Dispersion of a flow that passes a dynamic structure. 5*t 5*t. Time
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1 > # PROJECT > # ======= > # Principles of Modelling and Simulation in Epidemiology > # Laboratory exercise 1 > # > # DESCRIPTION > # =========== > # Suggestions of solutions > > # THOR AND DATE > # Andreas Karlsson, October 2013 > > require(desolve) > ## 2.1 Filling and draining compartments + dx1 <- c1 - c2 + dx2 <- c2 - c1 + return(list(c(dx1, dx2))) > init <- c(x1 = 17.3, X2 = 0) > parameters <- c(c1 = 10, c2 = 5) > times <- seq(0, 10, by = 0.01) func = deriv, + parms = parameters)) > matplot(times, out, type = 'l', xlab = '', ylab = '', + main = 'Dispersion of a flow that passes a dynamic structure', + lty = 1, bty = 'l', col = 2:3) > legend('topleft', c('5*t','-5*t'), lty = 1, col = 2:3) Dispersion of a flow that passes a dynamic structure *t 5*t
2 > ## 2.2 Possitive feedback > derivs <- function(time, state, parameters) { + dx1 <- X1/T1 + dx2 <- X2/T2 + dx3 <- X3/T3 + return(list(c(dx1, dx2, dx3))) > init <- c(x1 = 1, X2 = 1, X3 = 1) > parameters <- c(t1 = 5, T2 = 10, T3 = 20) > times <- seq(0, 10, by = 0.01) + func = derivs, parms = parameters)) > matplot(times, out, type = 'l', xlab = '', ylab = '', + main = 'Positive Feedback', lty = 1, bty = 'l', col = 2:4) > legend('topleft', c('t=5','t=10','t=20'), lty = 1, col = 2:4) Positive Feedback T=5 T=10 T=
3 > ## 2.3a Negative feedback > derivs <- function(time, state, parameters) { + dx1 <- -X1/T1 + dx2 <- -X2/T2 + dx3 <- -X3/T3 + return(list(c(dx1, dx2, dx3))) > init <- c(x1 = 100, X2 = 100, X3 = 100) > parameters <- c(t1 = 5, T2 = 10, T3 = 20) > times <- seq(0, 50, by = 0.01) + func = derivs, parms = parameters)) > matplot(times, out, type = 'l', xlab = '', ylab = '', + main = 'Negative Feedback', lty = 1, bty = 'l', col = 2:4) > legend('topright', c('t=5','t=10','t=20'), lty = 1, col = 2:4) Negative Feedback T=5 T=10 T=
4 > ## 2.3b Avarage Sojurn > derivs <- function(time, state, parameters) { + dx1 <- -X1/T1 + Av_ <- -time * dx1 / return(list(c(dx1, Av_))) > init <- c(x1 = 100, Av_ = 0) > parameters <- c(t1 = 10) > times <- seq(0, 100, by = 0.01) + func = derivs, parms = parameters)) > matplot(times, out, type = 'l', xlab = '', ylab = '', + main = 'Avarage sojourn time', lty = 1, bty = 'l', col = 2:3) > legend('topright', c('x','avarage '), lty = 1, col = 2:3) Avarage sojourn time X Avarage
5 > ## 2.4 Negative feedback over two compartments > oscillation <- function(time, state, parameters) { + dx2 <- -X1 + dx1 <- X2 + return(list(c(dx1, dx2))) > init <- c(x1 = 20, X2 = 0) > parameters <- NULL > times <- seq(0, 10, by = 0.01) + func = oscillation, parms = parameters)) > layout(matrix(c(1,1,2,3), 2, 2, byrow = TRUE)) > matplot(times, out, type = "l", xlab = "", ylab = "", + main = "Negative feedback over two compartments", + lwd = 1, lty = 1, bty = "l", col = 2:3) > plot(out$x1,out$x2, type = 'l',ylab='x2', xlab='x1') Negative feedback over two compartments X X1
