Maximizing the impact of malaria funding through allocative efficiency: using the right interventions in the right locations

Background The high burden of malaria and limited funding means there is a necessity to maximize the allocative efficiency of malaria control programmes. Quantitative tools are urgently needed to guide budget allocation decisions. Methods A geospatial epidemic model was coupled with costing data and an optimization algorithm to estimate the optimal allocation of budgeted and projected funds across all malaria intervention approaches. Interventions included long-lasting insecticide-treated nets (LLINs), indoor residual spraying (IRS), intermittent presumptive treatment during pregnancy (IPTp), seasonal mass chemoprevention in children (SMC), larval source management (LSM), mass drug administration (MDA), and behavioural change communication (BCC). The model was applied to six geopolitical regions of Nigeria in isolation and also the nation as a whole to minimize incidence and malaria-attributable mortality. Results Allocative efficiency gains could avert approximately 84,000 deaths or 15.7 million cases of malaria in Nigeria over 5 years. With an additional US$300 million available, approximately 134,000 deaths or 37.3 million cases of malaria could be prevented over 5 years. Priority funding should go to LLINs, IPTp and BCC programmes, and SMC should be expanded in seasonal areas. To minimize mortality, treatment expansion is critical and prioritized over some LLIN funding, while to minimize incidence, LLIN funding remained a priority. For areas with lower rainfall, LSM is prioritized over IRS but MDA is not recommended unless all other programmes are established. Conclusions Substantial reductions in malaria morbidity and mortality can be made by optimal targeting of investments to the right malaria interventions in the right areas. Electronic supplementary material The online version of this article (doi:10.1186/s12936-017-2019-1) contains supplementary material, which is available to authorized users.


Population and compartments
The compartments and indices are defined as:  Human compartments: S, susceptible; E, exposed (liver-stage); I, infectious (blood stage); R, recovered (temporarily immune);  Mosquito compartments: ̂, susceptible; ̂, exposed; ̂, infectious;  Subscript represents the different geopolitical regions in the setting of the application;  Superscript represent the different population groups (0-5 year old children, pregnant women, other).
For humans, deaths are assumed to occur at equal rates from the S, E and R compartments (equal to the all-cause mortality rates for that population group), but with an increased rate from the I compartment. Mosquito mortality is assumed to be constant in the absence of specific interventions or environmental changes. Total human population sizes and growth rates were taken from UN population data [1,2] for each population group and region, and population growth was assumed to occur in the susceptible compartment. Explicitly, these are given by:  = human mortality rate for population in the S, E and R compartments  i + Γ j = human mortality rate for population in the compartment (i.e., mortality due to infection is increased by an amount Γ )  ̿̿̿ = growth rate for population Mosquito population sizes were maintained by assuming birth rates were equal to death rates; however the total population size was scaled between seasons depending on a seasonality factor:  ̂= mosquito mortality rate from each compartment [1 / life expectancy]. It is also assumed that mosquitoes are born into the ̂ compartments at the same rate to maintain a constant population.
 ̂= * (1 + sin ( 2 365 )) is the total mosquito population size, where is a seasonality constant for region . represents the difference in mosquito population density between the wet and dry seasons.
For the northern regions of Nigeria, the entomological inoculation rate (infective bites per person per night) was estimated to be 0.37 in the dry season, 1.24 in the rainy season compared to an annual average of 0.83 [3]. Therefore was calibrated to ensure that in the northern regions: (peak_incidence -min_incidence) / average incidence = (1.24 − 0.37)/0.83 = 1.05 while was set to be zero in the southern regions based on considerably smaller seasonal effects [4].

Transmission
Transmission is defined in terms of bites per month, probability of transmission per bite, and the current infectious proportion of the population. Let  = bites per month among population group in region (calibrated to malaria incidence among population group in each region)  ̂= bites per month in mosquito population for region , where the total bites among all humans balances the total bites among mosquitoes:

Other parameters
Movements occur between compartments at the following rates:


= average duration of immunity among population group in region ; Because the duration of immunity is extended through repeated exposure to parasite, the waning of immunity needs to account for the background entomological inoculation rate. Malaria modelling literature suggests that it should be modelled in the form [5][6][7]:

Model equations
The equations describing the human and mosquito models are as follows.

