The interaction of the HIV-1 virus with the body's immune system can be modeled by a system of differential equations similar to a predator-prey system. After an individual is infected with the HIV-1 virus, the amount of the virus in the bloodstream rises dramatically and the person will often suffer from flu-like symptoms. However, these symptoms will disappear after a period of weeks or months as the body begins to manufacture antibodies against the virus. Tests have been developed to determine the presence of HIV-1 antibodies. If a individual has such antibodies, then they are said to be HIV-1 positive. Once infected with the HIV-1 virus, it can be years before an HIV-positive patient exhibits the full symptoms of AIDS. The body's immune system fights the HIV-1 virus with white blood cells. The CD4-positive T-helper cells, a specific type of white blood cell, is especially important since it helps other cells fight the virus. However, the HIV-1 virus can destroy CD4-positive T-helper cells.
This topic describes the differential equations that govern the classic deterministic SIR and SIRS compartmental models and describes how to configure EMOD, an agent-based stochastic model, to simulate an SIR/SIRS epidemic. In SIR models, individuals in the recovered state gain total immunity to the pathogen; in SIRS models, that immunity wanes over time and individuals can become reinfected. The EMOD generic simulation uses an SEIR-like disease model by default. You can modify the default SEIR model to an SIR model by turning off the incubation period.
As the first step in the modeling process, we identify the independent and dependent variables. The independent variable is time t, measured in days. We consider two related sets of dependent variables.
The first set of dependent variables counts people in each of the groups, each as a function of time:
-
S = S(t) is the number of susceptible individuals,
s the number of susceptible individuals,
-
I = I(t) is the number of infected individuals, and
-
R = R(t) is the number of recovered individuals.
The second set of dependent variables represents the fraction of the total population in each of the three categories. So, if N is the total population (7,900,000 in our example), we have
-
s = s(t) = S(t)/N is the susceptible fraction of the population,
-
i = i(t) = I(t)/N is the infected fraction of the population, and
-
r(t) = R(t)/N is the recovered fraction of the population.
% SIR Model parameters
beta = 5*10^-9; % rate of infection
gamma = 0.12; % rate of recovery (try also 0.07)
delta = 0.0; % rate of immunity loss
N = 7*10^7; % Total population N = S + I + R
I0 = 10; % initial number of infected
T = 300; % period of 300 days
dt = 1/4; % time interval of 6 hours (1/4 of a day)
R_0 = N*beta/gamma;
fprintf('Value of parameter R_0 is %.2f \n', R_0)
% Calculate the model
[S,I,R] = sir_model(beta,gamma,delta,N,I0,T,dt);
% Plots that display the epidemic outbreak
tt = 0:dt:T-dt;
% Curve
figure
plot(tt,S,tt,I,tt,R,'LineWidth',2); grid on;
xlabel('Days'); ylabel('Number of individuals');
legend('S','I','R');
title('SIR Model')
delta = 1/60; % rate of immunity loss
% Calculate the model
[S,I,R] = sir_model(beta,gamma,delta,N,I0,T,dt);
% Curve
figure
plot(tt,S,tt,I,tt,R,'LineWidth',2); grid on;
xlabel('Days');ylabel('Number of individuals');
legend('S','I','R');
title('SIRS Model')
function [S,I,R] = sir_model(beta,gamma,delta,N,I0,T,dt)
% if delta = 0 we assume a model without immunity loss
S = zeros(1,T/dt);
S(1) = N;
I = zeros(1,T/dt);
I(1) = I0;
R = zeros(1,T/dt);
for tt = 1:(T/dt)-1
% Equations of the model
dS = (-beta*I(tt)*S(tt) + delta*R(tt)) * dt;
dI = (beta*I(tt)*S(tt) - gamma*I(tt)) * dt;
dR = (gamma*I(tt) - delta*R(tt)) * dt;
S(tt+1) = S(tt) + dS;
I(tt+1) = I(tt) + dI;
R(tt+1) = R(tt) + dR;
end
end
% SIERDS Model parameters
alpha = 7; % latency period of 7 days
beta = 5*10^-1; % rate of infection
gamma = 0.12; % rate of recovery (try also 0.07)
delta = 0.0; % rate of immunity loss
mu = 0.0002; % rate of death (population)
mo = 0.001; % rate of death (disease)
N = 7*10^7; % Total population N = S + E + I + R + D
I0 = 10; % initial number of infected
T = 300; % period of 300 days
dt = 1/24; % time interval of 1 hours (1/24 of a day)
% Calculate the model
[S,E,I,R,D] = sir_model(alpha,beta,gamma,delta,mu,mo,N,I0,T,dt);
% Plots that display the epidemic outbreak
tt = 0:dt:T-dt;
% Curve
figure
plot(tt,S,tt,E,tt,I,tt,R,tt,D,'LineWidth',2); grid on;
xlabel('Days');ylabel('Number of individuals');
legend('S','E','I','R','D');
title('SEIRD Model')
delta = 1/60; % rate of immunity loss
% Calculate the model
[S,E,I,R,D] = sir_model(alpha,beta,gamma,delta,mu,mo,N,I0,T,dt);
% Curve
figure
plot(tt,S,tt,E,tt,I,tt,R,tt,D,'LineWidth',2); grid on;
xlabel('Days');ylabel('Number of individuals');
legend('S','E','I','R','D');
title('SEIRDS Model')
function [S,E,I,R,D] = sir_model(alpha,beta,gamma,delta,mu,mo,N,I0,T,dt)
% if delta = 0 we assume a model without immunity loss
S = zeros(1,T/dt);
S(1) = N;
E = zeros(1,T/dt);
I = zeros(1,T/dt);
I(1) = I0;
R = zeros(1,T/dt);
D = zeros(1,T/dt);
alpha = 1/alpha;
for tt = 1:(T/dt)-1
% Equations of the model
dS = (mu*N - mu*S(tt) - beta*I(tt)*S(tt)/N + delta*R(tt)) * dt;
dE = (beta*I(tt)*S(tt)/N - (mu+alpha)*E(tt)) * dt;
dI = (alpha*E(tt) - (mu+gamma)*I(tt) - mo*I(tt)) * dt;
dR = (gamma*I(tt) - mu*R(tt) - delta*R(tt)) * dt;
dD = (mo*I(tt)) * dt;
S(tt+1) = S(tt) + dS;
E(tt+1) = E(tt) + dE;
I(tt+1) = I(tt) + dI;
R(tt+1) = R(tt) + dR;
D(tt+1) = D(tt) + dD;
end
end
-
-
Bjørnstad, O.N., Shea, K., Krzywinski, M., The SEIRS model for infectious disease dynamics, Nature Methods, 2020, volume 17, pages 557–558. https://doi.org/10.1038/s41592-020-0856-2
-
SIR and SIRS Models, IDM
-
Smith, D. and Moore, L., The SIR Model for Spread of Disease - The Differential Equation Model, The American Mathematical Monthly,
-
Yano, T.K., Makinde, O.D., Malonza, D.M.,
Modelling Childhood Disease Outbreak in a
Community with Inflow of Susceptible and
Vaccinated New-born, Global Journal of Pure and Applied
Mathematics, 2016, Volume 12, Number 5, pp. 3895-3916.