Castillo–Chavez approach is used to find the global equilibrium point free of disease.According to this method, the proposed model (1) is divided into two subsystems:
$$left. begin{gathered} frac{{dell_{1} }}{dt} = G(ell_{1} ,ell_{2} ), hfill frac{{dell_{2} }}{dt} = H(ell_{1} ,ell_{2} ). hfill end{gathered} right}.$$
(20)
where (ell_{1}) and (ell_{2}) represent uninfected and infected individuals, respectively, that is, (ell_{1} = (S_{h}^{1} ,C_{h}^{1} ,S_{c}^{1} ) in R^{3}) and (ell_{2} = (I_{h}^{1} ,I_{c}^{1} ) in R^{2}). (E^{0}) represents disease free equilibrium and define as (E^{0} = left( {ell_{0} ,0} right)). Hence, global stability at disease-free equilibrium depends on two conditions:
(frac{{dell_{1} }}{dt} = G(ell_{1} ,0)), (x_{1}^{0}) is globally asymptotically stable.
-
1.
(H(ell_{1} ,ell_{2} ) = Bell_{2} – overline{H}(ell_{1} ,ell_{2} )),where (overline{H}(ell_{1} ,ell_{2} ) ge 0) for ((ell_{1} ,ell_{2} ) in k).
-
2.
In the second condition, (B = D_{{ell_{2} }} H(ell_{1}^{0} ,0)) is an M-matrix with positive diagonal entries, and (k) is the feasible region.
Lemma 2:
If (R_{0} < 1), then the equilibrium point (E^{0} = (ell_{1}^{0} ,0)) of the system (1) is said to be globally asymptotically stable, if the above conditions are satisfied36.
Theorem:
According to the proposed model (1), if (R_{0} < 1), it is globally asymptotically stable at disease-free equilibrium (E^{0}), and otherwise it is unstable.
Proof:
Proof Let (ell_{1} = left( {S_{h}^{1} ,C_{h}^{1} ,S_{c}^{1} } right)) and (ell_{2} = left( {I_{h}^{1} ,I_{c}^{1} } right)) and define (E^{0} = left( {ell_{0} ,0} right)), where
$$ell_{1}^{0} = left( {frac{{Lambda_{,h} }}{{phi_{,h} }},frac{{Lambda_{,c} }}{{phi_{,c} }}} right)$$
By using model system (1), we have
$$frac{{dell_{1} }}{dt} = Gleft( {ell_{1} ,ell_{2} } right),$$
$$frac{{dell_{1} }}{dt} = left( {begin{array}{*{20}c} {Lambda_{,h} + mu_{,h} C_{h}^{1} – frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} – phi_{,h} S_{h}^{1} } {delta_{,h} I_{h}^{1} – mu_{,h} C_{h}^{1} – phi_{,h} C_{h}^{1} } {Lambda_{,c} + mu_{,c} I_{c}^{1} P_{,c} – frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }} – phi_{,c} S_{c}^{1} } end{array} } right),$$
(21)
For (S_{h}^{1} = left( {S_{h}^{1} } right)^{0} ,S_{c} = left( {S_{c}^{1} } right)^{0}) and (Gleft( {ell_{1} ,0} right) = 0.) we get
$$Gleft( {ell_{1} ,0} right) = left( {begin{array}{*{20}c} {Lambda_{,h} – phi_{,h} S_{h}^{1} } {Lambda_{,c} – phi_{,c} S_{c}^{1} } end{array} } right) = 0.$$
(22)
From above equation as (t to infty ,ell_{1} to ell_{1}^{0} .) So (x_{1} = x_{1}^{0}) is globally asymptotically stable. Now
$$Bleft( {ell_{2} } right) – overline{H}left( {ell_{1} ,ell_{2} } right) = left( {begin{array}{*{20}c} { – delta_{,h} – phi_{,h} } & {2beta_{,h} left( {S_{h}^{1} } right)^{0} } 0 & {2beta_{,c} left( {S_{c}^{1} } right)^{0} – P_{,c} mu_{,c} – phi_{,c} } end{array} } right)left( {begin{array}{*{20}c} {I_{h}^{1} } {I_{c}^{1} } end{array} } right) – left( {begin{array}{*{20}c} {2beta_{,h} left( {S_{h}^{1} } right)^{0} I_{c}^{1} – frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }}} {2beta_{,c} left( {S_{c}^{1} } right)^{0} I_{c}^{1} – frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }}} end{array} } right)$$
(23)
As (2beta_{,h} left( {S_{h}^{1} } right)^{0} I_{c}^{1} ge frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }}) and (2beta_{,c} left( {S_{c}^{1} } right)^{0} I_{c}^{1} – frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }}). Hence (Hleft( {ell_{1} ,ell_{2} } right) ge 0). B is clearly an M-matrix. Therefore, Lemma 2 indicates that the equilibrium point for disease-free equilibrium is globally asymptotically stable.