Programmatic responses considered and their implementation
The following programs and variations of programs were considered. Where possible, parameters were estimated for individual regions and population groups, to account for differences in the cost and effectiveness of delivering programs in different areas and to different people.
Reviews of published and grey literature were undertaken to obtain data for each program. Where no or limited estimates were available for specific regions or population groups, country estimates were used. Parameter estimates and references are shown in Table A1 and Table A2.

Long lasting insecticide treated nets (LLINs)
For each region and population group, LLIN programs have two features: coverage (the percentage that have an LLIN) and utilization (of those who have an LLIN, the percentage that use it). Studies have shown that in areas where they are used, LLINs can reduce the mosquito biting rate the mosquito biting rate N due to barrier protection; and the mortality rate among mosquitoes ̂ due to insecticide treatment. If LLINs have effectiveness in reducing bites of and they are treated with chemicals that have effectiveness at killing mosquitoes, then the biting rate for each population group and region is given by and the mosquito mortality for each region is given by where the underlined and ̂ values represent the biting rate and mosquito mortality rate in the absence of LLINs. Note that the biting rate applies to each population group whereas the mosquito mortality is regional.
Therefore the biting rates and mosquito mortality varies when the coverage or utilization of LLINs changes. The coverage of LLINs (in each region and among each population group) can be increased with additional spending on nets, and a logistic curve was used to relate spending to coverage which assumed a saturation value of 95% of the total population. The utilization of LLINs can be increased with increased coverage of a behavioural change and communication (BCC) program. This in turn requires additional spending on the BCC program; see section on the BCC program.
LLINs are required to be replaced every 5 years, and so we modelled the coverage to reduce by 1/5th each year.

Indoor residual spraying (IRS)
Coverage of IRS acts on two parameters in the model: the mosquito biting rate N, due to targeted killing of mosquitoes who approach humans; and the mortality rate among mosquitoes . If IRS has an effectiveness in reducing bites of and the chemicals used have an effectiveness at killing mosquitoes, then the biting rate for each population group and region and mosquito mortality for each region are given by (similarly to LLINs): where the underlined and ̂ values represent the biting rate and mosquito mortality rate in the absence of IRS. Note that in comparison to LLINs, we would expect > , and < .
The coverage of IRS (in each region and among each population group) can be increased with additional spending, and a logistic curve was used to relate spending to coverage which assumed a saturation value of 95% of the total population. IRS needs to be renewed annually, and so every year the coverage reduces to zero unless ongoing spending occurs.

Chemoprophylaxis in pregnancy (IPTp)
Coverage of IPTp applies only to the population of pregnant women (i.e. j value corresponding to this population), and has both a coverage (the percentage of pregnant women who have one or more doses) and utilization (defined as the percentage of pregnant women who had at least three doses, among those who had at least one). It acts firstly to move a proportion from the and R compartments to the S compartment, and secondly to reduce the force of infection by a factor (1 − ) where is the effectiveness of IPTp in reducing new infections among pregnant women. Therefore, the biting rate among pregnant women in each region is given by where the underlined value represents the biting rate in the absence of IPTp. The coverage of IPTp (in each region) can be increased with additional spending, and a logistic curve was used to relate spending to coverage which assumed a saturation value of 95% of pregnant women. The utilization of IPTp can be increased with increased coverage of a BCC program. Increasing coverage of the BCC program in turn requires additional spending; see section on the BCC program. When utilized, IPTp is assumed to last for the duration of pregnancy.

Seasonal mass chemoprevention in children (SMC)
Coverage of SMC applies only to the population of children (i.e. j value corresponding to this population) and only applies to the North East and North West geopolitical zones, where malaria is highly seasonal. Similarly to IPTp, SMC acts firstly to shift a proportion of children from the and R compartments to the S compartment, and secondly to reduce the force of infection by a factor (1 − ) where is the effectiveness of SMC in reducing new infections among children. Therefore, the biting rate among children in each region is given by where the underlined value represents the biting rate in the absence of SMC. The coverage of SMC (in each region) can be increased with additional spending, and a logistic curve was used to relate spending to coverage which assumed saturations values of 78% of children in the NW region, 50% of children in the NE region, 4% of children in the NC region and 0% of children in other regions, based on the proportion of each region that is suitable for the program (which requires seasonal malaria). When utilized, SMC is assumed to give protection for a year.

