%% Modelling infectious disease outbreaks
% Hrothgar, October 2014
%%
% (Chebfun example ode-nonlin/ModellingDiseases.m)
% [Tags: #ode, #nonlinear, #ivp, #system, #ebola]
%%
% Many mathematical models exist for the spread of disease. This is partly
% because as epidemiology matured, more sophisticated models were developed.
% It is also because not all diseases spread in the same fashion. In this
% example we explore one of the best-known models of disease spread, the
% famous SIR model. The SIR model is called "compartmental" because it groups
% members of a population into compartments -- for example, "infected" and
% "uninfected" -- which interact according to a system of differential
% equations.
%%
% The problem posed in this example is an initial value problem, and its
% solution over a large time interval is made possible by the IVP
% capabilities released with version 5.1 of Chebfun in December 2014.
%% The SIR model
% The most famous mathematical model of epidemics is the SIR model. This model
% groups members of a fixed population as susceptible (S), infected (I), or
% recovered (R). The dynamics dictate a one-way track: susceptible members may
% become infected, and infected individuals may recover, but that is all. So
% beginning with a nonzero number of infected people, after enough time
% everyone ends up recovered.
%%
% The model is a great simplification of how most diseases actually spread: it
% ignores incubation period; it assumes full contact among the whole
% population, ignoring geographic constraints; it assumes everyone is equally
% susceptible to the disease; and it treats the population as being continuous
% rather than discrete. Nevertheless, its assumptions about immunity make it a
% good model for measles, mumps, and rubella, which are all highly contagious
% diseases that infected people eventually develop an immunity to.
%%
% The SIR equations are
% $$ \frac{d S}{d t} = -c I S, $$
% $$ \frac{d I}{d t} = c I S - r I, $$
% $$ \frac{d R}{d t} = r I. $$
% The positive constants $c$ and $r$ are called the contact rate (or
% transmission rate) and recovery rate, and are determined empirically for a
% given disease. Looking at them for a while, you'll see that these equations
% all make sense intuitively. For example, the rate of increase of the
% population of recovered individuals is proportional to the size of the
% population of infected individuals.
%%
% Here is a chebop for the SIR model.
contact_rate = .003;
recovery_rate = .3;
op = @(x,S,I,R) [ ...
diff(S) + contact_rate*I*S
diff(I) - contact_rate*I*S + recovery_rate*I
diff(R) - recovery_rate*I
];
N = chebop(op, [0, 30]);
%%
% The initial conditions will be that out of population of 501 there is a
% single infected individual.
N.lbc = @(S,I,R) [ ...
S - 500
I - 1
R
];
%%
% We will use chebop's nonlinear backslash syntax to solve the problem.
[S,I,R] = N\0;
%%
% Here is a plot of the solution.
plot([S I R])
legend('S','I','R')
title('SIR model')
xlabel('t')
%%
% So beginning from a small fraction of infected people, eventually the entire
% population gets the disease and recovers (or dies). Notice that if $I(0)=0$,
% the solution component for $I$ would be the steady function $I(t)=0$, which
% is an unstable equilibrium of the system.
%%
% What is the largest number of people infected at a particular time?
round(max(I))
%%
% Nearly half the population! At what time is the number of infected
% people equal to the number of recovered people?
t_eq = roots(I-R)
plot([S I R]), legend('S','I','R'), xlabel('t'), hold on
plot(t_eq*[1; 1], ylim(gca), 'k:')
plot(t_eq, I(t_eq), 'k.', 'MarkerSize', 15)
%%
% Chebfun makes such computations remarkably easy.
%%
% What about the instantaneous mortality rate? A natural measure of
% mortality rate is
% $$ M(t) = \frac{\rho R(t)}{\int_0^t I(\xi) d\xi}, $$
% where $0\leq\rho\leq 1$ denotes the average fraction of people who die from
% the disease. That is, the mortality rate at time $t$ is the number of people
% who have died from the disease divided by the total number of people who
% have been infected up to time $t$. Here is the instantaneous mortality rate
% as a function of time.
hold off
rho = .4; % 40 percent of infected people die
plot(rho*R/cumsum(I)) % The instantaneous mortality rate
ylim([0 1])
xlabel('t')
title('Instantaneous mortality rate for the SIR model')
%%
% It turns out that $M(t)$ is constant and equal to $\rho r$, the product of
% $\rho$ with the recovery rate.
%%
% For this model, it is perhaps unsurprising that the instantaneous mortality
% rate is constant. But it is important to note that in reality that is not
% always the case. In the case of the 2014 Ebola epidemic in West Africa,
% for instance, other factors are at play to make the transmission rate $c$
% variable, actually an increasing function of time. When the transmission
% rate $c$ is increasing so $dc(t)/dt > 0$, the disease spreads increasingly
% fast and the the instantaneous mortality rate actually _decreases_. Once the
% infection levels peak, however, the mortality rate skyrockets.
%% References
%
% 1. Daley, D. J. & Gani, J. _Epidemic Modeling: An Introduction._
% NY: Cambridge University Press (2005).
%
% 2. [Wikipedia: Compartmental models in
% epidemiology](https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology).