The Ross-Macdonald Model
Basic Dynamics
Infection
Ross wanted a basis for managing malaria through an understanding of the the factors that determine the fraction of humans who are infected with malaria parasites, especially factors related to mosquitoes. The fraction infected is a quantity, so we will need to use mathematics. In this case, we will understand malaria in populations as a dynamical system where we define variables that describe the states that matter – malaria prevalence in humans and mosquitoes – through the processes that change those states over time (see Ross’s a priori Pathometry). Here, we present the model in concise form.
Variables
We want to translate the ideas into mathematical formulas using basic quantitative logic, so we start by assigning variable names to the quantities we want to understand:
Let \(x\) denote the fraction of humans who are infected with malaria parasites;
Let \(y\) denote the fraction of mosquitoes who are infected with malaria parasites;
This might seem too simple – and perhaps it is – but we have to start somewhere.
Blood Feeding and Transmission
Mosquitoes and humans get infected through blood feeding, so we will need to understand the basics and then find a mathematical representation of it:
Humans become infected when an infectious mosquito bites a human. By infectious, we mean that the mosquito has got sporozoites in its salivary glands. For a human to become infected, at least some parasites must cross into the dermis in mosquito saliva, get into the blood and reach the liver, infect some liver cells, and then emerge into the blood stream as merozoites. These merozoites replicate in the blood. Some of the sporozoites that enter the dermis in saliva can get consumed when/if the mosquito blood feeds. It is possible the mosquito consumes all of them. There are many other reasons why a bite by an infectious mosquito might not cause an infection.
Mosquitoes become infected by blood feeding on an infectious human. To get infected, a mosquito has to consume at least two gametocytes – one male (microgametocyte) and one female (macrogametocyte) – in the bloodmeal. In a mosquito, the two gametocytes must mate to form a gamete in the mosquito midgut. After some development, the parasite forms an ookinete that embed in the mosquito midgut wall. At this point, we would say the mosquito is infected;
Mosquitoes blood feed to get the protein and other materials needed to make mosquito eggs. The eggs are laid in aquatic habitats, and the mosquito blood feeds again. We need a parameter that describes how often a mosquito blood feeds, and the overall blood feeding rate by mosquito populations:
Let \(a\) denote the human blood feeding rate, the number of bloodmeals, per mosquito, per day;
Let \(m\) denote the number of mosquitoes, per human.
The quantity \(ma\) is called the human biting rate (HBR); it is the number of bites / bloodmeals by mosquitoes, per human, per day.
Let \(c\) be the fraction of bites on an infected human that infect the mosquito.
In the mosquito population, uninfected mosquitoes (\(1-y\)) become infected after blood feeding on an infected mosquito, or in math:
\[a c x (1-y)\]
Humans become infected after getting bit by an infectious mosquito. Now, we have a problem writing down the equations, because we have variables describing the fraction of mosquitoes that are infected, but not the fraction that are infective. We need another variable, \(z\):
Let \(z\) denote the fraction of mosquitoes that are infectious. It has also been called the sporozoite rate. After becoming infected, it takes several days before the mosquito becomes infectious. After embedding in the midgut, the ookinete becomes an oocyst. Parasites replicate in the oocyst until it bursts,releasing sporozoites into the mosquito hemocoel. These sporozoites migrate into the salivary glands, when the mosquito is capable of transmitting, or infectious. Since mosquitoes tend to be short-lived, a large fraction of mosquitoes die before they become infectious.
Let \(b\) denote the fraction of bites by infectious mosquitoes that cause an infection.
In the human population, uninfected humans (\(1-x\)) become infected at the rate:
\[mabz(1-x)\] The quantity \(E=maz\) is called the entomological inoculation rate, and the quantity \(h = mabz\) is called the force of infection. These are important concepts, and we will have a lot more to say about them later, but for now, we keep it simple.
Malaria Dynamics
By infection dynamics, we mean changes in the fraction of mosquitoes and humans infected over time. We have already developed expressions for the rate that humans and mosquitoes get infected. How are infections lost?
Humans remain infected for some period of time before clearing parasites. After clearing, they are susceptible infection if they get re-exposed. Humans live a long time, so we ignore human mortality, and we let \(r\) denote the rate of parasite clearance.