Mass drug administration (MDA)
Similarly to IPTp and SMC, acts firstly to shift a proportion of people from the and R compartments to the S compartment, and secondly to reduce the force of infection by a factor (1 − ) where is the effectiveness of MDA in reducing new infections. Therefore, the biting rate among each population group and in each region is given by where the underlined value represents the biting rate in the absence of MDA. The coverage of MDA (in each region) can be increased with additional spending, and a logistic curve was used to relate spending to coverage which assumed saturations values of 78% of the population in the NW region, 50% of the population in the NE region, 4% of the population in the NC region and 0% of the population in other regions, based on the proportion of each region that is suitable for the program.

Larval source management (LSM)
Coverage of LSM acts to reduce the population density of mosquitoes in the model. This in turn will reduce the biting rate of mosquitoes on the human population. If LSM has an effectiveness in reducing bites of , then the biting rate for each population group is given by: where the underlined value represents the biting rate in the absence of LSM. The coverage of LSM (equal among all population groups) can be increased with additional spending, and a logistic curve was used to relate spending to coverage which assumed a saturation value of 25% of the total population given the need for areas to be suited.

Behavioural change and communication (BCC) program
Education campaigns in Nigeria are typically operated through churches and mosques or targeted to healthcare workers, and provide information on the proper use and importance of various malaria prevention and treatment programs. This includes correct usage and replacement of LLINs, in particular for high-risk groups such as children and pregnant women, as well as the benefits of programs like IPTp.
Changing the coverage of BCC programs influence several parameters in the model. For those who receive education, there has been a documented increase in utilization of LLINs of = 20%, and IPTp of = 30% [8].
Coverage of BCC program was assumed to be nested. This means that as the coverage increased, it was assumed to do so among individuals who had access to LLINs or IPTp first, before reaching the population who were not covered by these programs. This was done in preference to random interactions, as it was believed that having access to prevention programs was likely to be correlated with access to the education program. This also meant that if the coverage of the BCC program increased beyond the coverage of the LLIN or IPTp programs, then it had no additional effects on utilization in the model. A logistic curve was used to relate spending on BCC to coverage which assumed a saturation value of 95% of the total population.

Combining the effects of interventions
When multiple interventions were allocated to the same population they were assumed to do so at random; namely, the likelihood of an individual within a population receiving an intervention was independent of the coverage of other interventions. The exception to this was the BCC program, which was assumed to be preferentially provided to people who had LLINs or pregnant women who had started IPTp ( Figure A2). Therefore, when multiple program coverages were changed, the change in mosquito biting rate, mosquito mortality were updated iteratively.

Model calibration and validation
For each population group and region, an optimization algorithm [9] was used to calibrate parameters for the biting rate ( ), the proportion who develop immunity following infection ( ), the duration of immunity ( ), and the malaria specific mortality rate (Γ i ), so that at equilibrium the model outcomes best fit available data for the 2015 malaria incidence, mortality and Plasmodium falciparum prevalence. The algorithm was based on minimizing the sum of squared errors between the modelled outcomes and the data.
The short malaria lifecycle and rapid changes to intervention coverage over the previous five years dominates any background trends in incidence and mortality, making it suitable to start a model from equilibrium for forward projections. However, the effects of changes in intervention coverage on epidemiological outcomes still needed to be validated. This was done by separately calibrating the model to 2010 data and program coverages (the only other year data was available), and then linearly changing the coverage of programs to 2015 values while running the model over this fiveyear period. The resulting model estimates for 2015 incidence, prevalence and malaria-attributable deaths among each region and population group were then compared to 2015 data estimates.
Model calibration results are shown in Figure A2.

