Question

Recall the Kermack-McKendrick Model (also known as the SIR-model), which models the spread of an infection by looking at the population of susceptibles S, infected I, and recovered R at time t. A modification to this model, known as the SEIR-model1 , introduces a new group E that represents the exposed population. The idea behind E(t) is that when someone becomes exposed to the virus, the virus takes some time to incubate before the person becomes infective. I took the SEIR-model and made two additional assumptions that reflect the specific nature of COVID-19: 1. When someone recovers from the virus, they are not immune. 2. Someone that is exposed (and infected) by the virus might display no symptoms but is still infective. These assumptions lead to the following system of autonomous differential equations, where λ, µ1, µ2, µ3, β1, β2, γ, and δ are positive parameters.

dS/dt = (λ − µ1)S − β1SI − β2SE + γI (1)

dE/dt = +β1SI + β2SE − (δ + µ2)E (2)

dI/dt = δE − (γ + µ3)I (3)

Note that I’m ignorantly / naively assuming that no one else has done this exact model and hence I named it after myself. We can decipher the parameter’s meaning by setting assuming certain terms are zero; and/or comparing the parameters to the SIR-model. For instance, λ is the birth rate, and each µ represents a death rate.

(a) For each of β1, β2, γ, and δ, pick the most appropriate meaning from the following list: (i) incubation period of the virus (ii) infectivity of an exposed person (iii) infectivity of a fully-infected person (iv) recovery rate

(b) To simplify the model, let’s assume λ = µ1 = µ2 = µ3 = 0. Clearly, (S, ˆ E, ˆ ˆI) = (0, 0, 0) is a trivial equilibrium. It turns out there are infinitely more equilibria. Find them – you’ll need to use a free variable. Hint: start by setting equation (3) to zero, which will you give an equation involving E and I. Plug this equation into equations (1) and (2).

Answer 1