The Tank

APPENDIX: SEIR MODEL
In these equations, 𝑆 + 𝐸 + 𝐼 + 𝑅 = 𝑁 is the total population, with rate of spread, β >
0, incubation rate σ > 0, and recovery rate γ > 0. The value of
represents the rate of change 𝑆
with respect to time 𝑡. Same as
,
,
.
The rate of spread, β is the rate of infection from an infected individual to one of their
susceptible contacts on the unitary time step dt. For example, given two people A (infectious) and
B (susceptible), the probability of B becoming infected after contacting A during the unitary time
step is β. The term
∆t is the difference between two observation points. Thus, number of
individuals transferred from Susceptible state to Exposed state is

𝛥𝑡, (14)
where is the
is the Force of Infection in the SEIR model. Similarly, on the unitary time step,
there are σE∆t number of cases transferred from Exposed state to Infectious, and γI (t)∆t number
of cases transferred from Infectious to Removed.
Let
S(t) ,
E(t),

I(t) and
R(t) be the number of susceptible, exposed, infectious and
removed individuals at time t, then
𝑆(𝑡 + ∆𝑡) = 𝑆(𝑡) − ()()
∆𝑡, (15)
𝐸(𝑡 + ∆𝑡) = 𝐸(𝑡) + ()()∆
− 𝜎𝐸(𝑡)∆𝑡, (16)
𝐼(𝑡 + ∆𝑡) = 𝐼(𝑡) + 𝜎𝐸(𝑡)∆𝑡 − 𝛾𝐼(𝑡)∆𝑡, (17)
𝑅(𝑡 + ∆𝑡) = 𝑅(𝑡) + 𝛾𝐼(𝑡)∆𝑡, (18)
Based on the definition of the first-order derivative,
=
(∆)() ∆
, as ∆𝑡 → 0 + . Thus
equation (15) – (18) can be rewritten as equation (10) – (13).
Assumptions of SEIR model [24] are as follows.
i. The SEIR model assumes a closed population, which means that the total number of
populations is fixed, no births, no death, or introduction new individuals. From equation
(10) – (13), we see that
[𝑆(𝑡) + 𝐸(𝑡) + 𝐼(𝑡) + 𝑅(𝑡)] = 0, where the population N is
constant in any time 𝑡:𝑆(𝑡) + 𝐸(𝑡) + 𝐼(𝑡) + 𝑅(𝑡) = 𝑁 for any 𝑡 ≥ 0.
ii. The individuals in the Exposed state are infected but not yet infectious.
iii. Well-mixed population.
iv. SEIR model assumes that the latent and infectious times of the pathogen are exponentially
distributed.
In general, the dynamic SEIR is summarized as below [24].
a) Start the epidemic with a group of Susceptible individuals and at least one Infectious
individual.
b) The Infectious individuals mix with the Susceptible class and create Exposed individuals
following a probabilistic process.
c) Exposed individuals spend some days without spreading the virus and based on another
probabilistic process become additional Infectious class.
d) Newly Infectious class repeat #2 and create more Exposed class.
Infectious individuals based on a probabilistic process either recover or die and become Removed
class.