Property i: total population
Consider and . Then the addition of all three equations in (1) yields
Taking the Laplace transform of both sides of this equation yields
(4)
where is the Laplace transform . Taking the inverse Laplace transform of both sides of (4) gives
(5)
and therefore, the model (1) assumes that the total population is constant all the time.
Property ii: wellposedness
If the initial condition is nonnegative then the corresponding solution of the fractional model (1) is nonnegative all the time .
To show it, we can investigate the direction of the vector field on each coordinate plane and see whether the vector field points to the interior of or is tangent to the coordinate plane. Indeed:
-
(a)
On the coordinate plane IR, we have and
-
(b)
The same way, on the coordinate plane SR, we have and
-
(c)
Finally, on the coordinate plane SI, we have and
To conclude the property, we need the generalized mean value theorem proved in [16] and stated as follows.
Theorem 3.1 Let the function and its fractional derivative for , and then we have
where .
Thus, considering the interval for any , this theorem implies that the function is nonincreasing on if for all and nondecreasing on if for all . Then, from the points (a)-(c) above, the vector field on each coordinate plane is either tangent to the coordinate plane or and points to the interior of .
Property iii: extension of solutions on
The fundamental theory of differential equations (with integer or noninteger derivative order) tells us that their bounded solutions can be extended for all time . From Properties i and ii we have at least , , and , meaning that solutions are bounded in their maximal interval of existence. Thus, solutions to the fractional model (1) exist for .
Property iv: asymptotic behavior of solutions
We aim to show the existence of
Indeed, the first line of (1) yields
Thus, from Theorem 3.1, is nonincreasing (decreasing) on its domain of existence and since for all , we conclude that exists and .
In the same way, the third line of (1) gives
and from Theorem 3.1, is nondecreasing (increasing) on its domain of existence and since for all , we conclude that exists and .
Finally, the existence of is therefore obvious since and exist.