Four Basic Malaria Models
A Primer
In this malaria modeling primer, we present four basic models of malaria. Each one describes the same process using different species of mathematics.
Part 1: We start by building a deterministic discrete-time system (DTS). In this basic model, we use a 1-day time step. The model describes the processes by which malaria changes over time. One of the reasons we start here is that these models can be constructed using spreadsheets (for example, the models have been solved using this Google Sheet.
Part 2: We build a stochastic DTS. The system uses the same rules to determine how a system is updated as the deterministic DTS, but the state of the system is determined by drawing a random number.
Part 3: We build a system of ordinary differential equations. We show how to understand this as the limiting case of a deterministic DTS where the time step becomes very small.
Part 4: We present an asynchronously updated stochastic process. We use Gillespie’s algorithm [1]
Part 5: We compare the outputs of the four models and discuss some important differences.
Part 6: The code for parts 1-5 is fully internalized, but developing reusable code creates some enormous challenges. To solve these challenges, we developed SimBA for simulation-based analytics for malaria and other mosquito-transmitted pathogens. SimBA is a suite of R packages available from github.
Goals
We want to show how models work and draw attention to the fact that:
there are many ways of implementing the same model; and thus that
all models are approximations.
These models are very simple. We’re not making the claim that they are good approximations. They are, perhaps, too simple to be useful for anything other than teaching people the basic features of malaria models.
The Process
In each case, we start with a very basic set of assumptions about the infection dynamics. We will be careful to ensure that the parameter values we use match up, so we will be formulating the model using rates and translating these into probabilities or proportions. For example:
If mosquitoes die at a constant rate \(g,\) the fraction surviving one day is \(p=e^{-g}.\)
If humans gets \(h\) infectious bites, per person, per day, then the fraction getting infected is \(1-e^{-h}.\)
Blood Feeding and Transmission
Malaria parasites are transmitted through blood feeding. Parasites are transmitted from mosquitoes to humans in saliva through a bite. Parasites are transmitted from humans to mosquitoes in the blood meal. In these models, we are assuming that the ratio of mosquitoes to humans is constant, and we also assume that mosquitoes blood feed on humans at a constant rate:
The flood feeding rate \(a=1/3\) would mean that the interval between bites is 3 days. In discrete-time models, we convert rates into probabilities, so the fraction that have blood feed in a day is \(1-e^{-a}.\)
The ratio of mosquitoes to humans \(m=2\) would mean that there are two mosquitoes for every human.
One of the most important features of mosquito-borne systems is that the potential for transmission is inversely proportional to human population density.
Human Infections
Our model for human infections assumes that a person is either uninfected or infected and infectious. After recovering from infection, they are immediately susceptible to infection once again. We are further assuming that humans are long-lived, and so we are ignoring human demographic changes. Each model thus assumes:
infections clear at a constant daily rate, \(r.\) The fraction that remain infected after one day is thus \(s=e^{-r};\)
letting \(\kappa\) denote the fraction of bites that are infectious, humans would become infected at the rate \(h = ma\kappa,\)
Mosquito Infections
Our model for mosquito infections assumes that a mosquito is either uninfected or infected and infectious. Mosquitoes are comparatively short lived, so we assume that they will die before recovering from infection. In assuming that mosquito populatio size is constant, every mosquito that dies is replaced by a susceptible mosquito.