A standard epidemiological model of malaria was built, with transmission of malaria between a population of humans and a population of mosquitoes 24 . In order to simplify the framework, the usual assumptions have been made, i.e. constant population sizes (human and mosquito) over time, uniform contacts between human and mosquitoes, ignorance of superinfection and immunity. Within the life-time period of humans, malaria prevalence among humans and mosquitoes reaches a steady state. This leads, in absence of protection, to equations based on the McDonald and Ross malaria transmission model 24 .

The time variation of malaria prevalence among humans can be defined in a simplified way as:

X ˙ = mabZ 1 − X − rX

where m is the vector density (ratio of mosquitoes per human), a is the number of bits per unit of time and per mosquito, b is the proportion of infected bites that produce infection among humans, Z is the proportion of infectious mosquitoes, and r is the clearance rate of malaria in humans.

Similarly, the time variation of the proportion of infectious mosquitoes, can be written as:

Z ˙ = acX e − gn − Z − gZ

where c is the proportion of bites on infectious humans that produce infection among mosquitoes, g is the death rate of mosquitoes, and n is the length of sporogonic cycle.

Assuming that the time period of life is long enough, malaria prevalence reaches a steady state equilibrium defined by 24 :

Q X = mab acX e − gn g + acX r + mab acX e − gn g + acX = bEIR r + bEIR

where EIR is the entomological inoculation rate classically defined such as:

EIR = maZ = m a 2 cX e − gn g + acX

In what follows (after protection through ITN/LLIN), the parameter m will become itself a variable. The function Q(X, m) is concave, and characterized by the following properties:

Q 0 , m = 0 Q 1 , m < 1

And its slope at origin, is equal to

∂ Q 0 , m ∂ X = m a 2 bc e − gn rg = R 0

This number, R ₀, is classically called, in the McDonald and Ross tradition, the “basic reproduction number” 25 26 . As demonstrated elsewhere, if R ₀ is below or equal to 1, then Q(X,m) converges towards the trivial disease free stable steady state. This case is not considered in what follows, as it does not coincide with the persistence of malaria in large regions of the developing world. Conversely, if R ₀ is higher than 1, then Q(X,m) converges towards a stable steady state characterized by a strictly positive prevalence of malaria. In what follows, Q(X,m) will be considered as the functional relationship between the prevalence and contagious persons, at steady state, also depending on the vector density m. This vector density will depend on protection behavior against mosquitoes.

When the basic reproduction number R ₀ is higher than 1, using protection tools could nevertheless reduce malaria transmission 24 , and then, the trivial disease free stable steady state could be reached. This is the rationale of ITN/LLINs dissemination policies. In order to assess this possibility, a model of protection behavior has been added to the previous epidemiological model. This behavioral model, as described below, is based on economic mechanisms. Peoples adopt a certain behavior: use of an insecticide-treated net (h = 1) or exposure to malaria risk (h = 0). It is supposed that the only means by which a person can prevent himself from parasitic infection is to sleep under an ITN/LLIN (even if a person can be infected during the first part of the night). As a first assumption, the use of an ITN/LLIN was supposed to provide complete protection from malaria infection. This assumption has been relaxed in Additional file 1 without affecting the main findings of the model. At any time, depending on the use of ITN before, the health status of the individual, σ(h) can take one of two values: susceptible, σ(h) = S, or infected, σ(h) = I. The probability of being infected at any time, conditionally to the absence of protection before, can then be written as:

π I = P σ h = I / h = 0

If H is the proportion of population using ITN/LLIN, among the (1-X) uninfected persons, the proportion of infected persons can be simply written as:

