Property i: total population
Consider N(t)=S(t)+I(t)+R(t) and N(0)={N}_{0}={S}_{0}+{I}_{0}+{R}_{0}. Then the addition of all three equations in (1) yields
{D}_{t}^{\alpha}N(t)=0\phantom{\rule{1em}{0ex}}\text{for all}t0.
Taking the Laplace transform of both sides of this equation yields
{s}^{\alpha}\tilde{N}(s){s}^{\alpha 1}N(0)=0,
(4)
where \tilde{N}(s) is the Laplace transform \mathcal{L}(N(t),s). Taking the inverse Laplace transform of both sides of (4) gives
N(t)={N}_{0}\phantom{\rule{1em}{0ex}}\text{for all}t0
(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 (S(t),I(t),R(t)) of the fractional model (1) is nonnegative all the time t>0.
To show it, we can investigate the direction of the vector field (\beta g(I)S,\beta g(I)S\gamma I,\gamma I) on each coordinate plane and see whether the vector field points to the interior of {\mathbb{R}}_{+}^{3}=\{(S,I,R)\in {\mathbb{R}}^{3}:S\ge 0,I\ge 0,R\ge 0\} or is tangent to the coordinate plane. Indeed:

(a)
On the coordinate plane IR, we have S=0 and
{D}_{t}^{\alpha}{S}_{{}_{S=0}}=0.

(b)
The same way, on the coordinate plane SR, we have I=0 and
{D}_{t}^{\alpha}{I}_{{}_{I=0}}=\beta g(0)S=0.

(c)
Finally, on the coordinate plane SI, we have R=0 and
{D}_{t}^{\alpha}{R}_{{}_{R=0}}=\gamma I\ge 0.
To conclude the property, we need the generalized mean value theorem proved in [16] and stated as follows.
Theorem 3.1 Let the function U\in C[{t}_{1},{t}_{2}] and its fractional derivative {D}_{t}^{\alpha}U\in C({t}_{1},{t}_{2}] for 0\le \alpha <1, and {t}_{1},{t}_{2}\in \mathbb{R} then we have
U(t)=U({t}_{1})+\frac{1}{\mathrm{\Gamma}(\alpha )}{D}_{t}^{\alpha}U(\tau ){(t{t}_{1})}^{\alpha}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t\in ({t}_{1},{t}_{2}],
where 0\le \tau <t.
Thus, considering the interval [0,{t}_{2}] for any {t}_{2}>0, this theorem implies that the function U:[0,{t}_{2}]\to {\mathbb{R}}^{+} is nonincreasing on [0,{t}_{2}] if {D}_{t}^{\alpha}U(t)\le 0 for all t\in (0,{t}_{2}) and nondecreasing on [0,{t}_{2}] if {D}_{t}^{\alpha}U(t)\ge 0 for all t\in (0,{t}_{2}). 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 {\mathbb{R}}_{+}^{3}.
Property iii: extension of solutions on (\mathrm{\infty},+\mathrm{\infty})
The fundamental theory of differential equations (with integer or noninteger derivative order) tells us that their bounded solutions can be extended for all time t\in \mathbb{R}. From Properties i and ii we have at least {N}_{0}\ge S\ge 0, {N}_{0}\ge I\ge 0, and {N}_{0}\ge R\ge 0, meaning that solutions (S(t);I(t);R(t)) are bounded in their maximal interval of existence. Thus, solutions to the fractional model (1) exist for t\in (\mathrm{\infty},+\mathrm{\infty}).
Property iv: asymptotic behavior of solutions
We aim to show the existence of
\underset{t\to \mathrm{\infty}}{lim}S(t)={S}_{\mathrm{\infty}},\phantom{\rule{2em}{0ex}}\underset{t\to \mathrm{\infty}}{lim}I(t)={I}_{\mathrm{\infty}},\phantom{\rule{2em}{0ex}}\underset{t\to \mathrm{\infty}}{lim}R(t)={R}_{\mathrm{\infty}}.
Indeed, the first line of (1) yields
{D}_{t}^{\alpha}S=\beta g(I)S\le 0,\phantom{\rule{1em}{0ex}}\text{since}g(I)\ge 0.
Thus, from Theorem 3.1, S(t) is nonincreasing (decreasing) on its domain of existence and since 0\le S(t) for all t\in (\mathrm{\infty},+\mathrm{\infty}), we conclude that {S}_{\mathrm{\infty}} exists and 0\le {lim}_{t\to \mathrm{\infty}}S(t)={S}_{\mathrm{\infty}}.
In the same way, the third line of (1) gives
{D}_{t}^{\alpha}R=\gamma I\ge 0
and from Theorem 3.1, R(t) is nondecreasing (increasing) on its domain of existence and since {N}_{0}\ge R(t) for all t\in (\mathrm{\infty},+\mathrm{\infty}), we conclude that {R}_{\mathrm{\infty}} exists and {N}_{0}\ge {lim}_{t\to \mathrm{\infty}}S(t)={R}_{\mathrm{\infty}}.
Finally, the existence of {I}_{\mathrm{\infty}}\phantom{\rule{0.25em}{0ex}}(={N}_{0}{S}_{\mathrm{\infty}}{R}_{\mathrm{\infty}})\ge 0 is therefore obvious since {S}_{\mathrm{\infty}} and {R}_{\mathrm{\infty}} exist.