# SEIR model need help solving first order linear ODE

I've recently been working on simulating an SEIR (susceptible, exposed, infected, recovered) project for an endemic disease using matlab solving via Euler's method.

I have taken a picture of the model I'm using:

Basically the variables can be explained as follows:

• $a$ is the infection rate from susceptible to exposed
• $b$ is the progression rate for exposed to infected
• $c$ is the recovery rate from infected to recovered
• $M$ is a constant which represents both the birth rate and death rate
• $N$ is a constant, the total population $(S+E+I+R)$ still alive (due to birth rate = death rate)

The birth rate is added to the susceptibles and death rate subtracted from each of the 4 categories.

Anyhow, all I wanted to ask was that I'm currently solving this using Euler's method but wanted to verify my results against the real results. Does anyone know the analytical (real) solutions to the set of differential equations on the picture in the link?

• M is a constant which represents birth rate and death rate – bob Mar 1 '15 at 19:38
• That is a non-linear system. – mvw Mar 1 '15 at 21:42

SEIR models are not typically "solved". Finding a nice solution might not be possible. You could make some assumptions which reduce the number of variables (like close the system so $S+E+I+R=1$), but still the nonlinear parts $S(t)I(t)$ will remain a pain to deal with. Here is my advice:
• Alternatively you might want to look at things like equilibrium points for this SEIR model as well as the $R_0$ (reproductive number). These are often of interest to mathematical biologists and may corroborate your numerical solutions.