Mosquitoes might clear parasite infections, but since they are short lived, they are much more likely to die. We let \(g\) denote the mosquito death rate. To make sense of this, this is equivalent to assuming that the average mosquito lifespan is \(1/g\) days, and the probability of surviving one day is \(p=e^{-g}.\)
The number of days it takes from the blood meal to the appearance of sporozoites in mosquito saliva is called the extrinsic inbubation period: \(\tau\) days. The probability a mosquito survives \(\tau\) days is \(p^\tau = e^{-g\tau}.\)
The Sporozoite Rate
Macdonald’s discussion of the sporozoite rate is focused mainly on the biology of transmission and the science that had informed it [1]. The formulas are relegated to appendices with the promise that an analysis would be forthcoming.
If Macdonald’s analysis of the sporozoite were presented as a dynamical system, it would almost certainly look something like the following pair of equations.
- The change in the fraction of infected mosquitoes is a dynamical equation that describes mosquito infection and mortality: \[\frac{dy}{dt} = a c x (1-y) - g y\]
- To describe infectious mosquitoes, we will need a delay differential equation. Let \(y_\tau\) denote the value of \(y\) at time \(t-\tau,\) and \(x_\tau\) the value of \(x\) at time \(t-\tau.\) The equation below says that mosquitoes entering the infectious class were the ones infected, \(\tau\) time units ago that survived over \(\tau\) days (with probability \(e^{-g\tau}\)). The fraction of infectious mosquitoes is described by the delay differential equation:
\[\frac{dz}{dt} = e^{-g\tau} a x_\tau (1-y_\tau) - g z\]
Macdonald’s formula for the sporozoite rate can be derived by holding \(x\) constant, and so the equations have a steady state for the fraction infected: \[\bar y = \frac{a x} {a x + g}\] The fraction infectious, also called the sporozoite rate, is \[\bar z = \frac{a x} {a x + g}e^{-g\tau}.\]
In reporting the formula, Macdonald used the probability of surviving one day, \(p,\) so he showed the formula : \[\bar z = \frac{a x} {a x -\ln p}e^{-p\tau}\]
Before going on, there is one more term we would like to define because it is important, and because it will help us to make sense of Macondald’s formula later.
First, we note that if we the death process is constant, per-capita, then it is from the exponential family, and the mean lifespan of a mosquito is \(1/g\) days. Since human blood feeding drives malaria parasite transmission, we let \(S\) denote the expected number of human bloodmeals taken by a mosquito over its whole life.
\[S=\frac a g\]
Macdonald called it the stability index.
Malaria Prevalence
In Macdonald’s second paper from 1952, he discussed malaria, thresholds, and the concept of an equilibrium. All the math was found in footnotes and appendices.
Before proceeding, there is one more historical note. Macdonald had introduced a model for superinfection in 1950 [2], but \(R_0\) describes malaria transmission when malaria is rare in a population and superinfection is not an issue. Under these conditions, his model for superinfection can be reduced to a standard SIS compartmental model. Macdonald’s model for superinfection was flawed [3], so we will just present the SIS model here.
The change in the fraction of infected and infectious humans is a balance between human infection and clearance:
\[\frac{dx}{dt} = m a b z (1-y) - r x.\]
Notably, Macdonald’s model assumes that the fraction of mosquitoes becoming infected after blood feeding on a human is exactly \(x.\)
The term \(ma\) is called the human biting rate (HBR). We have already mentioned that it is defined as the expected number of bites by malaria vectors, per person, per day. The variable \(z\) was derived as the sporozoite rate (SR). The entomological inoculation rate is their product:
\[ E = \mbox{EIR} = \mbox{HBR} \times \mbox{SR} = maz.\]
In this model, the hazard rate for infection, also called the force of infection (FoI, \(h\)), is \[h=b E.\]
Taking \(h\) as an input, prevalence is given by: \[ x = \frac{h}{h+r}\] Alternatively, we note that the odds of infection is the ratio of the FoI to the clearance rate:
\[\frac{x}{1-x} = \frac h r\]
The Basic Reproductive Number
We define \(R_0\) as the quantity: \[R_0 = \frac{m b a^2}{gr} e^{-g\tau} = \frac{m b a^2}{(-\ln p)r} e^{-p\tau}\]
It can be shown that if \(R_0>1,\) then at the steady state, the prevalence of infection is:
\[\bar x = \frac{R_0-1}{R_0 + S}\] and
\[\bar z = \frac{R_0-1}{R_0} \frac{S}{1+S} e^{-g\tau}\]