Equilibria of the model
Obviously, the equilibrium points are positive solutions of the system
$$ \textstyle\begin{cases} 0= rN(1\frac{N}{k})(\lambda +a(S+I))N(S+I), \\ 0= e(\lambda +a(S+I))N(S+I)\beta SI \mu S, \\ 0= \beta SI \eta I \mu I. \end{cases} $$
(6)
As a first remark we can derive that (6) has \(\Lambda _{0}=(0,0,0)\), \(\Lambda _{1}=(k,0,0)\) as equilibrium points. Now we look for the diseasefree equilibrium (DFE) which is written as \(\Lambda _{3}=(N,S,0)\), where \((N,S)\) is the positive solution of the following system:
$$ \textstyle\begin{cases} 0= erN(1\frac{N}{k})e(\lambda +aS)NS, \\ 0= e(\lambda +aS)NS\mu S. \end{cases} $$
(7)
By summing the two equations in (7), we get
$$ S=\frac{erN}{\mu }\biggl(1\frac{N}{k}\biggr). $$
By replacing this result in the first equation of (6), we find the following thirdorder polynomial:
$$ \frac{ae^{2}r}{k\mu ^{2}}N^{3}\frac{ae^{2}r}{\mu ^{2}}N^{2} \frac{\lambda e}{\mu }N+1=0. $$
(8)
As a first remark, and using the Descartes rule, we can deduce that equation (8) has either two or no positive solutions in the positive quadrant. The number of positive solutions in the interval \([0,k]\) dominates the number of the DFEs. Now we set
$$ f(N)=\frac{ae^{2}r}{k\mu ^{2}}N^{3}\frac{ae^{2}r}{\mu ^{2}}N^{2} \frac{\lambda e}{\mu }N+1. $$
At first, we can highlight that f verifies
$$ f(0)=1>0,\qquad f(k)=\frac{\lambda k e}{\mu }+1, $$
also,
$$ f'(N)=\frac{3ae^{2}r}{k\mu ^{2}}N^{2} \frac{2ae^{2}r}{\mu ^{2}}N \frac{\lambda e}{\mu } $$
and
$$ f'(0)=\frac{\lambda e}{\mu }. $$
Using the fact that \(f(x)\) is a thirdorder polynomial and using \(f'(0)<0\), we can deduce that f has a unique global minimum at \(N_{\min }\), where
$$ N_{\min }= \frac{aerk+\sqrt{a^{2}e^{2}r^{2}k^{2}+3\lambda a e r k \mu }}{3aer}. $$
By a simple discussion on the positivity of \(f(N_{\min })\) and \(f(k)\), we resume the existence conditions for the DFEs (see also Fig. 1) in the following theorem.
Theorem 2.1
The existence of the DFE for system (5) is arranged in the following aspects:

(i)
System (5) has no DFE if
$$ \biggl(N_{\min }< k\textit{ and }\frac{ae^{2}r}{k\mu ^{2}}N^{3} _{\min }+1> \frac{ae^{2}r}{\mu ^{2}}N^{2} _{\min }+ \frac{\lambda e}{\mu }N_{\min } \biggr). $$

(ii)
System (5) has one DFE denoted by \(\Lambda _{3} =(N_{3},S_{3},0)\) if \(\mu <\lambda ke\) or
$$ \biggl(N_{\min }< k\textit{ and }\frac{ae^{2}r}{k\mu ^{2}}N^{3} _{\min }+1= \frac{ae^{2}r}{\mu ^{2}}N^{2} _{\min }+ \frac{\lambda e}{\mu }N_{\min } \biggr). $$

(iii)
System (5) has two DFEs denoted by \(\Lambda _{3} =(N_{3},S_{3},0)\) and \(\Lambda _{4} =(N_{4},S_{4},0)\) if \(\mu >\lambda ke\), and
$$ \biggl(N_{\min }< k \textit{ and }\frac{ae^{2}r}{k\mu ^{2}}N^{3} _{\min }+1< \frac{ae^{2}r}{\mu ^{2}}N^{2} _{\min }+ \frac{\lambda e}{\mu }N_{\min } \biggr). $$
