# Question:how do you assume a positive solution for a system of differential equations?

## Question:how do you assume a positive solution for a system of differential equations?

Maple

I was working on a project about optimal strategies for HIV treatment, models used from [Butler, Kirschner, and Lenhart] 1997. This model explains the spread of HIV viruses in the human body, where there is one control function u(t). My work is following pontryagin maximum principle. But i have a problem solving the differential equation system, where there are 6 differential equations with 6 initial conditions. Everything works normally and I get a numerical solution for the system

```restart;
with(linalg);
with(DEtools);
with(plots);
```

Model declaration

```dx := -T*V*k*u+B*T*r-T*m1+A; dx := T*V*k*u-Ti*m2; dx := N*Ti*m2-V*m3; H := A*T-(1-u)^2+add(dx[i]*L[i], i = 1 .. 3); satu := -(diff(H, T)); dua := -(diff(H, Ti)); tiga := -(diff(H, V)); empat := diff(H, L); lima := diff(H, L); enam := diff(H, L)

eq1 := diff(L1(t), t) = -A-(-V(t)*k*u(t)+B*r-m1)*L1(t)-u(t)*k*V(t)*L2(t); eq2 := diff(L2(t), t) = -N*m2*L3(t)+m2*L2(t); eq3 := diff(L3(t), t) = T(t)*k*u(t)*L1(t)-T(t)*k*u(t)*L2(t)+m3*L3(t); eq4 := diff(T(t), t) = -T(t)*V(t)*k*u(t)+B*T(t)*r-T(t)*m1+A; eq5 := diff(Ti(t), t) = T(t)*V(t)*k*u(t)-Ti(t)*m2; eq6 := diff(V(t), t) = N*Ti(t)*m2-V(t)*m3

```

Value for parameter

``` u(t):=-1/(2)*L1(t)*k*V(t)*T(t)+1/(2)*L2(t)*k*V(t)*T(t)+1; s:=10;  m1:=0.02;  m2:=0.5;  m3:=4.4;  r:=0.03;  Tm:=1500;  k:=0.000024;  N:=300;    A:=1;  B:=1-(T(t)+Ti(t))/(Tm):
```

Numerical Solution with BVP

`fcns := {L1(t), L2(t), L3(t), T(t), Ti(t), V(t)}; a := dsolve({eq1, eq2, eq3, eq4, eq5, eq6, L1(20) = 0, L2(20) = 0, L3(20) = 0, T(0) = 800, Ti(0) = 0.4e-1, V(0) = 1.5}, fcns, type = numeric, method = bvp[midrich])`

The plot

`odeplot(a, [[t, u(t)], [t, V(t)]], 0 .. 20, numpoints = 1000)`

Output Blue: V(t)
Red: u(t)
Based on graphs v (t) and u (t) have negative values, whereas in fact v (t) shows many viruses where it is never negative (this is irrational) and u (t) is limited in interval [0,1]. My question are:
How to provide positive assumptions for the system solution?
So v (t), T (t), Ti (t) are never negative? ﻿