HIV-1 Dynamics in Blood Streamwith Euler’s method & RK4

Team 7

Louis, Lee Young June, Heo jisu

Outline

1. Preliminary

Ⅰ. Improved Euler’s method

Ⅱ. Runge-Kutta method2. Main topic

Ⅰ. Problem : HIV-1 Dynamics in the Blood Stream

Ⅱ. Codes & result of Improved Euler’s method

Ⅲ. Codes & result of RK4

3. Summary - Comparison

4. Future work - Multistep method

Preliminary

Ⅰ. Improved Euler’s method

Ⅱ. The 4th order Runge-Kutta method

Main topicⅠ.Problem

T : concentration of target cells in plasmaT* : concentration of infected cells in plasmaV1 : concentration of infectious viral RNA in plasmaVx : concentration of non-infectious viral particles in plasmaδ : the rate of cell loss by lysis, apoptosis, or removal by the immune systemc : rate at which viral particles are cleared from the blood stream

target cells→

infected→

infectious virons ->Vx V1

T*(+) *kT

(-) *δ

(-) *cV1

(+) *Nδ

(-) *c

T

non-infectious virons ->

Problem : HIV-1 Dynamics in the Blood Stream

Equation

dT* / dt = kV1T −δT*

dV1 / dt = −cV1

dVx / dt = NδT*−cVx

Initial value

V1 (t = 0) = 100 /µlVx (t = 0) = 0 /µlT (t = 0) = 250non-infected cells /µlT* (t = 0) = 10infected cells /µl k = 2 ·10−4µl /day/virions (infection rate)N = 60virions produced per cellδ= 0 .5/day c = 3 .0/day

Assumptions*No natural death of healthy cells*No production of new healthy cells

Problem.

Find the number of infected cells, infectious viral particles, and non-infectious viral particles at t = 5 days using improved Euler’s method and fourth-order RK method.

Anaylitic solution We can find anaylitic solution of V1(t), T*(t) and Vx(t) by intergrating

dV1 / dt = −cV1 V1 (t = 0) = 100 /µl

dT* / dt = kV1T −δT* T* (t = 0) = 10 infected cells /µl

dVx / dt = NδT*−cVx Vx (t = 0) = 0 /µl

Then we got these equations.

V1(t) = 100e-3t-

T*(t) = 12e-0.5t-2e-3t

Vx(t) = -144e-3t-60*t*e-3t+144e-0.5t

So we can get exact values at each t. (t=1,2,3,4,5)

Ⅱ. MATLAB Codes & results of Improved Euler’s method

MATLAB Codes & results of Improved Euler’s method

MATLAB Codes & results of Improved Euler’s method

Ⅲ. MATLAB Codes & results of RK4

MATLAB Codes & results of RK4

MATLAB Codes & results of RK4

Summary

As you can see, we get some pattern in the tables.In the table 3, about two zeros are added as h is multiplied by 0.1.On the other hand, about 4 zeros are added as h is muliplied by 0.1 in the

table 5.So we may say Global errors is for the improved Euler’s method,and Global errors for the Runge-Kutta method is .

* In RK4, as the number of iteration becomes large, other types of errors may creep in.

Summary

Summary

It is clear that Numerical values obtained with RK4 are closer to the real values than values obtained with IE.

Future workMULTISTEP METHODSEuler’s method, IE, RK4 are examples of single-step or starting methods. In these methods each succesive value is computed based only on information about the immediately preceding value . On the other hand, multistep methods use the values from several computed steps to obtain the value of . Here is one example of multistep method. The fourth-order Adams-Bashforth-Moulton method.

- Adams-Bashforth-Moulton MethodLike the improved Euler’s method, it is a predictor-corrector method. One formula is used to predict a value , which in turn is used to obtain a corrected value . The predictor in this method is the Adams-Bashforth formula

for The value of is then substituted into the Adams-Moulton corrector

THANK YOU

Q&A