Next, we investigate the existence of endemic equilibrium EE which highlights the persistence of the three populations. From the third equation of (6) we deduce that the susceptible density equilibrium is
$$ S^{*}=\frac{\mu +\eta }{\beta }. $$
Note that the infected density equilibrium and the resource density equilibrium are the solution of the system
$$ \textstyle\begin{cases} 0= r(1\frac{N}{k})(\lambda +a(S^{*}+I))(S^{*}+I), \\ 0= e(\lambda +a(S^{*}+I))N(S^{*}+I)(\beta I+\mu ) S^{*}. \end{cases} $$
(9)
From the first equation of (9) we get
$$ N=f_{1}(I)=k \biggl(1\frac{\lambda }{r}S^{*} \frac{a}{r}\bigl(S^{*}\bigr)^{2} \biggr)( \lambda +2a)\frac{kI}{r}\frac{ka}{r}I^{2}. $$
(10)
It is easy to see that \(f_{1}\) is a strictly decreasing concave function; also, for guaranteeing the positivity of \(f_{1}\) for some values of I, we assume that \(1>\frac{\lambda }{r}S^{*}+\frac{a}{r}(S^{*})^{2}\) and intersects the horizontal axis at
$$ I_{\mathrm{int}}=a^{1} \bigl[(\lambda +2a)+\sqrt{(\lambda +2a)^{2}+4a\bigl(r \lambda S^{*}a\bigl(S^{*} \bigr)^{2}\bigr)} \bigr]. $$
Using the second equation of (9), we have
$$ N=f_{2}(I)= \frac{(\beta I+\mu ) S^{*}}{e(\lambda +a(S^{*}+I))(S^{*}+I)}, $$
(11)
which is a positive functional. Under the condition \(\beta <\frac{\mu }{S^{*}}\) and \(a< a^{*}:=\frac{\mu \lambda \beta S*\lambda }{\beta (S^{*})^{2}}\), we get that \(f_{2}\) is strictly decreasing in I. Hence, we can draw the following result.
Theorem 2.2
Assume that \(1>\frac{\lambda }{r}S^{*}+\frac{a}{r}(S^{*})^{2}\), \(\beta <\frac{\mu }{S^{*}}\), \(a< a^{*}:=\frac{\mu \lambda \beta S^{*}\lambda }{\beta (S^{*})^{2}}\), then system (5) has an interior equilibrium denoted by \(\Lambda ^{*}=(N^{*},S^{*},I^{*})\) where
$$ S^{*}=\frac{\mu +\eta }{\beta },\qquad N^{*}= \frac{(\beta I^{*}+\mu ) S^{*}}{e(\lambda +a(S^{*}+I^{*}))(S^{*}+I^{*})}, $$
and \(I^{*}\) is the positive intersection between the graphical representation of \(f_{1}\) and \(f_{2}\).
Asymptotic behaviors (5)
In this part, we are interested in determining the asymptotic stability of the equilibria obtained in the previous section.
Letting \((N,S,I)\) be an equilibrium for system (5), the Jacobian matrix (Jmatrix) of (5) at \((N,S,I)\) is
$$ J(N,S,I)= \begin{pmatrix} r(1\frac{2N}{k})(\lambda +a(S+I))(S+I) & N(\lambda +2a(S+I)) & N( \lambda +2a(S+I)) \\ e(\lambda +a(S+I))(S+I) & eN(\lambda +2a(S+I))\mu \beta I & eN( \lambda +2a(S+I))\beta S \\ 0 & \beta I & \beta S(\mu +\eta ) \end{pmatrix}. $$
(12)
For defining the concept of the local stability for the fractionalorder system, we set the following theorem.
Theorem 2.3
At a random equilibrium for fractional system (5), the local stability of the equilibrium occurs if the eigenvalues θ of the Jmatrix (12) verify \(\vert \arg (\theta ) \vert >\frac{\alpha \pi }{2} \) for each eigenvalue λ of J. The equilibrium for (6) is unstable if \(\vert \arg (\theta ) \vert <\frac{\gamma \pi }{2}\) for some eigenvalues θ.
