Multi-year optimization of malaria intervention: a mathematical model

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by: Harry J. Dudley, Abhishek Goenka, Cesar J. Orellana and Susan E. Martonosi

Abstract

Background

Malaria is a mosquito-borne, lethal disease that affects millions and kills hundreds of thousands of people each year, mostly children. There is an increasing need for models of malaria control. In this paper, a model is developed for allocating malaria interventions across geographic regions and time, subject to budget constraints, with the aim of minimizing the number of person-days of malaria infection.

Methods

The model considers a range of several conditions: climatic characteristics, treatment efficacy, distribution costs, and treatment coverage. An expanded susceptible-infected-recovered compartment model for the disease dynamics is coupled with an integer linear programming model for selecting the disease interventions. The model produces an intervention plan for all regions, identifying which combination of interventions, with which level of coverage, to use in each region and year in a 5-year planning horizon.


Results

Simulations using the model yield high-level, qualitative insights on optimal intervention policies: The optimal intervention policy is different when considering a 5-year time horizon than when considering only a single year, due to the effects that interventions have on the disease transmission dynamics. The vaccine intervention is rarely selected, except if its assumed cost is significantly lower than that predicted in the literature. Increasing the available budget causes the number of person-days of malaria infection to decrease linearly up to a point, after which the benefit of increased budget starts to taper. The optimal policy is highly dependent on assumptions about mosquito density, selecting different interventions for wet climates with high density than for dry climates with low density, and the interventions are found to be less effective at controlling malaria in the wet climates when attainable intervention coverage is 60 % or lower. However, when intervention coverage of 80 % is attainable, then malaria prevalence drops quickly in all geographic regions, even when factoring in the greater expense of the higher coverage against a constant budget.


Conclusions

The model provides a qualitative decision-making tool to weigh alternatives and guide malaria eradication efforts. A one-size-fits-all campaign is found not to be cost-effective; it is better to consider geographic variations and changes in malaria transmission over time when determining intervention strategies.


Keywords

Malaria policy Operations research Compartment model Integer programming

Background

Malaria remains a lethal disease affecting an estimated 200 million people and killing 627,000 in 2012 [1]. There are a variety of interventions for treating or preventing malaria infection, but the use of these interventions is hindered by scarcity of resources. Mathematical models provide a useful tool for evaluating intervention strategies and studying the relative effectiveness of interventions. These evaluations will become increasingly useful as success with malaria elimination is predicted to change transmission dynamics. In fact, the WHO Global Malaria Programme cites the specific need for operations research models to determine the best intervention strategies in areas where transmission dynamics are changing as malaria is being eliminated [2].

In this paper, an integer linear program (ILP) and a coupled susceptible-infected-recovered (SIR) compartment model are developed to create a decision-making tool for planning future interventions. The model suggests the best strategy for minimizing person-days of malaria infection over a 5-year period given an initial population, cost of each intervention, and a budget constraint. The model allows for the possibility of a malaria vaccine in combination with other interventions. Simulations are performed in which the budget, the efficacy of the interventions, and their cost are varied to determine the sensitivity of the optimal policy to these parameters.


Interventions

There are many existing methods to prevent or treat malaria infection. The model will consider the following five interventions and their combinations.

Long-lasting insecticidal nets (LLINs) cover sleeping individuals during the night when mosquito biting can be highest. When intact, the nets block mosquitoes from reaching humans. The insecticides work by deterring mosquitoes from feeding and by killing female mosquitoes that come in contact with the net. LLINs can remain effective for multiple years [3]. In fact, the WHO Pesticide Evaluation Scheme 2005 guidelines state that LLINs should survive at least 3 years of recommended washing and use [4].

Indoor residual spraying (IRS) is another insecticidal prevention method. IRS is believed to deter mosquitoes from entering sprayed areas and to kill female Anopheles mosquitoes that rest on sprayed surfaces after feeding. (Resting after feeding is a hallmark of some mosquito species while others prefer to rest outdoors [5]). Historically, IRS with an insecticide called dichlorodiphenyltrichloroethane was effective in reducing malaria in Europe, Asia, and Latin America. However, as insecticide use increases, insecticide resistance has been observed in some mosquito populations in Africa, and new insecticides must be used [1].

