Logistic from SEIR

This notebook derives the logistic epidemiologic model from the SEIR posted by Enrique Garcia Moreno E.
This is the set of differential equations he posted recently.
Out[]=
It looks like we can derive the logistic model from this SEIR model if we make an assumption that at the beginning of the epidemic s[t] is large and relatively constant. Below I substitute S for s[t]/p and solve for the e[t] and i[t].
In[]:=
solution=DSolve[{e'[t]-σe[t]+βi[t]S,i'[t]σe[t]-γi[t]},{e[t],i[t]},t];
Now take the solution for i[t]
In[]:=
isol=FullSimplify[i[t]/.solution[[1]]]//ExpandAll
Out[]=
-
-
tγ
2
-
tσ
2
-
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ

σ

1
2
γ
+4Sβσ-2γσ+
2
σ
+
-
tγ
2
-
tσ
2
+
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ

σ

1
2
γ
+4Sβσ-2γσ+
2
σ
+
1
2
-
tγ
2
-
tσ
2
-
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ


2
+
1
2
-
tγ
2
-
tσ
2
+
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ


2
+
-
tγ
2
-
tσ
2
-
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ

γ

2
2
2
γ
+4Sβσ-2γσ+
2
σ
-
-
tγ
2
-
tσ
2
+
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ

γ

2
2
2
γ
+4Sβσ-2γσ+
2
σ
-
-
tγ
2
-
tσ
2
-
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ

σ

2
2
2
γ
+4Sβσ-2γσ+
2
σ
+
-
tγ
2
-
tσ
2
+
1
2
t
2
γ
+4Sβσ-2γσ+
2
σ

σ

2
2
2
γ
+4Sβσ-2γσ+
2
σ
Simplify by creating a simpler constant -- a little trickery is needed to do this all at once.
In[]:=
isol1=isol/.
2
γ
+4Sβσ-2γσ+
2
σ
w,1
2
γ
+4Sβσ-2γσ+
2
σ
1w//FullSimplify
Out[]=
1
2w
-
1
2
t(w+γ+σ)

((w+
tw

(w-γ)+γ)

2
+(-1+
tw

)σ(2

1
+

2
))
Now make another substitution where
t/2

 x and
t


2
x
In[]:=
isol2=isol1/.
-
1
2
t(w+γ+σ)

ax,
tw

bx^2
Out[]=
ax((w+b
2
x
(w-γ)+γ)

2
+(-1+b
2
x
)σ(2

1
+

2
))
2w
Now take the first derivative of the solution for i[t] and you get a quadratic equation consistent with the logistic model.
In[]:=
Collect[D[isol2,x],x]
Out[]=
a

2
2
+
aγ

2
2w
-
aσ(2

1
+

2
)
2w
+
2
x
3ab(w-γ)

2
2w
+
3abσ(2

1
+

2
)
2w