Cost-coverage curves
The key assumptions of resource optimization are the relationships between (a) the cost of prevention and treatment programs; (b) the resulting coverage levels among targeted populations; and (c) how these coverage levels influence clinical and epidemiological outcomes. The relationship between (b) and (c) is described above for each program, however the link between (a) and (b) is also required to understand how incremental changes in spending are likely to affect outcomes (e.g. total incidence and total mortality).
Most programs typically have a period of effective scale-up as funding is increased, but attaining very high coverage levels involves reaching the most difficult to reach groups, which requires increased incremental investment for demand generation and related activities (i.e. there is a saturation effect with increased funding). These effects are incorporated in the model using logistic function cost-coverage curves ( Figure A3). Logistic functions allow coverage to saturate at high spending levels, thus better reflecting the program reality. The relationship between spending and coverage was therefore calculated as where C(x) is the number of people covered for a given amount of budget x; B is the saturation coverage (upper asymptote of Figure A3), P is the target population size and u is the unit cost. Changes in coverage expected from changes in program funding are assumed by interpolating and extrapolating available data using the fitted logistic curve. These are in turn related to changes in the model parameters as described in the programmatic responses section above. Figure A3: Example logistic cost-coverage curve. When spending on a program becomes very high, only marginal gains in coverage can be made as a saturation effect occurs.

Regional specific optimizations
For each region, an existing optimization algorithm [9] was used to incrementally shift the current available budget (Table C1) between programs (treatment, LLINs, IRS, IPTp, SMC, MDA, LSM) and population groups (children 0-5 years, pregnant women, general population) in order to achieve (a) a minimum 5-year incidence, and (b) a minimum 5-year mortality. This was then repeated for different total available budgets.

Geographical optimization
Once the regional estimates had been completed, a cost-effectiveness curve could be formed for each region. These curves describe the additional incidence (or mortality) averted for incremental increases in total funding to each region ( Figures D8-D9). The same optimization algorithm was then used to minimize (a) total country-level incidence and (b) total country-level mortality by shifting funding between regions according to the relative cost-effectiveness ratios for each region. Where total funding within a region changed, it was allocated to programs and population groups optimally according to the region specific optimizations.

Sensitivity analysis
Once the geospatially optimal allocation of funding was determined for each scenario, a multivariate sensitivity analysis was conducted to estimate bounds for the impact of allocating funding in this way (i.e. bounds for the number of deaths and cases prevented). 95% bounds were obtained for the number of deaths or cases averted using Monte Carlo sampling for the model's structural parameters ( ,, ,̂), unit costs and the effects of programs, with samples taken from Normal distributions around point estimates (standard deviations of 5%, truncated at 10% above and below).  [19], and during pregnancy [20]. Note Cochrane review has LLIN reducing incidence by 50% [21], not necessary bites.

Effectiveness of LLIN in killing mosquitoes Global parameter 19%
Treatments kill 41% of the 46% of mosquitoes not repelled that landing on net [19,22]. Note that this is independent of the reduction in mosquito bites.
Efficacy in randomized controlled trials ranges from 75-87% [27,28]. Proportion carrying parasites who clear them Global parameter 95% [19] Effectiveness of MDA in preventing new infections Global parameter 90% [19] Effectiveness of LSM in reducing the biting rate Global parameter 52% [29] Effectiveness of BCC in increasing LLIN utilization Global parameter 20% Effects would be highly variable based on local context and implementation, with these values cautiously estimated from the higher reported success of one faith-based program [8] Effectiveness of BCC in increasing IPTp utilization Global parameter 30% Effects would be highly variable based on local context and implementation, with these values cautiously estimated from the higher reported success of one faith-based program [8]  Median cost of a years worth of protection was $4.03; however only 13% of this cost was for the treatment (the rest being for infrastructure and personnel) [37].
LSM $1.65 [38] Cost of a years' worth of maintenance in suitable areas. Unit cost for larviciding includes some other potentially cheaper larval control methods like reengineering of irrigation to have less open standing water.
BCC $0.03 [8] NIFAA program trained 6,500 faith leaders; when the total NIFAA budget is divided by 2/3 (for the 2 out of 3 states where implementation took place), the cost to reach and train each faith leader was estimated to be $0.65. Assuming 20 contacts per faith leader, then the cost per exposure would be $0.65/20 = $0.033.

MDA $5.25 [19]
Walker et al. [19] found the cost in 2012 of three rounds of seasonal malaria chemoprevention using door-to-door delivery was estimated at $6.10 in Senegal and $4.40 in Mali; we used the average of these two estimates.

