Constructing Lyapunov function for system of ODEs

In summary: Finally, you can combine these individual Lyapunov functions to obtain a composite Lyapunov function for the entire system, which would be of the form $L = L_A + L_D + L_G$. This function should satisfy the conditions for a Lyapunov function and provide an insight into the stability of the system.I hope this suggestion is helpful to you. Please keep in mind that this is just one possible approach, and there may be other ways to construct a Lyapunov function for your system. I also encourage you to seek feedback from other researchers and experts in this field. Good luck with your research.
  • #1
kalish1
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**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
 
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  • #2
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^* =
 

FAQ: Constructing Lyapunov function for system of ODEs

What is a Lyapunov function?

A Lyapunov function is a mathematical function used to analyze the stability of a dynamical system. It measures the energy or the distance of the system from a stable equilibrium point, and helps determine whether the system will converge to or diverge from the equilibrium point.

Why is constructing a Lyapunov function important?

Constructing a Lyapunov function is important because it helps us determine the stability of a system of ordinary differential equations (ODEs). It allows us to analyze the behavior of the system and predict whether it will reach a steady state or oscillate around the equilibrium point.

What are the steps for constructing a Lyapunov function?

The steps for constructing a Lyapunov function for a system of ODEs are as follows:

  1. Choose a candidate Lyapunov function that is positive definite, continuous, and differentiable.
  2. Compute the derivative of the candidate Lyapunov function along the trajectories of the system.
  3. Choose a control law that ensures that the derivative of the Lyapunov function is negative definite.
  4. Verify that the Lyapunov function is decreasing along the trajectories of the system and converges to zero as time goes to infinity.

Can a Lyapunov function be used for any system of ODEs?

No, a Lyapunov function can only be used for certain types of systems, such as nonlinear autonomous systems or linear time-invariant systems. It is not applicable to all types of systems since the function requires specific properties to ensure stability analysis.

Are there any limitations to using a Lyapunov function for stability analysis?

Yes, there are some limitations to using a Lyapunov function for stability analysis. For example, it may not be able to determine the stability of non-autonomous or time-varying systems. Additionally, the function may not be able to provide information on the transient behavior of the system, and may only be able to determine the stability of the equilibrium point.

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