1.2 The Quantitative Approach
The goal of malaria control is to prevent malaria. In the long-term, the expectation is that malaria will be eliminated from an ever growing list of places, and that it will eventually be eradicated. All of this revolves around the study of malaria transmission by mosquito populations through blood feeding.
Since malaria is a mosquito-transmitted diseases, malaria control programs need some basic theory to understand how malaria (i.e. any disease that is caused by infection with the parasite) is related to exposure risk, malaria transmission intensity, and the prevalence of infection. In particular, programs should should have a basic understanding that relates the factors driving malaria in a place, the metrics that we use to measure malaria, mosquito population densities and bionomics, and the actions taken by malaria control programs.
A vision for doing all this was articulated by Ross in the first decade of studying malaria transmission. Ross was not the only one to grapple with the complex, multi-factorial nature of malaria transmission and vector control, but what set him apart was his attempt to use of mathematics to illustrate his ideas about malaria. We will use Ross’s writings throughout this chapter to introduce the nature of the challenges of managing malaria control. With the benefit of decades of research and cheap high-speed computing, we are in a better position to fulfill that vision than Ross ever could have.
At the end of this chapter, we make some attempt to put Ross into a broader historical narrative.
In 1899, Ross announced a new phase of his career in a pair of essays on exterminating or extirpating malaria through mosquito control [35,36]. Ross’s initial focus was on larval source management:
…in order to eliminate malaria wholly or partly from a given locality, it is necessary only to exterminate the various species of insect which carry the infection. This will certainly remove the malaria to a large extent and will almost certainly remove it altogether. It remains only to consider whether such a measure is practical [35].
Over the decade, Ross began to use quantitative concepts to understand some very basic questions. Ross was interested in using mathematics to help him reason through problems. Among Ross’s early writings, we find a question posed in 1902, in Mosquito Brigades and How to Organise Them [37]:
It may now be asked, what percentage of diminution in mosquito-borne diseases may be expected to follow a given percentage of reduction in the number of mosquitoes? I regret that I cannot as yet give any actual statistics on the point, but we may perhaps attempt an estimate on a priori grounds. We ask, are we to expect a decrease in the same ratio as the decrease in the number of mosquitoes; or in a dupblicate ratio? The disease will probably diminish in a duplicate ratio? —pp. 56 in [37]
Ross was thus trying to develop quantitative intuition to establish reasonable expectations about malaria transmission and the responses to mosquito control. His guess was that the responses should be non-linear because two bites are required for transmission – one to infect a mosquito and another to transmit it back to humans. In paragraph that follows, Ross explains what he meant by a duplicate ratio.
Now, if we reduce the number of mosquitoes in the locality by one-half, the mosquito bites also will be reduced by one half; and, consequently, only half as many people will now become infected as was formerly the case. But, since the mosquitoes themselves are infected by biting previously infected persons, the percentage of infected mosquitoes … will also be reduced in its turn, because the insects will now find fewer infected persons to bite. Hence, ultimately, the number of infected persons in the locality will be reduced by much more than one-half. In fact, we may perhaps assume that the number of infected persons will be reduced to one-quarter, that is, in the duplicate ratio of the percentage of reduction of the mosquitoes. —pp. 56 in [37]
While Ross was right that there should be a non-linearity, his quantitative logic failed when he tried to use it without going the rigorous process of developing and analyzing a mathematical model. Using mathematical models, we would now argue that, in fact, the expected reduction varies from place to place. The response would depend on the intensity of transmission before doing any control, which scales linearly with the density of mosquitoes. When transmission is very intense, halving mosquitoes would scarcely change the fraction of humans infected. In some places, halving mosquitoes might be enough to end local malaria transmission. If malaria transmission were sustained through malaria importation, then Ross’s answer would not be too far off.
Despite getting the logic wrong – or at the very least, underestimating the complexity of the question – Ross was asking relevant questions. In this case, we was asking about scaling relationships in malaria transmission and their causes. The search for an a priori approach would eventually lead to development of his first mathematical model for malaria transmission in 1908, in Report on the Prevention of Malaria in Mauritius [17]. A clear basis for giving an answer would finally come when Lotka analyzed Ross’s models in 1923 [38–41], and in Macdonald’s analysis of the sporozoite rate [42] and his formula for the basic reproductive number for malaria [43]. The study of these scaling relationships is still evolving.
Ross also discused response timelines. He was aware of differences in duration of infection for yellow fever, lypmhatic filariasis, and malaria. Compared to yellow fever, the responses to vector control should be slow because,
…the parasites remain alive for years after the first moment of introduction by the mosquito…. We must not, therefore, expect to see malaria vanish, as if by magic, immediately after our campaign against mosquitoes. – pp. 53 of [37]
In all this, Ross was asking thus questions about what would determine the outcomes of malaria control.
What other factors could affect what was observed? One of the arguments against the “mosquito hypothesis” had been that malaria was sometimes found in places where there was no evidence for malaria transmission. In The Prevention of Malaria, Ross spends a great deal of time discussing the importance of imported malaria. In the chapters that follow, we will follow this idea through to its logical conclusion.