CarnotCycle is a thermodynamics blog but occasionally it ventures into new areas. This post concerns the modeling of disease transmission.
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In 1867, a violent epidemic of malaria broke out on the island of Mauritius in the Indian Ocean. In the coastal town of Port Louis 6,224 inhabitants out of a local population of 87,000 perished in just one month. Across the island as a whole there were 43,000 deaths out of a total population of 330,000. It was the worst calamity that Mauritius has ever suffered, and it had a serious impact on the island’s economy which in those days was principally generated by sugar cane plantations.
At the time, Mauritius was ruled by the British. The island had little in the way of natural resources, but perhaps because of its strategic position for Britain’s armed forces, the government was keen to keep the malaria problem under observation. Medical statistics show that following the great epidemic of 1867, deaths from malaria dropped to zero by the end of the century.
In the first years of the 20th century however, a small but significant rise in deaths from malarial fever was observed. And in May 1907 the British Secretary of State for the Colonies requested Ronald Ross, Professor of Tropical Medicine at Liverpool University, to visit Mauritius in order to report on measures for the prevention of malaria there. Ross sailed from England in October 1907 and arrived in Mauritius a month later.
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Just south of Port Louis, on the west coast of Mauritius, lies the township of Albion. Today it is home to a Club Med beach resort, but in 1907 when Ronald Ross visited the island, there were sugar plantations here – Albion Estate and Gros Cailloux estate, employing considerable numbers of Indian laborers. This part of the sea-coast was known for its marshy localities and it was here that the first sporadic cases of malaria were observed in 1865, two years before the great epidemic broke out.
Ronald Ross no doubt toured this area, his mind occupied with the genesis of the outbreak. Just a handful of cases in 1865, then in 1866 there were 207 cases on Albion Estate and 517 cases on Gros Cailloux Estate. From these estates the disease spread north and south, and during 1867 the epidemic broke out with such severity along sixty miles of coastline that those who survived were scarcely able to bury the dead.
How could this rapid increase in cases be explained? Ronald Ross was probably better placed than anyone to furnish an answer. Not only was he the discoverer of the role of the marsh-breeding Anopheline mosquito in spreading malaria (for which he received a Nobel Prize in 1902), he was also a thinker with a mathematical turn of mind.
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Soon after the malaria epidemic broke out on Mauritius, the British government appointed a commission of enquiry, which published a bulky report in 1868. This was followed by numerous other publications, giving Ronald Ross an abundance of statistical data with which to chart the course of the epidemic.
I can picture Ross studying the monthly totals of malaria cases as the epidemic unfolded, and noting how they followed an exponential curve. And I can imagine him seeing the list of figures as the terms of a mathematical sequence, with the question forming in his mind “What is the formula that generates the numbers in this sequence?”.
Although trained in medicine rather than mathematics, Ross nevertheless knew that one route to finding the formula was to construct a first-order difference equation which expresses the next term in a sequence as a function of the previous term. In his 1908 report he adopts this approach, finds a formula, and demonstrates some remarkable results with it. Although at times loosely worded, his pioneering elaboration of what he calls the ‘malaria function’ displays original thinking of a high order.
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Ronald Ross was a mathematician by nature but not by training, which explains the absence of formal rigor in his mathematical argument. The style of exposition is somewhat saltatory; in fact he never actually states the difference equation, but instead leaps straight to its general solution (the malaria function) without showing the intermediate steps.
Ross begins with the argumentation leading to his famous ‘fsbaimp’ expression (familiarity is assumed; otherwise see Appendix 1), but it is not particularly conducive to understanding his overall scheme since he presents it as an algebraic thing-in-itself rather than a component variable in a first-order difference equation.
To apprehend the architecture of Ross’s thinking, one has to work backwards from the malaria function to obtain the difference equation, which can be expressed in words as
Infections (month n+1) = Infections (month n) – number of recoveries + number of new cases
Now although Ross did not address the matter of dimensions at any point in his argumentation, it was nonetheless a crucial consideration in formulating the above equation. Equality is symmetric, so the dimensions of each RHS term must be the same as the LHS term, which according to Ross’s terminology for infected people is mp. Since Ross is seeking to obtain a difference equation of the form
where α is the growth/decay constant, each of the three RHS terms must be the product of mp and a dimensionless coefficient k:
Clearly k1 is a dimensionless 1 since the total infections in month n is simply m(n)p. The coefficient k2 is the dimensionless recovery constant for the infected population (Ross uses the symbol r), whose value lies in the range 0–1. The real difficulty is with k3 – how to transform fsbai into a dimensionless quantity. Ross achieved this (see Appendix 1) by introducing a one-to-one correspondence constraint which had the effect of changing the units of a from mosquitoes to people, thereby cancelling out the units of b (1/people) and rendering fsbai dimensionless. This could with some justification be regarded as an exercise in artifice, but Ross really had no alternative to employing facilitated convenience if he was to solve this equation.
Putting all these pieces together, the difference equation Ross arrived at, but did not state, was:
where all terms except m (called the malaria rate) are considered constant. In his 1908 report, Ross skipped directly from the above equation, which is of the form
to its solution
which enabled him to compute his malaria function explicitly in terms of the initial value m(0)
or as Ross actually rendered it (by substituting f/p for b; see Appendix 1)
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In the above equation, Ross found an explanation not only of the outbreaks of malaria epidemics, but also of why malaria can diminish and even die out – as had happened for example in Europe – despite the continued presence of mosquitoes capable of carrying the disease.
Ross recognized that m(n) would increase or diminish indefinitely at an exponential rate as n increases, according to whether the contents of the parentheses were greater or less than unity, i.e.
