Constructing Lyapunov function for system of ODEs

[kalish1]

**Background:** I have been working on this problem for my research for months now, and I am in dire need of help. That is why I have come here to seek help.

I have a system of nine ODEs that describe the dynamics of HIV and Tuberculosis co-infection in a population. The disease-free equilibrium is $E_0=(\frac{\Lambda}{\mu},0,0,0,0,0,0,0).$ I also have the expression for the reproductive number, $\Re_0$.

**Problem:** I need to find a Lyapunov function that will establish negative-definiteness if and only if $\Re_0 \leq 1.$

Here is the setup:
\begin{array}
$
\frac{dA}{dt}=\Lambda-\mu A-\beta(C+D+E+F)\frac{A}{N}-\tau(B+D)\frac{A}{N} \\
\frac{dB}{dt}=\tau(B+D)\frac{A}{N}-\beta(C+D+E+F)\frac{B}{N}-(\mu+\mu_T)B, \\
\frac{dC}{dt}=\beta(C+D+E+F)\frac{A}{N}-\tau(B+D)\frac{C}{N}-(\mu+\mu_A)C, \\
\frac{dD}{dt}=\beta(C+D+E+F)\frac{B}{N}+\tau(B+D)\frac{C}{N}-(\mu+\mu_T+\psi\mu_A+\lambda_T)D, \\
\frac{dE}{dt}=\lambda_TD-(\mu+\mu_A+\rho_1+\eta_1)E, \\
\frac{dF}{dt}=\rho_1E-(\mu+\mu_A+\rho_2+\eta_2)F, \\
\frac{dG}{dt}=\eta_1E-(\mu+\rho_1+\gamma)G, \\
\frac{dH}{dt}=\eta_2H+\rho_1G-(\mu+\rho_2+\frac{\gamma\rho_1}{\rho_1+\rho_2})H, \\
\frac{dN}{dt}= \Lambda-\mu A - (\mu+\mu_T)B - (\mu+\mu_A)C -(\mu+\mu_T+\psi\mu_A)D - (\mu+\mu_A)E - (\mu+\mu_A+\rho_2)F -(\mu+\gamma)G -(\mu+\rho_2+\dfrac{\gamma\rho_1}{\rho_1+\rho_2})H
%$
\end{array}

where $$\Re_0 = \max\left\{\frac{\beta}{\mu+\mu_H},\frac{\tau}{\mu+\mu_T}\right\}.$$

I have tried many classes of functions: quadratic, composite quadratic, logarithmic, etc. to no avail. My advisor tells me that a logarithmic Lyapunov function should work. Something like $$L(t)=\left(A-A^{*}-\ln\frac{A}{A^{*}}\right)+(B+C+D+E+F+G+H), $$ where $A^{*}=\frac{\Lambda}{\mu}.$

Please let me know what looks feasible. Thank you.

I have crossposted this question on MSE: http://math.stackexchange.com/questions/986704/constructing-lyapunov-function-for-system-of-odes

 

[jvicens]

Dear researcher,

Thank you for reaching out for help with your research problem. From the information provided, it seems like you have a well-defined system of nine ODEs that describe the dynamics of HIV and Tuberculosis co-infection in a population. It is great that you have also obtained the expression for the reproductive number, $\Re_0$, as it is an important parameter in understanding the stability of the system.

To address your problem, let us first recall the definition of a Lyapunov function. A Lyapunov function is a scalar function of the state variables of a dynamical system that is positive definite, continuously differentiable, and satisfies certain conditions that guarantee the stability of the system. In your case, you are looking for a Lyapunov function that will establish negative-definiteness if and only if $\Re_0 \leq 1$.

As you have mentioned, you have already tried several classes of functions, including quadratic and logarithmic functions. It is important to note that the choice of the Lyapunov function depends on the specific system and its dynamics. Therefore, it is possible that the functions you have tried did not work for your system. However, I would like to suggest a possible approach that you can try.

First, let us consider the components of the state variables in your system. From the setup, we can see that the state variables can be divided into three groups: susceptible individuals (A, B, C), infected individuals (D, E, F), and recovered individuals (G, H). This division can provide some insight into the dynamics of the system.

Next, as you have mentioned, a logarithmic Lyapunov function may be suitable for your system. However, instead of using a logarithmic function for the entire system, you can try constructing a Lyapunov function for each group of state variables separately. For example, for the susceptible group, you can try a function like $L_A = A - A^* - \ln(A/A^*)$, where $A^* = \Lambda/\mu$. Similarly, for the infected group, you can try a function like $L_D = D - D^* - \ln(D/D^*)$, where $D^* = \lambda_T/\mu$. For the recovered group, you can try $L_G = G - G^* - \ln(G/G^*)$, where $G^* =