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for each positive solution of model (1.1).
where ) is the positive periodic solution of model (1.1).
This indicates that u is a solution of model (2).
Let be an any solution of model (2.1) and.
The solution of model (3.5) is globally attractive.
Suppose that (x t)) is a solution of model (4) with any given initial value.
end{aligned} Therefore, ((w x,t),sigma (t))) is an upper solution of model (1.3).
First, we show that the local existence and uniqueness of the solution of model (1.3).
We first investigated the positivity and boundedness of the solution of model (1.3).
Let ((S(n),E(n),I(n),R(n))) be any positive solution of model (2).
Figure 2 Periodic solution of model (3.2): the coefficients are given in Example 3.2.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com