The linear SEIR model with a constant susceptible-to-exposed intensity is

\dot{S} = - q_{SE} S, \dot{E} = q_{SE} S - q_{E I} E, \dot{I} = q_{E I} E - q_{I R} I, \dot{R} = q_{I R} I .

S + E + I + R = N

ii

Solution

The separable equation

\dot{S} = - q_{SE} S

with

S (0) = S_{0}

has solution

S (t) = S_{0} e^{- q_{SE} t} .

Substitute part (ii) into

\dot{E} + q_{E I} E = q_{SE} S_{0} e^{- q_{SE} t}

e^{q_{E I} t}

gives

\frac{d}{d t} (e^{q_{E I} t} E (t)) = q_{SE} S_{0} e^{(q_{E I} - q_{SE}) t} .

Since

E (0) = 0

, integration and rearrangement yield

E (t) = \frac{S _{0} q _{SE}}{q _{E I} - q _{SE}} (e^{- q_{SE} t} - e^{- q_{E I} t}) .

At an interior maximum,

\dot{E} (t) = 0

, equivalently

q_{E I} e^{- q_{E I} t} = q_{SE} e^{- q_{SE} t} .

Thus

t_{m a x} = \frac{l o g ( q _{E I} / q _{SE} )}{q _{E I} - q _{SE}} .

This is positive for distinct positive rates. Its continuous limit when

q_{E I} = q_{SE} = q

1/ q

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