Intermittent preventive therapy (IPT) is the regular administration of a drug like sulfadoxine–pyrimethamnine to decrease morbidity due to malaria in infants, children, and pregnant women. IPT decreases the chance of developing symptoms after being bitten by an infected mosquito [6]. There is evidence that children withstand acute infection better than adults. However, in endemic areas, adults develop acquired immunity from repeated exposures, and children remain more susceptible to high levels of parasitaemia (parasite density in the blood) [7]. Most of the 627,000 people killed by malaria in 2012 were children in Africa, so giving IPT to infants, children, and pregnant women treats the most vulnerable population while limiting the risk of spreading drug resistance [1].

Artemisinin combination therapy (ACT) can be used to treat a patient after they contract malaria. This is the best treatment for uncomplicated P. falciparum malaria when confirmed by rapid diagnostic tests (RDT) [1, 8]. ACT kills the parasites that cause symptoms and may destroy or disable the gametocytes that are responsible for infecting mosquitoes [9]. Both these factors mean that ACT increases the recovery rate.

Many malaria vaccines are in development, and one has gone through Phase III clinical trials. The complex life-cycle of the malaria parasite makes it possible to intervene at many stages. Vaccines that target different forms of the parasite will operate by different mechanisms, but in general, a vaccine would decrease the chance of developing symptoms and increase the recovery rate if infected. The leading malaria vaccine candidate is the RTS,S malaria vaccine. It is an antigen composed of the RTS and S proteins. The RTS,S vaccine is a pre-erythrocytic vaccine that presents circumsporozoite protein (CSP) from malaria sporozoites to the immune system. CSP is a parasitic surface protein that is an important part of the invasion of hepatocytes by sporozoites [10]. Such a vaccine will decrease the probability that a susceptible person becomes infected after a bite from an infectious mosquito. Moreover, it is believed the vaccine could increase a person’s recovery rate by increasing their exposure to asexual blood-stage parasites, thereby boosting their immunity [10]. (By contrast, a transmission-blocking vaccine that acts in mosquitoes would decrease the probability of transmission from an infectious mosquito but would not change the human recovery rate).


Literature review

This paper extends a single-stage optimization model of Dimitrov et al. Their model divides the country of Nigeria into approximately 270,000 cells and chooses one action (either a single intervention or no intervention) for each cell over a year, subject to budget constraints, to minimize societal costs caused by malaria infection. The model also identifies optimal locations for supply distribution centres. They treat the societal benefit of each intervention as an exogenous parameter that depends on geographic characteristics. This allows their model to consider geographic variability in malaria dynamics [11].

However, because malaria dynamics depend on the fraction of the population that is infectious, a quantity that the interventions are themselves trying to reduce, the framework of Dimitrov et al. does not permit the examination of multiyear efforts against malaria in which the optimal policy might vary over time as the malaria dynamics shift. This paper extends the optimization model above to select interventions (or combinations thereof) over multiple years by explicitly incorporating malaria disease dynamics over time in response to those interventions. This is a novel approach that combines two areas of mathematics that do not regularly interact: ILP from the area of operations research and differential equations modelling from the area of mathematical epidemiology.

There is a long history of mathematical models of malaria transmission, going back to the work of Sir Ronald Ross in the early 1900s [12, 13]. In recent years, malaria has drawn significant attention from the academic community. Epidemiologists have traditionally modelled the spread of malaria in a population using variations on the SIR model to capture different aspects of the disease. Mandal et al. survey the models found in the literature and offer a hierarchy based on model complexity [13].

