16.3 Infection
16.3.3 Superinfection
From early on in malaria epidemiology it was clear that exposure to malaria differed among populations, and that in some places, the rate of exposure was far higher than the rate of clearance. Ross emphasized a need to measure exposure both entomologically, through metrics that are known today as the EIR and the FoI, and parasitologically, through the prevalence of infection by light microscopy (or more commonly today, through RDTs), which was called the malaria parasite rate . There was no good reason to believe that people in highly malarious areas would be exposed faster than they would clear infections, so they would carry infections that could be traced back to many infectious mosquitoes [86]. This phenomenon was called superinfection.
Macdonald was the first to develop a mechanistic model of superinfection [57], but the mathematical formulation was at odds with his description [55]. It is an interesting bit of history for a different time.
A mathematical basis for understanding superinfection was worked out as a problem in the study of stochastic processes as part of queueing theory. This may seem strange, but understanding how many people are queueing involves understanding how people come in and how fast they are processed. One of these queuing models has become a mainstay of malaria epidemiology; in queuing theory, it is called \(M/M/\infty\).
The model tracks the multiplicity of infection (MoI). It assumes that infections arrive through exposure at a rate \(h\) (the FoI), and that they clear independently. Without clearance, the MoI, denoted \(\zeta\), would just go up. The model assumes that each infection clears at the rate \(r\); if the MoI were \(3\) then infections would clear at the rate \(3r\). Regardless of how fast infections arrive, the fact that the pressure for the MOI to go down increases with MoI means that the MoI will reach a stable state. The mean MoI is \(h/r.\) The following diagram illustrates and provides the equations:
\[\begin{equation*} \begin{array}{c} % \begin{array}{ccccccccc} \zeta_0 & {h\atop \longrightarrow} \atop {\longleftarrow \atop r} & \zeta_1 & {h\atop \longrightarrow} \atop {\longleftarrow \atop {2r}} & \zeta_2 & {h \atop \longrightarrow} \atop {\longleftarrow \atop {3r}} & \zeta_3 & {h \atop \longrightarrow} \atop {\longleftarrow \atop {4r}}& \ldots \end{array} \\ \\ \begin{array}{rl} d\zeta_0/dt &= -h \zeta_0 + r \zeta_1 \\ d \zeta_i /dt &= -(h+r) \zeta_i + h \zeta_{i-1} + r(i+1) \zeta_{i+1} \\ \end{array} \end{array} \end{equation*}\]
If one is willing to abandon compartment models, then it is possible to formulate more elegant solution using hybrid models. The mean MoI, \(m\) changes according to the equation:
\[\frac{dm}{dt} = h - r m.\] Using queuing models, it is easy to show that the distribution of the MoI is Poisson, and in these hybrid models, if the initial distribution is not Poisson, then it will converge to the Poisson distribution asymptotically. The complex dynamics of superinfection can thus be reduced to this simple equation.
Unfortunately, things become more complex if we add simple features such as treatment with drugs, or heterogeneous exposure. The distribution of the MoI is no longer Poisson [87]. Superinfection is an important part of malaria epidemiology, and we will use these models for superinfection in developing some adequate models for infection and immunity.
In the Garki Model (see below), the waiting time to clear an infection used these queuing models to formulate an approximate clearance rate: \[ \frac{h}{e^{h/r}-1}\]