X = 1 − H π I

Furthermore the density of mosquitoes in contact to humans, is affected by the presence of ITN/LLIN used by a proportion H of the population. First, as the contact between mosquito and human is more difficult, the denominator of the mosquito density decreases, being now the proportion 1 – H of non-protected population. Second, as ITN/LLINs do not only protect humans, from anopheles bites, but also kill mosquitoes (knock down effect), the numerator (the number of mosquitoes) decreases with H. Hence m can be written as follows:

m H = m 0 1 − H 1 − γ H

Where γ(H) is the proportion of mosquitoes killed by the use of ITN/LLINs, an increasing function of H. Note that, the EIR is then a decreasing function of H. It follows that, at the steady state:

π I = Q X , m H

In order to complete the model, the determinants of H were specified in a next step. At the microeconomic level, the choice of protection is determined by maximizing the expected utility of each individual. The decision h of protection (h = 1 for protection, h = 0 for non-protection) affects individuals’ utility through two channels: (i) an expected positive impact on his/her health status in case of protection and (ii) a private cost, called κ. This cost can be interpreted as the shortfall of paying for protection. This broad definition includes the opportunity cost of protection and depends on the marginal utility of personal income. In other words, the private cost of the ITN/LLIN, includes direct, indirect and opportunity costs, such as paying the market price of ITN/ILLNs (direct costs), transportation costs to get them or costs related to social ostracism (indirect costs). Opportunity costs are the costs of using them for protection rather than using them for an economically productive activity, or the cost equal to what an individual must give up in order to use an ITN/LLIN, which he/she would otherwise never use. Hence protection decision is described through the following maximization program:

max h E u σ h − κW ω h

Where u (S) or u (I) are the utility levels attached to the health status (susceptible, σ(h) = S, or infected, σ(h) = I, thus depending on h, the use of a protection), with 0 < u(I) < u(S); ω is the individual income; W (ω) is the marginal utility of the income, supposed as usual to decrease with income 27 , and κ is the private cost of the ITN/LLIN.

The expected utility (the expected positive impact of using ITN/LLIN on the health status) can be estimated using the following probabilities of being susceptible or infected, conditionally to the use of protection:

st P σ h = S / h = 1 = 1 P σ h = S / h = 0 = 1 − π I P σ h = I / h = 0 = π I

In addition, it is assumed that there exists a minimum subsistence level such as in the case a Stone-Geary utility function 28 29 30 . This implies that the marginal utility of income W (ω) goes to infinity for all individuals at (or below) the minimum subsistence level, which is classically called the extreme poverty line Ω (i.e. the minimum level of income deemed adequate in a given country for an individual or a household). In other words, the extreme poverty line is an income level below which nobody can afford an ITN/LLIN, i.e. h = 0.

As in standard economic epidemiological models, the individual will use protective tools when W (ω) is lower than the expected utility loss associated with the risk of infection that occurs in the absence of protection:

E u σ 1 − u σ 0 ≥ κW ω

According to Equation (9) and the three probabilities of Equation (10) it follows that:

h = 1 if and only if u S − 1 − π I u S − π I u I ≥ κW ω

A person will use ITN/LLIN if the utility of being non-infected is greater than the utility of paying for a protective tool, according to the income and the probability of being infected without using any protection. Hence, protection occurs if and only if:

π I ≥ κW ω u S − u I

This Equation shows that there is a threshold probability of infection above which a person engages in protection. The key point in this approach is that the threshold probability of infection depends on the marginal income utility loss associated with using the ITN/LLIN, κW(ω), with respect to the net value attached to susceptible health status, u(S)–u(I). This threshold depends on the individual income ω. The threshold function, linking π _I to ω, termed C(ω), is monotonic and C ' (ω) < 0, as the function W() is monotonic and W ' (ω) < 0. In addition, the function C() is increasing with κ. Consequently:

h = 1 if ω ≥ C − 1 π I h = 0 else

and the income threshold conditioning protection, C ^− 1(π _I ), decreases with κ. Knowing individual protection behaviors, the aggregated level of protection H (the percentage of protected persons) can be computed by integration as follows:

H = ∫ C − 1 π I + ∞ f ω dω = 1 − F C − 1 π I

Where f is the probability density function of ω (and F the associated cumulative density function), describing the income distribution of the population. Equations (6), (8) and (15) fully describe the dynamics of H and π _I as a function of X.