Global stability analysis of endemic equilibria
When a system has global stability, it indicates that it may be started in any position and yet return to its original state of equilibrium.
Theorem:
If (R_{0} > 1),(E^{*}) the asymptotic behavior of the model (1) is globally asymptotical37.
Proof:
With the help of the following Lyapunov function, we can calculate (E^{*}) global asymptotic stability:
$$begin{aligned} L(S_{h}^{1*} ,I_{h}^{1*} ,C_{h}^{1*} ,S_{c}^{1*} ,I_{c}^{1*} ) = & left( {S_{h}^{1} – S_{h}^{1*} – S_{h}^{1*} ln frac{{S_{h}^{1} }}{{S_{h}^{1*} }}} right) + left( {I_{h}^{1} – I_{h}^{1*} – I_{h}^{1*} ln frac{{I_{h}^{1} }}{{I_{h}^{1*} }}} right) + left( {C_{h}^{1} – C_{h}^{1*} – C_{h}^{1*} ln frac{{C_{h}^{1} }}{{C_{h}^{1*} }}} right) & + left( {S_{c}^{1} – S_{c}^{1*} – S_{c}^{1*} ln frac{{S_{c}^{1} }}{{S_{c}^{1*} }}} right) + ,,left( {I_{c}^{1} – ,,I_{c}^{1*} – ,,I_{c}^{1*} ln frac{{I_{c}^{1} }}{{I_{c}^{1*} }}} right). end{aligned}$$
(24)
Calculating the derivative of L directly along solution (1), we have:
$$frac{dL}{{dt}} = left( {1 – frac{{S_{h}^{1*} }}{{S_{h}^{1} }}} right)frac{{dS_{h}^{1} }}{dt} + left( {1 – frac{{I_{h}^{1*} }}{{I_{h}^{1} }}} right)frac{{dI_{h}^{1} }}{dt} + left( {1 – frac{{C_{h}^{1*} }}{{C_{h}^{1} }}} right)frac{{dC_{h}^{1} }}{dt} + left( {1 – frac{{S_{c}^{1*} }}{{S_{c}^{1} }}} right)frac{{dS_{c}^{1} }}{dt} + left( {1 – frac{{I_{c}^{1*} }}{{I_{c}^{1} }}} right)frac{{dI_{c}^{1} }}{dt}.$$
(25)
$$begin{gathered} frac{dL}{{dt}} = left( {1 – frac{{S_{h}^{1*} }}{{S_{h}^{1} }}} right)(Lambda_{,h} + mu_{,h} C_{h}^{1} – frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} – phi_{,h} S_{h}^{1} ) + left( {1 – frac{{I_{h}^{1*} }}{{I_{h}^{1} }}} right)(frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} – delta_{,h} I_{h}^{1} – phi_{,h} I_{h}^{1} ) + left( {1 – frac{{C_{h}^{1*} }}{{C_{h}^{1} }}} right)(delta_{,h} I_{h}^{1} – mu_{,h} C_{h}^{1} – phi_{,h} C_{h}^{1} ) hfill ,,,,,,, + left( {1 – frac{{S_{c}^{1*} }}{{S_{h}^{1} }}} right)(Lambda_{,c} + mu_{,c} I_{c}^{1} P_{,c} – frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }} – phi_{,c} S_{c}^{1} ) + left( {1 – frac{{I_{c}^{1*} }}{{I_{h}^{1} }}} right)(frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }} – mu_{,c} I_{c}^{1} P_{,c} – phi_{,c} I_{c}^{1} ). hfill end{gathered}$$
(26)