Here was the riposte to those who claimed that malaria should persist wherever Anopheline mosquitoes continued to exist, and that anti-malarial strategies which merely reduced mosquito numbers would never eradicate the disease.
Ross could now show that it was the relation of the mosquito-human population ratio in a locality to its threshold value (a/p = r/f2si) that determined growth or decay of the malaria rate m(n), and that mosquito reduction measures, if sufficiently impactful, could indeed result in the disease diminishing and ultimately disappearing. He could even provide a rough estimate of the threshold value of a/p by assigning plausible values to s, i, f and r. In his 1908 report, Ross calculated this value to be 39.6, or about 40 mosquitoes per individual.
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Ross’s malaria function was a remarkable result of some brilliantly original thinking, but as with most early forays into uncharted territory it had its limitations. Principal among them was that the equation was valid on the restrictive assumption that infected mosquitoes bit only uninfected human beings.
This clearly lacked credibility in the circumstances of a developed epidemic where a substantial proportion of the local population would be infected. So Ross was forced to preface his equation with the words ‘if … m is small’, which meant that the equation was strictly invalid for charting log phase growth or decay – thereby weakening support for his argument that total eradication of mosquitoes was unnecessary for disease control.
Another significant assumption in Ross’s equation was that the local population p was regarded as constant*, something wildly at variance with the actuality of the Mauritius epidemic of 1867, where a great many deaths occurred in the absence of any significant immigration.
*Although p cancels out from the mp term on both sides of the equation, it remains present in the third coefficient which is a component part of the growth/decay constant.
With limitations like these, it is evident that in his 1908 report Ross had not yet achieved a convincing mathematical argument to support his controversial views on how to control malaria. Ross was well aware of this, and over the next eight years he developed his ideas considerably – both in refining his model and advancing his mathematical approach.
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The next phase of Ross’s mathematical thinking was published in a book entitled The Prevention of Malaria (1911) wherein Ross addresses the malaria rate issue using iterated difference equations, from which he computes a limiting value of m. In an addendum to the 2nd edition of this work, under the heading Theory of Happenings, Ross addresses the population variation issue using a systematized set of difference equations, and in the closing pages of the addendum makes the transition from the discrete time period of his difference equations to the infinitesimal time period of a corresponding set of differential equations. This allows him to address variations from the perspective of continuous functions.
Ross could have stopped there, but the instinctive mathematician in him had more to say. This resulted in a lengthy paper published in parts in the Proceedings of the Royal Society of London between July 1915 and October 1916. In this paper, Ross continues from where he left off in 1911, but in a more generalized form. He considers a population of whom a number are affected by something (such as a disease) and the remainder are non-affected; in an element of time dt a proportion of the non-affected become affected and a possibly different proportion of the affected revert to the non-affected group. He then supposes that both groups are subject to possibly different birth rates, death rates, immigration and emigration rates, and asks: What will be the number of affected individuals, the number of new cases, and the number of people living at time t?
To answer these questions, Ross attempts to integrate his differential equations; this forms the substance of Part I. For Parts II and III, Ross enlists the assistance of “Miss Hilda P. Hudson, MA, ScD”, a 34-year-old Cambridge mathematician, whom he acknowledges as co-author. In Part II they examine cases where the something that happens to the population (such as a disease) is not constant during the considered period. This propels them into the study of what they call hypometric happenings. In Part III they turn their attention to graphing some of the functions they have obtained, and note the steadily rising curve of a happening that gradually permeates the entire population, the symmetrical bell-shaped curve of an epidemic that dies away entirely, the unsymmetrical bell curve that begins with an epidemic and settles down to a steady endemic level, the periodic curve with its regular rise and fall due to seasonal disturbances, and the irregular curve where outbreaks of differing violence occur at unequal intervals. The conclusion they reach is that “the rise and fall of epidemics as far as we can see at present can be explained by the general laws of happenings, as studied in this paper.”
In summary then, it can be said that having resolved the issues that restricted the applicability of the malaria function, Ross and Hudson found that their generalized model – taking the happening to be a malaria outbreak – endorsed Ross’s original assertions, with the attendant implications for management and prevention.
But all this lay ahead of Ronald Ross in February 1908 as he completed the groundwork for his first report. We leave him as he packs his bags to depart Mauritius, his mind full of island impressions, malaria statistics and mathematical ideas that he will contemplate at leisure on the month-long journey home.
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What f.s.b.a.i.m.p means
(terms as defined in the 1908 report; note that Ross later revised some of these definitions)
p = the average population in the locality (units: people)
m = the proportion of p which are already infected with malaria in the start month (dimensionless)
i = the proportion of m which are infectious to mosquitoes (dimensionless)
a = the average number of mosquitoes in the locality (units: mosquitoes)
b = the proportion of a that feed on a single person (units: 1/people)
hence baimp = the average number of mosquitoes infected with malaria in the month
s = the proportion of mosquitoes that survive long enough to bite human beings (dimensionless)
f = the proportion of a which succeed in biting human beings (dimensionless)
hence fsbaimp = the average number of infected mosquitoes which succeed in biting human beings
If the constraint is applied that each of these mosquitoes infects a separate person and only one person, then fsbaimp will denote the average number of persons infected with malaria during the month. Since the constraint imposes a one-to-one correspondence, the units of fsbaimp may equally be taken as ‘infected mosquitoes’ and ‘infected people’.
Note also that, given p, either b or f is technically redundant since p = f/b
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Ronald Ross, Report on the Prevention of Malaria in Mauritius (1908)
Paul Fine, Ross’s a priori Pathometry – a Perspective (1976)
Smith DL et al., Ross, Macdonald, and a Theory for the Dynamics and Control of Mosquito-Transmitted Pathogens (2012)
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