SIR Epidemics: How to model its behavior? Soonmook Lee 1

To model the behavior of epidemics in terms of the rate at which infections occur and individuals leave the infective class through recovery or death in the next unit of time. To predict the change of susceptible population affected by the infective occurrence and the effectiveness of public health system ensuring recovery or yielding unfortunate death. Purpose

Faced with frequent and unexpected occurrence of epidemics that are crucial to individual health, societal well-being, and national economy there is a compelling need to develop a system for figuring out the trajectories of the epidemics. We are seriously interested in how the rates of infective occurrence and removal(recovery+death) are combined leading to the change of susceptible population. Motivation

Conceptual Model: S → I → R Change of Susceptibles, Infectives, and Removed class as time(t) goes on. Assumption The population is closed, so that S(t)+I(t)+R(t)=A, where A is the constant population size. The duration of the epidemic is short compared to the lifetime of its hosts, so that we can ignore birth and disease-unrealated death. Disease individuals leaving the infective class play no further role in the disease. They may be immune, or dead, or removed by an isolation policy or otherwise. Conceptual Scheme for Modeling Epidemics

Closed Population: S(t)+I(t)+R(t)=A Differential Equations Representing Change of Status Nonlinear differential equations can be solved by Euler's method which is the most elementary approximation technique for solving initial-value problems. Mathematical Model

Approximation to y(t) will be generated at various values, called mesh points, in the interval t=[0, T], where T=49 and there are 100 points between each unit of T in the present problem. Distance between points or step size is h=t(2)-t(1). Given a dependent variable y(t) Euler's method constructs Mathematical Model

Application of Euler's method to approximate the dynamics of S, I, and R variables given initial values I(1), S(1), R(1), and constant population size "a". a= I(1)=2 R(1)=0 S(1)=a-I(1)-R(1) S(i+1)=S(i) - b(i)S(i)I(i)h, where f(I, S)=-bIS. I(i+1)=I(i)+(b(i)S(i)I(i)-kI(1))h, where f(I, S)=bIS-kI. R(i+1)=kI(i)h. Mathematical Model

file: MERS.m clear; clc; clf; % initial values n=49*100; T=49; t=linspace(0, T, n+1); h=t(2)-t(1); a= ; I(1)=2; R(1)=0; S(1)=a-I(1)-R(1); % little change of the value of b given in the class b= *exp(-t/8.); k=0.0348; % Euler method to approximate data values for S, I, R for i=1:n S(i+1)=S(i)-b(i)*S(i)*I(i)*h; I(i+1)=I(i)+(b(i)*S(i)*I(i)-k*I(i))*h; R(i+1)=R(i)+k*I(i)*h; end Coding of the Model

% Experimental values for occurrence of infection(exi0), death, recovery exi0=[ ]; death=[ ]; recover=[ ]; Coding of the Model

% removal from the infective class(exr=death+recover) % remaining infectious class(exi=exi0-exr) exr=death+recover; exi=exi0-exr; tt=linspace(0,T, 50); % plotting experimental data plot(tt, exi, 'ro-', tt, exr, 'md-'); hold on % plotting model estimates plot(t, I, 'b.', t, R, 'k') Coding of the Model

legend('data-infec', 'data-recovry+death', 'model-infec', 'model-recovry+death') % computation of error sum=0 ; for i=1:50 sum=sum+ (I(i)-exi(i))^2+ (R(i)-exr(i))^2; end error=sqrt(sum/50) Coding of the Model

The graph of fitting between data and model with error= Data and Model

1.How to estimate b fitting best to experimental data? 2.How to estimate k fitting best to experimental data? Issues

1)Estimation of b with k value fixed at the given value bx: estimate of b We treat the MERS.m as a function having "bx" as the input and "error" as the output. Develop an algorithm that provides bx leading to the minimum value of error as the output of the function fit(bx). Ideas for Estimating b and k function error=fit(bx) T=49; b=bx*exp(-t/8); k= ⋮ error= end

2)Estimation of k with b value fixed at bx. kx: estimate of k We may follow the similar procedure to the above (1) for bx. Idea to Estimate b and k

Thank you 16