Data estimates for each region and population group
The data used to populate the model (population size, population growth rates and all-cause mortality rates) and calibrate the model (incidence, prevalence, testing, treatment and malariaattributable deaths) is shown for each region and population group in Table B3. This data was estimated based on information from a range of sources, as described below.
The total population of each state (and hence geopolitical region) in 2015 was obtained from the Malaria Atlas Project [39,40]. This was then divided among population groups: the number of children 0-4 years (inclusive) as a percentage of the total Nigerian population was obtained from UN Population Data [1,2] from 2000 to 2014, and used to estimate the number of children in each region. This was between 17.2% and 17.5% of the total population depending on the year. The number of pregnant women was estimated using the crude birth rate for Nigeria [41]. The general population size was then taken as the remaining population. UN Population Data [1, 2] from 2000 to 2014 was used to estimate a linear annual population growth rate.
Incidence data (for calibration purposes) was estimated using different methods for each population group. For incidence among children, methods from Cameron et al. [42] were used, which establishes a relationship between the infection prevalence of P. falciparum malaria in Children 2-10 (as reported in the Malaria Atlas Project) and incidence in Children 0-5. This worked out to a linear conversion where for example 60% prevalence converted to 1.5 cases of malaria per child per year. For pregnant women, this was based on the assumption that pregnant women have the same incidence to prevalence ratio as children from Cameron et al. [42], with the prevalence among pregnant women determined by the ratio of 1/1.44 for prevalence in children 0-5 to prevalence in pregnant women implied in van Eijk et al. [43]. Finally, the general population incidence was taken as the total incidence for each region from the Malaria Atlas Project [39,40] minus the incidence among children and pregnant women.
Prevalence among children 0-5 years was assumed to be equal to 2015 P. falciparum prevalence among children 2-10 years from the Malaria Atlas Project [39,40]. Prevalence among pregnant women was then calculated by the ratio of 1/1.44 for prevalence in children 0-5 to prevalence in pregnant women implied in van Eijk et al. [43]. No estimates were available for prevalence among the general population, and in particular, Cameron et al. found no correlation between incidence and prevalence [42]. Therefore, prevalence in the general population was estimated as: (proportion of national child cases in this region * (number of estimated national cases not accounted for by child and pregnant women estimates) / region population.
Number of tests was estimated as incidence multiplied by the percentage of children 0-5 years with a fever who are tested in each state [44] and a weighting factor for each population (we assume that adults with a fever are one third as likely to be tested as children and pregnant women), then normalised across regions to the known total number of tests for all of Nigeria in 2015 from the World Malaria Report [44].
Treatment numbers were estimated as incidence multiplied by the percentage of children 0-5 with a fever who are given (any) malaria treatment in each region [44] and a weighting factor for each population (we assume that adults with a fever are a third less likely to be tested than children and pregnant women), then normalised across regions to the known total number of treatments for Nigeria in 2015 from the World Malaria Report [44].
All-cause mortality rates were taken from UN Population Division data [2]. Regional adult 15-60 mortality rates were scaled for general population 6+ to match national all-cause mortality rate after removing estimated maternal and child mortality. Maternal mortality added for pregnant women.
Malaria-attributable deaths were based on the WHO World Malaria Report 2015 [44] estimates for Nigeria in 2014 (119,000) and their estimate that 73% (292,000 out of 400,000) of deaths in the African Region are in children. These estimated 86,870 child 0-5 deaths were distributed across regions according to the relative P. falciparum prevalence values reported in the Malaria Atlas Project [39,40] (not taking into account that some states are likely to have better treatment and lower mortality for the same number of cases, as no data were available). The remaining deaths were distributed among the general population and pregnant women populations in each region according to their population sizes and relative all-cause mortality rates (which included malariaattributable deaths). Appendix C: Estimated current spending Based on the unit costs of each malaria prevention or treatment program (Error! Reference source not found.), the estimated annual direct cost associated with the current coverage of programs (estimated via the logistic curves, Figure A3) was US$175,351,471 (Table C1) LLINs we have assumed that the cost of achieving this level of coverage was spread evenly over the past five years (given their five-year lifespan). If the majority of currently owned LLINs were purchased in more recent years, then the expenditure in these years would be considerably higher. Third, given the funding for malaria in Nigeria is entirely from donors, there is likely to be substantial variability between years.
The program receiving the greatest amount of funding was LLINs, followed by treatment, while the North West region was the one receiving the most current funding for programs.