6 > ## 2.5 Dispertion by a dynamic structure + import <- sigimp(time) + dx1 <- (import - X1)/T0 + dx2 <- (X1 - X2)/T0 + dx3 <- (X2 - X3)/T0 + return(list(c(dx1, dx2,dx3))) > init <- c(x1 = 0, X2 = 0, X3 = 0) > parameters <- c(t0=10) > dt < > times <- seq(0, 100, by = dt) > ## external signal with triangle impulse with area 100 > signal <- as.data.frame(list(times = times, import = rep(0,length(times)))) > #The 2 since it is a triangle > signal$import[signal$times >= 0 & signal$times <= 0] <- 100 / dt * 2 > sigimp <- approxfun(signal$times, signal$import, rule = 2) + func = deriv, parms = parameters)) > matplot(times, cbind(sigimp(times),out), type = 'l', xlab = '', ylab = '', + main = 'Dispersion of a flow that passes a dynamic structure', lwd = 1, + lty = 1, bty = 'l', col = 2:5, ylim=c(0,12)) > legend('topright', c('pulse','x1', 'X2','X3'), lty = 1, col = 2:5) Dispersion of a flow that passes a dynamic structure Pulse X1 X2 X3
7 > ## 2.6 A logistic and an SI structure + dx <- c1 * X - c2 * X * X + ds <- - r * I * S + di <- r * I * S + return(list(c(dx, ds, di))) > init <- c(x = 1, S=99, I = 1) > parameters <- c(c1=1, c2=0.01, r=0.01) > times <- seq(0, 20, by = 0.01) + func = deriv, parms = parameters)) > out$s <- NULL > matplot(times, out, type = c('o','l'), xlab = '', ylab = '', + main = 'Logistic Model & SI-model', pch = 1, lwd =4, col = 2:4) > legend('topleft', c('logistic Model','SI-model'), + pch = c('o','-'), lwd =c(0,4), col = 2:4) Logistic Model & SI model o Logistic Model SI model
8 > ## 3 Equation behind a Forrester diagram + dx <- c1 -X/T1 + return(list(c(dx))) > init <- c(x = 0) > parameters <- c(c1=1, T1 = 10) > times <- seq(0, 100, by = 0.01) func = deriv, + parms = parameters, method='euler')) > matplot(times, out, type = 'l', xlab = '', ylab = '', + main = 'Logistic Model & SI-model', pch = 1, col = 2) > legend('topright', c('x'), pch ='-', col = 2) Logistic Model & SI model X
9 > ## 5 Equation behind a Forrester diagram + dx <- c1 -X/T1 + return(list(c(dx))) > init <- c(x = 0) > parameters <- c(c1=1, T1 = 10) > d_t <- c(5, 2, 0.1, 0.01) > for (tt in 1:length(d_t)){ + times <- seq(0, 100, by = d_t[tt]) + out <- as.data.frame(ode(y = init, times = times, func = deriv, + parms = parameters, method='euler')) + out$time <- NULL + if (tt==1) + matplot(times, out, type = 'l', xlab = '', ylab = '', + main = 'Choosing time-step', pch = 1, col = 1) + else + lines(times, t(out), col = tt) > legend('topright', sprintf("dt=%.2f", d_t), pch ='-', col = (1:(length(d_t)))) Choosing time step dt=5.00 dt=2.00 dt=0.10 dt=0.01
10 > ## 6 Stage and Compartment + import <- sigimp(time) + # One compartment + dx1 <- (import - X1)/T1 + # Two compartments + dx2_1 <- (import - X2_1)/T2 + dx2_2 <- (X2_1 - X2_2)/T2 + # Three compartments + dx3_1 <- (import - X3_1)/T3 + dx3_2 <- (X3_1 - X3_2)/T3 + dx3_3 <- (X3_2 - X3_3)/T3 + return(list(c(dx1, dx2_1, dx2_2, dx3_1, dx3_2, dx3_3))) > init <- c(x1 = 0, X2_1 = 0, X2_2 = 0, X3_1 = 0, X3_2 = 0, X3_3 = 0) > parameters <- c(t1 = 10, T2 = 10/2, T3 = 10/3) > d_t < > times <- seq(0, 100, by = d_t) > ## external signal with triangle impulse with area 1 > signal <- as.data.frame(list(times = times, import = rep(0,length(times)))) > #The 2 since it is a triangle > signal$import[signal$times >= 0 & signal$times <= 0] <- 1 / d_t * 2 > sigimp <- approxfun(signal$times, signal$import, rule = 2) func = deriv, parms = parameters)) > matplot(times, out[c('x1','x2_2','x3_3')], type = 'l', xlab = '', ylab = '', + main = 'Multiple compartments', lwd = 1, + lty = 1, bty = 'l', col = 2:5, ylim=c(0,0.1)) > legend('topright', c('1 Compartment', '2 Compartments','3 Compartments'), + lty = 1, col = 2:5) Multiple compartments Compartment 2 Compartments 3 Compartments
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