Nearby the steady-state, the dynamics corresponds to a standard prevalence-elastic behaviour of protection (positive malaria prevalence elasticity), where H is an increasing function of X, because it is increasing with π _I (Equations (8) and (15)). Note that as a consequence, nearby the steady-state, X is not necessarily monotonic in π _I : protection behaviors and epidemiological dynamics go in opposite directions. Indeed, combining Equations (6) and (15) it follows that:

X = F C − 1 π I π I

As a result, this is consistent with standard results in economic epidemiology 31 . Thus, Equations (15) and (16) provide us with some economic determinants of protection at individual and aggregated levels, that could be possibly tested (as studied in next section). For a given probability of infection in absence of protection, protection decreases with the unit costs of ITN/LLIN, κ (through the function C ^− 1). It also decreases with poverty, as the poorer the individuals, the higher their marginal utility of income.

The main question to be solved, concerning the long-term properties of this model at the steady-state, is whether a malaria trap can persist in the long run, in spite of the availability of ITN/LLINs as protection tools since the higher the unit cost κ of ITN/LLINs, the lower the protection. This is why ITN/LLINs programs are usually based on subsidized ITN/LLINs prices. Let us then consider the best case of almost full subsidization, when κ → 0 (i.e. the extreme case being free distribution). Conditions under which, for any positive unit cost κ, the malaria trap persists are given below:

For any κ > 0, when κ → 0 the long term equilibrium corresponds to a malaria trap, if and only if:

R 0 > 1 F Ω 1 − m F Ω

where F(Ω) is the proportion of persons under the extreme poverty line in a population, also called the extreme poverty incidence. Note that m depends on H, the proportion of protected persons (Equation (7)), which depends itself on income (Equation (15)), and, thus, on the extreme poverty incidence.

For κ → 0, individuals use protection if and only if they are above the extreme poverty line. Given Eq. (6), (8) and (15), it follows that:

H → 1 − F(Ω), i.e. only persons over the extreme poverty line will be protected then, X → F(Ω)π _I i.e. only persons under the extreme poverty line will be infected at rate π _I

and

π I → Q X , m 1 − F Ω

Thus, when κ → 0, the long term equilibrium can be mathematically analyzed along the same line as in the pure epidemiological model at the steady-state, after substitution of function Q by:

F Ω Q X , m 1 − F Ω

This implies that the malaria trap persists if and only if the slope of this function at origin (X = 0) is higher than 1, i.e.:

R 0 = F Ω m 1 − F Ω a 2 bc rg > 1 QED

Given κ → 0 and H → 1 − F(Ω), as the vector density m is a decreasing function of H, the higher the incidence of extreme poverty F(Ω), the higher the risk of persistence of a malaria trap. Hence, a malaria trap will persist for high enough values of the basic reproduction number R ₀ and of the extreme poverty incidence F(Ω), even when ITN/LLINs are highly subsidized. In the extreme case where all the population is at or below the extreme poverty line, the condition above corresponds exactly to the basic reproduction number, and hence this policy is certainly ineffective (R ₀ > 1).

One could argue that if ITN/LLINs were provided at no cost to individuals (κ = 0), then all individuals including the extreme poor, would use them. Distribution of ITN/LLINs for free would then possibly be a much more efficient policy to reduce malaria, compared to selling ITN/LLINs at a subsidized price. This is in line with randomized experiments that found that free distribution dramatically increases use of ITN/LLINs (as well as other important products for the poor), compared to charging even very small user fees 32 .

