Stability and Asymptotical behavior of a nonlinear system

Question

A simple mathematical model to describe how the HIV/AIDS virus infects healthy cells is given by the following equations:

$$ \begin{align} \frac{dT}{dt} &= s - dT - \beta Tv \\ \frac{dT^*}{dt} &= \beta Tv - \mu T^* \\ \frac{dv}{dt} &= kT^* - cv \end{align} $$

I have found two equilibrium points (DFE and EE), given by

$$ (T_0, T_0^*, v_0) = \left(\frac{s}{d}, 0, 0\right) $$

and

$$ (T_0, T_0^*, v_0) = \left(\frac{c\mu}{\beta k}, \frac{s}{\mu} - \frac{cd}{\beta k}, \frac{sk}{c\mu} - \frac{d}{\beta}\right)=\left(\frac{c\mu}{\beta k}, (R_0-1)\frac{dc}{\beta k},(R_0-1)\frac{d}{\beta} \right) $$ Where $R_0$ is the basic reproduction number.

I am asked to discuss

• the stability of these equilibria and
• the asymptotical behavior of this system

However, I am wondering how to do that because the system is not linear, and I typically use the Jacobian matrix (the only method I am aware of), linearizing method to determine stability. Any solution or hints will be appreciated. Thanks in advance.

Answer 1

The system as a whole isn't linear; but the local stability of each equilibrium should be determined by the linearization of the system of equations around that point. In general, for any fixed point $(T_0, T^*_0, v_0)$, you can write $T=T_0+\delta T$, $T^*=T^*_0+\delta T^*$, $v=v_0+\delta v$, and then expand the system of equations to linear order in the $\delta$-variables: $$ \dot{\delta T}=\dot{T}=s-d(T_0+\delta T)-\beta(T_0+\delta T)(v_0+\delta v)\approx -(d + \beta v_0)\delta T-\beta T_0\delta v \\ \dot{\delta T^*}=\dot{T^*}=\beta(T_0+\delta T)(v_0+\delta v)-\mu(T_0^*+\delta T^*)\approx\beta v_0 \delta T+\beta T_0 \delta v-\mu \delta T^* \\ \dot{\delta v}=\dot{v}=k(T_0^*+\delta T^*)-c(v_0+\delta v) = k\delta T^{*}-c\delta v. $$ Substituting the values from each fixed point lets you write these (linear) equations purely in terms of the free parameters $(d,c,k,s,\mu,\beta)$, and the Jacobian matrix will tell you the local stability. How far does that get you?