Coronavirus 的數學模型(4):Delay Differential Equations

Delay Differential Equations

We follow the lecture notes by Professor Richard Bertram https://www.math.fsu.edu/~bertram/lectures/delay.pdf to present a short introduction to DDE.

A simple example

Recall that for the linear ODE

where 

Consider positive initial value, i.e., 

As a simple example of the delay differential equation (DDE), we change the time variable to 

To solve it, we need function value on 

Let's try to solve this DDE. For simplicity, we consider 

For 

With this is known, we can move to the next interval 

We thus get the quadratic function 

It is easy to conclude the general solution is:

where the constant 

So this is just the 

Stability analysis

We assume 

This is a transdental equation. When 

When 

The critical point 

How about the case 

Therefore if 

Now there are three unknowns 

We compute another critical point 

We conclude that for 

Numerical examples

We include numerical approximation using MATLAB dde23 for several choices of parameter 

For real negative eigenvalue,  the solution monotonily decrease to zero and behaves like 

For complex eigenvalue with negative real part, the solution decays to zero but with oscillations. Here we chose the parameter is slightly less than the critical one 

If we cross the critical point and chose  

When  

Delay Logistic Equation

We first recall the logistic ODE

whose solution is the logist function

The constant 

Now we introduce delay feedback into the logistic equation. Denoted by 

We are interested in if 

The stability analysis is to assume 

Substitute into the delay logistic equation and skip the quadratic part, we obtain a simple DDE

for which we have done the stability analysis before. Note that the sign of parameter is changed.

The conclusion is: for