At the origin \(\Lambda _{0}\), Jmatrix (12) becomes
$$ J(0,0,0)= \begin{pmatrix} r & 0&0 \\ 0 & \mu & 0 \\ 0 & 0 & (\mu +\eta ) \end{pmatrix} . $$
(13)
Jmatrix (13) has the eigenvalues \(\theta _{1}=r>0\), \(\theta _{2}=\mu <0\), \(\theta _{3}=(\mu +\eta )\). This equilibrium is always unstable.
Now, we evaluate the Jmatrix at \(\Lambda _{1}\), we get
$$ J(k,0,0)= \begin{pmatrix} r & k\lambda &k\lambda \\ 0 & ek\lambda \mu & ek\lambda \\ 0 & 0 & (\mu +\eta ) \end{pmatrix} . $$
(14)
Jmatrix (14) has the eigenvalues \(\theta _{1}=r<0\), \(\theta _{2}=ek\lambda \mu \), \(\theta _{3}=( \mu +\eta )\). Then the sign of \(\theta _{2}\) dominates the stability/instability of the equilibrium \(\Lambda _{1}\). Hence,
$$ \theta _{2}= \textstyle\begin{cases} < 0 &\text{for } k< \frac{\mu }{e\lambda } , \\ >0 &\text{for } k>\frac{\mu }{e\lambda } . \end{cases} $$
The stability conditions are summarized in the following lemma.
Lemma 2.4
\(\Lambda _{1}\) is locally stable if \(k<\frac{\mu }{e\lambda }\) and unstable if \(k>\frac{\mu }{e\lambda }\).
Now, we focus on analyzing the stability of \(\Lambda _{i}\), \(i=3,4\),
$$ J(N_{i},S_{i},0)= \begin{pmatrix} \frac{rN_{i}}{k} & N_{i}(\lambda +2aS_{i}) & N_{i}(\lambda +2aS_{i}) \\ \frac{\mu S_{i}}{N_{i}} & eN_{i}aS_{i} & eN_{i}(\lambda +2aS_{i}) \beta S_{i} \\ 0 & 0 & \beta S_{i}(\mu +\eta ) \end{pmatrix},\quad i=3,4 . $$
(15)
Obviously, \(\theta _{3}=\beta S_{i}(\mu +\eta )\) can be considered as an eigenvalue of Jmatrix (15), hence, if \(\beta S_{i}>(\mu +\eta )\), then these equilibria are unstable. Now, we assume that \(\beta S_{i}<(\mu +\eta )\), which means that \(\theta _{3}<0\), hence the other two eigenvalues dominate the stability of the DFEs. The other two are the solution of the equation
$$ \theta ^{2}\operatorname{Tr}_{i} \theta +\mathrm{Det}_{i}=0,\quad i=3,4, $$
(16)
where
$$ \textstyle\begin{cases} \operatorname{Tr}_{i}= \frac{rN_{i}}{k}+ eN_{i}aS_{i} \\ \mathrm{Det}_{i}= \frac{eraS_{i}N_{i} ^{2}}{k} + \mu S_{i} (\lambda +2aS_{i}) \end{cases}\displaystyle i=3,4. $$
(17)
Clearly, if \(\mathrm{Det}_{i}<0\), \(i=3,4\), hence we get the instability of the equilibria \(\Lambda _{i}\), \(i=3,4\). Now we consider that \(\mathrm{Det}_{i}>0\) \(i=3,4\), then if \(\operatorname{Tr}_{i}<0\), \(i=3,4\), we conclude that \(\Lambda _{i}\), \(i=3,4\), are stable. Now we consider that \(\mathrm{Det}_{i}>0\), \(\operatorname{Tr}_{i}>0\), \(i=3,4\), hence, these equilibria are unstable for the firstorder derivative. However, for the FOD we have a possibility of the stability of this equilibrium. Note that in this situation (16) has two complex roots \(\theta _{j}=\tilde{\theta }_{j} \pm \bar{\theta }_{j} i\), \(\theta _{j} \in \mathbb{R}\), \(j=3,4\). Then these roots verify the following:
$$ \tan ^{2} \bigl( \arg \{ \theta _{i} \} \bigr) = \frac{4 (\frac{eraS_{i}N_{i} ^{2}}{k} + \mu S_{i} (\lambda +2aS_{i}) )}{ (\frac{rN_{i}}{k}+ eN_{i}aS_{i} )^{2}}1,\quad i=3,4. $$
For guaranteeing the stability of \(\Lambda _{i}\), \(i=3,4\), we must obtain \(\tan ^{2} ( \arg \{ \theta _{i} \} ) >\tan ^{2} ( \frac{\alpha \pi }{2} ) \), \(i=3,4\), which is equivalent to
$$ (H_{1}):\quad \frac{4 (\frac{eraS_{i}N_{i} ^{2}}{k} + \mu S_{i}(\lambda +2aS_{i}) )}{ (\frac{rN_{i}}{k}+ eN_{i}aS_{i} )^{2}}>1+ \tan ^{2} \biggl( \frac{\alpha \pi }{2} \biggr). $$
Then, if \((H_{1})\) holds, then we get the stability of the DFEs, else it is instable. The obtained results are resumed in the following theorem.