However, the assumption κ = 0 is merely theoretical, even though it can be possibly obtained in controlled experiments, as in practice κ is not merely the price of ITN/LLINs: it involves also all opportunity costs attached to using them for other productive activities. Selling, exchanging, discarding or re-using the material from ITN/LLINs is not uncommon. For instance, misuse of ITN/LLINs for profit (drying fish and fishing) has been observed by Lake Victoria 33 and in Zambia 34 . In some cases, nets have even been turned into wedding dresses and water filters.

The previous model describes protection behaviours and the existence of theoretical conditions under which a malaria trap persists. As stated above, protection should (a) increase with prevalence of malaria (i.e. positive malaria prevalence elasticity), (b) decrease with an increase of economic cost of protection and (c) decrease with an increase of the incidence of extreme poverty. Prediction (a) is testable with existing individual data as examined below. It corresponds to relation (1) of Figure 1. Prediction (b) is hardly testable for lack of data on cost of protection at individual level. However, prediction (c) is testable, and provides a way to study economic determinants of protection. Finally, the model predicts that, in a country with high extreme poverty incidence, the policy of dissemination of subsidized ITN/LLINs, sold at a small but strictly positive unit price, can be ineffective, insofar as the corrected basic reproduction number is increasing with extreme poverty incidence. As classically done, three regression models (ordinary least square estimators, OLS) have been used to estimate explanatory variables of (i) poverty, (ii) malaria prevalence, (iii) use of protection. Each of these 3 models includes, as predictor, the two remaining variables, plus cofactors to be adjusted on. But, none of these models takes into account endogeneity, which leads to estimation bias. Endogeneity is used in economics to describe the presence of an endogenous explanatory variable in a multiple regression model, i.e. a variable that is correlated with the error term, either because of an omitted variable, measurement error or reverse causality 35 . Indeed, classical regression models explaining malaria prevalence and protection use are certainly biased due to endogeneity. As illustrated in Figure 1 reverse causality is actually a highly plausible bias. In order to solve this problem, instrumental variables techniques have been used to deal with these potential biases 35 36 37 38 . Consequently, given the strong possibility of endogeneity of poverty incidence, malaria prevalence and protection, and given that the error terms are not necessarily independent of each other, the 3SLS estimates are useful, providing estimates that are free of endogeneity bias. The complete system of structural equations was estimated with a heteroskedastic-efficient 3SLS two step generalized method of moments (3SLS GMM) described by Wooldridge 35 . The system is illustrated in a simplified way with Equation (21) below and Figure 1:

F Ω = α 1 + β 1 a X + β 1 b Poverty Incidence IVs + β 1 c Regions + ϵ 1 X = α 2 + β 2 a F Ω + β 2 b H + β 2 c Malaria IVs + β 2 d Regions + ϵ 2 H = α 3 + β 3 a X + β 3 b F Ω + β 3 c Protection IVs + β 3 d Regions + ϵ 3

Where the βs are the coefficients associated with the corresponding factors (or vectors of factors) included in each equation, IVs stand for specific Instrumental Variables enabling to identify a causal effect (described below), Regions are a set of dummy variables in order to control for confounders linked to malaria regional control programs, and ϵs are the standard disturbance terms. Other variables remain unchanged with respect to previous notations.

