1.7 Mathematical Epidemiology
Despite the heroic efforts of Ross and Hudson on a priori pathometry, they were missing a key element in the modern mathematical study of epidemics. In 1920, a new mathematical model was published by Martini, which described epidemics in which recovery from infection was followed by lifelong immunity. Martini’s model would be explored in great detail in a series of papers by Kermack, and Mckendrick. In modern language, Ross’s models would be studied as SIS compartmental models, while Martini’s equations would be called SIR compartmental models. The two systems of equations have strikingly different features, and so do the kinds of epidemics they mimic.
In Ross’s model, a person would become susceptible to infection after recovering, so a person could be infected many times, and prevalence is a naturally meaningful statistic. In fact, Ross’s equations can be solved exactly: after Ross published the equations in Nature in 1911 [32], Alfred Lotka published a closed-form solution in 1912 [54].
There is no closed form solution to the SIR model, and prevalence is an ephemerally changing quantity. The natural summary statistic for SIR epidemic is called \(S_\infty\), the portion remaining uninfected asymptotically. In simple epidemics – an epidemic in a single population with no replenishment of susceptible hosts – the dynamics differ depending on how many cases that first case would tend to generate, called \(R_0\). If \(R_0 < 1\), then there is a closed form solution: \(S_\infty = S_0 - I_0(1-R_0)^{-1}\). If that first case would generate more than one other case, and the number of cases would initially rise to a peak, when each new case would generate exactly one case. Thereafter, the number of cases would decline. The time course of an SIR epidemic does not have a closed form solution.
The differences between SIS and SIR models would provide a mathematical basis for answering Ross’s question and point to an important role for the concept of population immunity. In SIS models, there is no immunity, so prevalence tends to approach an endemic equilibrium with very little tendency to cycle. When susceptible populations the SIR models are replenished by birth or migration, there is a natural tendency to cycle.
The SIS model is a useful starting point, but Ross was aware of evidence that immunity to malaria would develop in humans. The role of immunity is more complex, but a critical feature is that infection with one parasite would not prevent reinfection with another. In places where exposure rates were high, reinfection – also called superinfection – was quite common. The first model of malaria that was not an SIS model – published by George Macdonald but relying on mathematics by P. Armitage – considered a role for superinfection, or reinfection of susceptible individuals. In models with superinfection, an interesting statistic is the multiplicity of infection (MoI). A useful discussion of this superinfection model and its mathematical flaws was written by Paul EM Fine [55]. What is, perhaps, more important is that superinfection in malaria facilitates sexual recombination for the parasites. Sexual recombination facilitates development of parasite diversity, which could partly explain population immunity in malaria.
Sticking with Ross’s core challenge of how to study epidemics mathematically to understand the tendency to cycle, we must acknowledge the complex natural history of population immunity. In malaria, a model for the mathematical talents of Klaus Dietz combined with the epidemiological skills of Louis Molineaux to produce the first model of population immunity to malaria. That model described infection dynamics in a population with partial immunity. In malaria epidemiology, a core challenge is the problem of malaria immunity and its relationship to parasite genetic diversity, disease, and infectiousness. In the mathematical study of malaria epidemiology, the formulation of adequate models is among the most pressing and most difficult to address. Measles and a few other pathogens are well-described by the SIR model. For these acute immunizing infections, infection is followed by a life-long immunity. The dengue viruses has four (or maybe five) functionally distinct virues with complex patterns of cross immunity. For other pathogens, population immunity is undermined when immune-escape variantes evolve in the pathogen populations. For the influenza viruses and the coronavirus descendents of the viruses that sparked the global COVID-19 epidemic, escape variants arise at a rate that is high enough to sustain yearly outbreaks.
Adding to the complexity, immunity is not the only factor affecting a tendency to cycle. For arthropod-transmitted pathogens, fluctuations in arthropod population densities and behaviors driven by weather affect trasmission rates. For directly transmitted pathogens, humidity and survival in the environment and the tendency for populations to congregate amplifies transmission. In malaria, fluctuating mosquito one factor that must a priori force transmission, and since population immunity probably has a weak effect, seasonal malaria is largely driven by fluctuating mosquito populations. In SIR models, seasonal fluctuations are sustained by an interaction between environmental factors and the depletion of susceptible individuals.