Theorem 2.5
Assume that condition (ii) or (iii) in Theorem 2.1is verified, then we get:

(i)
If \(\beta S_{i}>(\mu +\eta )\), then DFE is unstable.

(ii)
If \(\beta S_{i}<(\mu +\eta )\) and \((H_{1})\) holds, then DFEs is stable, otherwise it is unstable.
Now, determining the stability of the endemic equilibrium (EE), the Jmatrix at this equilibrium takes the following form:
$$ J\bigl(N^{*},S^{*},I^{*}\bigr)= \begin{pmatrix} \frac{rN^{*}}{k} & N^{*}(\lambda +2a(S^{*}+I^{*})) & N^{*}( \lambda +2a(S^{*}+I^{*})) \\ e(\lambda +a(S^{*}+I^{*}))(S^{*}+I^{*}) & eN^{*}(\lambda +2a(S^{*}+I^{*})) \mu \beta I^{*} & eN^{*}(\lambda +2a(S^{*}+I^{*}))\beta S^{*} \\ 0 & \beta I^{*} & 0 \end{pmatrix} , $$
(18)
the characteristic equation corresponding to Jmatrix (18) is
$$ \Delta =\theta ^{3}+\omega _{2}\theta ^{2}+ \omega _{1}\theta +\omega _{0}, $$
where
$$\begin{aligned}& \omega _{2}= \frac{rN^{*}}{k}\bigl(eN^{*}\bigl( \lambda +2a\bigl(S^{*}+I^{*}\bigr)\bigr) \mu \beta I^{*}\bigr), \\& \begin{aligned} \omega _{1}={}& {} \frac{rN^{*}(S^{*}+I^{*})(eN^{*}(\lambda +2a(S^{*}+I^{*}))\mu \beta I^{*})}{k}\\ &{}+eN^{*}\bigl( \lambda +a\bigl(S^{*}+I^{*} \bigr)\bigr) \bigl(S^{*}+I^{*}\bigr) \bigl(\lambda +2a \bigl(S^{*}+I^{*}\bigr)\bigr) \\ &{}\beta I^{*} \bigl( eN^{*}\bigl(\lambda +2a \bigl(S^{*}+I^{*}\bigr)\bigr)\beta S^{*} \bigr) , \end{aligned} \\& \begin{aligned} \omega _{0}={}& {} \frac{r\beta I^{*}N^{*}}{k} \bigl( eN^{*} \bigl(\lambda +2a\bigl(S^{*}+I^{*}\bigr)\bigr)\\ &{} \beta S^{*} \bigr) \beta I^{*} N^{*} \bigl(e\bigl( \lambda +a\bigl(S^{*}+I^{*}\bigr)\bigr) \bigl(S^{*}+I^{*}\bigr) \bigr) \bigl(\lambda +2a \bigl(S^{*}+I^{*}\bigr) \bigr). \end{aligned} \end{aligned}$$
Put \(\Delta =18\omega _{2}\omega _{1}\omega _{0}+ ( \omega _{2} \omega _{1} ) ^{2}4\omega _{0} \omega _{2} ^{3}4 \omega _{1} ^{3}27 \omega _{0} ^{3}\). By using the Routh–Hurwitz criterion defined in [2, 3], we find the local stability of the EE, which is highlighted in the following theorem.