In order for the model presented here to make informed choices about which interventions to distribute, the dynamics of how disease transmission change after treatment interventions must first be understood. Lindblade et al. and Killeen et al. study the protective effect of insecticide-treated nets or LLINs [16]. Bousema et al. investigate how ACT reduces the circulation time of gametocytes, thereby reducing infectiousness [16]. Garner and Graves examine the community benefits of ACT [17]. Chandramohan et al., Grobusch et al., and Aponte et al. quantify the protective effects of IPT for infants [6, 18, 19]. Pluess et al. review the effects of IRS [5]. These results are used to inform the model’s choice of disease transmission parameters, as described later under “Effects of interventions on SIR parameters” section.

The model presented here includes in its portfolio of interventions a vaccine that is currently in development. Prosper et al. model the interaction between vaccine- and naturally-acquired immunity using a five-compartment model. Their model augments the S, I, and R classes with a partially-immune (due either to vaccination or natural immunity) susceptible class and a moderately-infectious class for infected, partially-immune individuals. They find that disease burden can be decreased only if a highly effective vaccine is coupled with a policy of actively treating asymptomatic infections in partially immune individuals [20]. Bojang et al. report there is minimal potential effect for a malaria vaccine given to adult men, and Asante et al. study the positive potential protective benefits of administering the vaccine to children [21, 22].

There are extensions to the SIR framework that are not considered here. Koella and Antia model the reduced efficacy of interventions due to the spread of drug-resistant strains of malaria [23]. The model presented here does not incorporate drug-resistance, so any policy recommended by the model should be evaluated in this context. Other researchers, for example Dawes et al. [24] and Koudou et al. [25], focus on the mosquitoes’ plasmodial transmission dynamics by analysing the effects of interventions on mosquito morbidity and mortality rates and the usefulness of the resulting manipulation of said rates. The changing mosquito population is not modelled explicitly; instead the effects of interventions on the mosquito population are represented as changes in the parameter values used in the human SIR model.

While the above references provide detailed models of malaria’s complex dynamics, this paper presents a simple SIR model that accommodates the effects of several types of interventions, while maintaining the computational tractability required by the optimization model. In the next section, the model and simulation approach are described in greater detail.

Methods

This paper considers the problem of allocating malaria treatments to many regions when limited by scarce resources. There is assumed to be a fixed annual budget shared across several geographic regions having different initial incidences and transmission rates of malaria and different unit costs for distributing treatment. A portfolio of interventions can be selected, including some in combination, each having its own effects on malaria transmission. Each intervention is selected at a particular coverage, which is the percentage of the population that receives the intervention and uses it correctly. Social and economic losses are assumed to be proportional to the time spent infectious, so person-days of malaria infection is the chosen measure of the malaria burden. The model identifies the optimal sequence of interventions and corresponding coverage percentages for each region and each year that minimizes the total infected person-days over a fixed time horizon.

An integer linear programming optimization model (ILP) suggests the best set of interventions in each year to minimize person-days of malaria infection over all time steps. The ILP takes as input the number of person-days of malaria infection that occur when a given intervention is used on a population with a given initial prevalence of malaria. The person-days of malaria infection is estimated by a SIR differential equations model of malaria transmission dynamics.

Integer linear programming (ILP) model

The ILP relies on several sets, parameters, and decision variables, which are defined here.

Sets

Geographic regions Because the cost of distributing an intervention to a particular district depends on its infrastructure and ease of access to treatment, and the malaria transmission dynamics depend on its climate, districts are grouped into geographic regions, denoted by index g. The optimization model determines the number of districts in each geographic region to receive a particular sequence of interventions.

Population states A population state, p, is a triplet, (S, I, R), that indicates the percentage of a district’s population susceptible to (S), infected by (I), or recovered from and temporarily immune to (R) malaria. Each district begins a year in a particular population state and ends in a new population state that depends on how the chosen intervention affects the malaria disease dynamics. (The model for determining the disease progression is described in the “Differential equations (DE) model” section).

Actions The set of actions is the set of possible choices of intervention (including certain combinations of interventions, or the possibility of applying no intervention). The choice of intervention at a determined coverage level in a district is referred to as an action, denoted by index i.


Link to Website:https://malariajournal.biomedcentral.com/articles/10.1186/s12936-016-1182-0


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