Instrumental variables for endogenous variables are distance to nearest market and remote location (for poverty: the poorest of the poor), altitude, longitude, latitude (for malaria prevalence), distance to nearest health center, % of houses sprayed against mosquitoes in last 12 months (for ITN use). Similar geographic instruments were already used for malaria prevalence 5 20 and the availability of health facilities is generally a good candidate as instrumental variable for endogenous health indicators 39 . Household distance to nearest market is supposed to independently affect economic activity and resources as in other examples. In a three-equation model, the order condition for identification requires that there is at least two endogenous or exogenous variables excluded from each equation, which is the case here 40 . The Hansen J test was used to test the over-identifying restrictions i.e. the validity of the instrumental variables 35 . A rejection of the null hypothesis implies that the instruments are not satisfying the orthogonality conditions required for their employment (i.e. they are uncorrelated with the error term of the estimated equation). The predictions of the model have been tested on DHS (Demographic and Health Surveys) household members data from Uganda (2009) aggregated at the community/village level. Note that DHS are generally not longitudinal surveys. Unfortunately, the selected clusters or individuals within a same country generally change from one survey to another. To date, MIS (Malaria Indicator Surveys), that provide at the same time malaria infection prevalence (through Paracheck tests) and ITNs use for children under five, are extremely recent and cover only five African countries 41 : Liberia (2008–2009) Angola (2006), Senegal (2008–2009), Uganda (2009) and Tanzania (2007–2008), hence Ugandan data were used 23 . The choice of the survey highly depends on the kind of information available in each survey. Potential reliable instrumental variables were not available in Angola, Liberia, Tanzania and Senegal. In Uganda, 28.67% of the population lives with less than the threshold of $1.25 a day (in PPP 42 ). The extreme poverty line was, therefore, defined using this threshold and the wealth index provided by DHS. Poverty incidence was calculated/estimated relatively to this extreme poverty line. Malaria prevalence was defined as the percentage of people tested positive for malaria with RDTs amongst the population tested in the survey. An ever-treated net is (i) a factory-treated long-lasting insecticidal mosquito net that does not require any further treatment, OR (ii) a factory net, with or without an insecticide kit, which has subsequently been soaked with insecticide at any time, OR (iii) a homemade net which has subsequently been soaked with insecticide at any time. Note that ordinary least-square models (OLS) have been kept in order to highlight the endogeneity bias.

Table 1 shows that reasons generally advanced in the literature for not using the nets, do not hold in the Uganda DHS report where these questions were asked. The main reason of not using nets was the heat (for 15% of the sample). Note that, ITN/LLIN are more used against nuisance of Culex or Aedes than Anopheles. OLS regression (Table 2), not taking into account endogeneity, showed no significant prevalence-elastic behaviour (i.e. relationship (1) of Figure 1): protection behaviour was not significantly related to malaria prevalence, nor poverty incidence (Table 2 column 1), adjusted on regions, but malaria prevalence was significantly related to poverty incidence (β=0.376, p<0.001) (Table 2 column 2 and 3). Furthermore, protection behaviour changed through region, as well as malaria prevalence and poverty incidence. Conversely, once the endogeneity problems were solved, a prevalence-elastic relationship between malaria and protection was found: protection behaviour was significantly related to malaria prevalence (β=0.438, p=0.012), and negatively related to poverty incidence (β=−0.323, p=0.096) (Table 2, column 4). Malaria prevalence remains significantly related to poverty incidence at a higher magnitude (β=0.543, p<0.001). Prevalence-elastic behaviour was defined in economic epidemiology as an increased protection behaviour in response to an increase in disease prevalence. Hence one of the questions tackled previously was how changing malaria affects protection? Regression (4) of Table 2 thus confirmed a significant causal positive effect of malaria on protection (ITN use), as far as the instrumentation strategy (use of instrumental variable for adjustments) is validated by usual tests. The Hansen J Test of over-identifying restrictions showed that the instruments were valid in 3SLS GMM regressions (Table 2).

Percentage of households with at least one mosquito net that was not slept under the previous night, and among those, percentage reporting various reasons for not using a net for sleeping the previous night, by background characteristics 23 .

The coefficients attached to each variable are presented (standard errors, adjusted for heteroscedasticity in parentheses). All regressions include regional dummies. ***denotes statistical significance at the 1% level, ** at the 5% level, * at the 10% level. The Hansen J Test of overidentifying restrictions shows that the instruments are well identified in 3SLS GMM regressions (Hansen's J chi2= 12.247; p value = 0.140). A rejection of the null hypothesis implies that the instruments are not satisfying the orthogonality conditions required for their employment (i.e. that they are uncorrelated with the error term of the estimated Equation).