Theorem 2.6
Assume that the condition mentioned in Theorem 2.2holds, then the EE is stable if

(i)
\(\Delta >0\), \(\omega _{2}>0\), \(\omega _{0}>0\), \(\omega _{2}\omega _{1}>\omega _{0}\), or

(ii)
\(\Delta <0\), \(\omega _{2}\geq 0\), \(\omega _{1}\geq 0\), \(\omega _{0}\geq 0\), and \(\alpha <\frac{2}{3}\).
Numerical scheme
Our goal in this subsection is to build a numerical scheme for our graphical representations for confirming the results found in the previous section. At first, we consider the following fractional problem:
$$ \frac{d^{\alpha }\chi }{dt}= {G}\bigl(t, \chi (t)\bigr). $$
(19)
The employment of the principal theorem of fractional calculus on (5) yields
$$ \chi (t)  \chi (0) = \frac{1}{\Gamma (\alpha )} \int _{0}^{t} {G}\bigl( \upsilon , \chi ( \upsilon )\bigr) (t\upsilon )^{\alpha 1}\,d\upsilon . $$
(20)
We set \(t=t_{n}=n\hslash \) in (20), we find
$$ \chi (t_{n}) = \chi (0)+\frac{1 }{\Gamma (\alpha )}\sum _{i=0}^{n1} \int _{t_{i}}^{t_{i+1}} {G}\bigl(\upsilon , \chi ( \upsilon )\bigr) (t_{n} \upsilon )^{\alpha 1}\,d\upsilon . $$
(21)
By approximating the functional \({G}(t, \chi (t))\) by
$$ {G}\bigl(t, \chi (t)\bigr)\approx {G}(t_{i+1}, \chi _{i+1} )+ \frac{tt_{i+1}}{\hslash } \bigl( {G}(t_{i+1}, \chi _{i+1} )\bigr) {G}(t_{i}, \chi _{i} ) ) , \quad t\in [t_{i},t_{i+1}], $$
(22)
where \(\chi _{i}= \chi (t_{i})\).
The substitution of Eq. (22) into (21) gives (for more details, see [70])
$$ \chi _{n}= \chi _{0} +\hslash ^{\alpha } \Biggl(A _{n} {G}(t_{0}, \chi _{0} )+\sum_{i=1}^{n}B_{ni} {G}(t_{i}, \chi _{i} ) \Biggr), $$
(23)
where
$$ \begin{gathered} A _{n}= \frac{(n1)^{\alpha +1}n^{\alpha }(n\alpha 1)}{\Gamma (\alpha +2)} \\ B_{n}=\textstyle\begin{cases} \frac{1}{{\Gamma (\alpha + 2)}},& n = 0 \\ \frac{{{{(n  1)}^{\alpha }}  2{n^{\alpha }} + {{(1+n)}^{\alpha }}}}{{\Gamma ( \alpha + 2)}}, & n = 1,2, \ldots \end{cases}\displaystyle \end{gathered} $$
(24)
The application of the numerical method (23) to solve (5) yields
$$ \begin{gathered} N_{n} = N_{0} +\hslash ^{\alpha } \Biggl(A _{n} {G}_{1}( N_{0},S_{0},I_{0})+ \sum _{i=1}^{n}B_{ni} {G}_{1}( N_{i},S_{i},I_{i}) \Biggr), \\ S_{n} = S_{ 0} +\hslash ^{\alpha } \Biggl(A _{n} {G}_{2}( N_{0},S_{0},I_{0})+ \sum_{i=1}^{n}B_{ni} {G}_{2}( N_{i},S_{i},I_{i}) \Biggr), \\ I_{n} = I_{0} +\hslash ^{\alpha } \Biggl(A _{n} {G}_{3}( N_{0},S_{0},I_{0})+ \sum_{i=1}^{n}B_{ni} {G}_{3}( N_{i},S_{i},I_{i}) \Biggr), \end{gathered} $$
(25)
with
$$ \begin{gathered} {G}_{1} ( N,S,I ) = rN\biggl(1 \frac{N}{k}\biggr)\bigl(\lambda +a(S+I)\bigr)N(S+I), \\ {P}_{2} (N,S,I ) = e\bigl(\lambda +a(S+I)\bigr)N(S+I)\beta SI  \mu S, \\ {P}_{3} ( N,S,I )=\beta SI \eta I \mu I. \end{gathered} $$
(26)