Epistrains

A multi-strain SIR model

Installation

git clone https://github.com/SABS-R3-Epidemiology/epistrains.git
cd epistrains
pip install .

Usage

Below is an example of how use the package to define, solve and visualise the multi-strain SIR model. This model uses the default birth rate and models two strains with different death, recovery, and transmission rates, along with a different number of individuals initially infected with each strain. This code can also be found in example.py along with additional information about the parameters which can also be found in the package documentation.

Import the relevant classes and functions from the epistrains package.

from epistrains import Population, make_br, Strain, Solver

Generate a birth rate function with exponential form: $Nae^{-kN}$

birth_rate_function = make_br(a=10.0, k=0.001)

Instantiate the population class. Additional optional inputs are the rate of waning immunity and the percentage of the initial population who are immune at the start of the simulation.

population = Population(death=0.000006, size=150000, birth_function=birth_rate_function)

Generate two strains to model. Additionally, a time delay can be input to define the number of days after the start of the simulation at which each strain is introduced to the population.

I1 = Strain(CFR=0.00007, recovery_time=7, R0=3.14, infected=150)
I2 = Strain(CFR=0.001, recovery_time=8, R0=4.22, infected=10)

Instantiate the Solver class.

model = Solver(pop=population, strains=[I1, I2], time=70)

Solve the set of equations for the 4 compartments and plot the results to be displayed and saved. To save the plot, provide the path denoting where to save the plot to save_compartments().

model.solve()
model.plot_compartments()
model.save_compartments('epistrains_example.png')
model.plot_death()
model.save_death('epistrains_death_example.png')

It is also possible to implement an alternative birth rate function instead of the default exponential. For example:

population = Population(death=0.000006, size=150000, birth_function= lambda N: 0.0005*N)

These two plots display 1) the number of individuals in each compartment over time, alongside the number of deaths, and 2) the number of deaths per day and the cumulative number of deaths over time.

Background

SIR models are a form of compartment model used for the mathematical modelling of infectious disease. Here, we implement the model developed by Bremermann and Thieme (1989)[^1], in which there are multiple infected compartments, one for each strain.

We assume that the total population size, $N$ is governed by a population growth model with a per capita exponential birth rate by default and a constant per capita death rate. This is represented as follows: $$\frac{dN}{dt} = N\left(ae^{-kN}-b\right)$$ where $a$ and $k$ are constants of the exponential birth function, $b$ is the per capita death rate, and $(a>b)$ such that the carrying capacity of the population is $\frac{1}{k}\log(\frac{a}{b})$. This birth function can be changed by the user, for example to uniform growth $aN$ or logistic growth $aN\left(1-\frac{N}{k}\right)$ where $k$ is the carrying capacity.

The total population size obeys the following equation: $$N = S + R + \sum_{j} I_j$$
where $S$ is the number of susceptible individuals, $R$ is the number of recovered individuals, and $I(_j)$ is the number of individuals infected with strain $j$. Therefore, for our model we use a system of SIR ODEs with the following compartments: $S$ (susceptible), $R$ (recovered) or $I_j$ (infected with the $j^{th}$ strain of the disease). The ODEs that govern the time evolution of the system are:
$$\frac{dS}{dt} = Nae^{-kN}-\sum_j \beta_j I_j S - bS$$

$$\frac{dI_j}{dt} = I_j(\beta_j S - (b + \nu_j + \alpha_j))$$

$$\frac{dR}{dt} = -bR + \sum_j \nu_j I_j$$

where $a$ and $k$ are parameters relating to the exponential birth rate, $b$ is the per capita death rate, $\beta_j$ is the transmission rate of strain $j$, $\nu_j$ the rate of recovery of individuals infected with strain $j$, and $\alpha_j$ the rate of death of individuals infected with strain $j$.

[^1]: Bremermann HJ, Thieme HR. A competitive exclusion principle for pathogen virulence. J Math Biol. 1989;27(2):179-90. doi: 10.1007/BF00276102. PMID: 2723551.

Future Development

An extension allowing multiple strains to be modelled with individual per strain immunity is under development. We aim to allow a cross-immunity matrix to be defined which governs the probabilty of becoming infected by certain strain given a previous infection with another strain.