Question

S’ = -σSI – β(t)SL

L’ = σSI + β(t)SL – 1/τL L
I’ = 1/τL L – 1/ τR I and R’ = 1/ τR I
Show S+L+I+R is constant, allowing us to consider only the (S,L,I) system.

Answer 1

By summing the derivatives of S, L, I, and R as per the provided differential equations and simplifying, it's demonstrated that the total population S+L+I+R is constant over time. This allows us to consider a reduced system of equations for (S, L, I) as R can be implicitly determined by the conservation law.

Step-by-step explanation:

The question involves differential equations and conservation laws within a mathematical or physical model. To show that S+L+I+R is constant, we take the derivatives of each function and sum them up:

S' = -σSI − β(t)SL
L' = σSI + β(t)SL − 1/τL L
I' = 1/τL L − 1/ τR I
R' = 1/ τR I

Now, we calculate the derivative of the total population S+L+I+R:

(S+L+I+R)' = S' + L' + I' + R' = (-σSI − β(t)SL) + (σSI + β(t)SL − 1/τL L) + (1/τL L − 1/τR I) + (1/ τR I) = 0

This shows that the equation expresses a conservation law where the total population is constant in time, therefore, we can reduce the system to consider only the (S, L, I) variables for analysis, as R can be determined implicitly.