will be 0.56 × 0.44 times the availability rate of the unprotected humans. The mosquito survival probabilities, P B i and P C i will decrease by a factor of 0.73 as defined for ITNs, and P D i will decrease by a factor of 0.76 as defined for IRS-DDT.