$$begin{gathered} frac{dL}{{dt}} = Lambda_{,h} + mu_{,h} C_{h}^{1} + frac{{2beta_{,h} S_{h}^{1*} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} + S_{h}^{1*} phi_{,h} + frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} + delta_{,h} I_{h}^{1*} + phi_{,h} I_{h}^{1*} + mu_{,h} C_{h}^{1*} + phi_{,h} C_{h}^{1*} + Lambda_{,c} + mu_{,c} I_{c}^{1} P_{,c} hfill + delta_{,h} I_{h}^{1} + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} S_{c}^{1*} }}{{S_{c}^{1} (S_{c}^{1} + I_{c}^{1} )}} + phi_{,c} S_{c}^{1*} + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }} + mu_{,c} I_{c}^{1*} P_{,c} + phi_{,c} I_{c}^{1*} – (frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }} + phi_{,h} S_{h}^{1} + frac{{S_{h}^{1*} Lambda_{,h} }}{{S_{h}^{1} }} + frac{{S_{h}^{1*} mu_{,h} C_{h}^{1} }}{{S_{h} }} hfill + delta_{,h} I_{h}^{1} + phi_{,h} I_{h}^{1} + frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} I_{h}^{1*} }}{{I_{h}^{1} (S_{h}^{1} + I_{c}^{1} )}} + mu_{,h} C_{h}^{1} + phi_{,h} C_{h}^{1} + frac{{C_{h}^{1*} delta_{,h} I_{h}^{1} }}{{C_{h}^{1} }} + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }} + phi_{,c} S_{c}^{1} hfill + frac{{S_{c}^{1*} Lambda_{,c} }}{{S_{c}^{1} }} + frac{{S_{c}^{1*} mu_{,c} I_{c}^{1} P_{,c} }}{{S_{c}^{1} }} + mu_{,c} I_{c}^{1} P_{,c} + phi_{,c} I_{c}^{1} + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} I_{c}^{1*} }}{{I_{c}^{1} (S_{c}^{1} + I_{c}^{1} )}}). hfill end{gathered}$$
(27)
$$frac{dL}{{dt}} = C – D.$$
(28)
where
$$begin{gathered} A = Lambda_{,h} + mu_{,h} C_{h}^{1} + frac{{2beta_{,h} S_{h}^{1*} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} + S_{h}^{1*} phi_{,h} + frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} + delta_{,h} I_{h}^{1*} + phi_{,h} I_{h}^{1*} + mu_{,h} C_{h}^{1*} + phi_{,h} C_{h}^{1*} + Lambda_{,c} + mu_{,c} I_{c}^{1} P_{,c} hfill ,,,,,,,,, + delta_{,h} I_{h}^{1} + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} S_{c}^{1*} }}{{S_{h}^{1} (S_{c}^{1} + I_{c}^{1} )}} + phi_{,c} S_{c}^{1*} + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c} + I_{c} }} + mu_{,c} I_{c}^{1*} P_{,c} + phi_{,c} I_{c}^{1*} . hfill B = frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} }}{{S_{h}^{1} + I_{c}^{1} }} + phi_{,h} S_{h}^{1} + frac{{S_{h}^{1*} Lambda_{,h} }}{{S_{h}^{1} }} + frac{{S_{h}^{1*} mu_{,h} C_{h}^{1} }}{{S_{h}^{1} }} + delta_{,h} I_{h}^{1} + phi_{,h} I_{h}^{1} + frac{{2beta_{,h} S_{h}^{1} I_{c}^{1} I_{h}^{1*} }}{{I_{h}^{1} (S_{h}^{1} + I_{c}^{1} )}} + mu_{,h} C_{h}^{1} + phi_{,h} C_{h}^{1} + frac{{C_{h}^{1*} delta_{,h} I_{h}^{1} }}{{C_{h}^{1} }} hfill ,,,,,,,, + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} }}{{S_{c}^{1} + I_{c}^{1} }} + phi_{,c} S_{c}^{1} + frac{{S_{c}^{1*} Lambda_{,c} }}{{S_{c}^{1} }} + frac{{S_{c}^{1*} mu_{,c} I_{c}^{1} P_{,c} }}{{S_{c}^{1} }} + mu_{,c} I_{c}^{1} P_{,c} + phi_{,c} I_{c}^{1} + frac{{2beta_{,c} S_{c}^{1} I_{c}^{1} I_{c}^{1*} }}{{I_{c}^{1} (S_{c}^{1} + I_{c}^{1} )}}. hfill end{gathered}$$
(29)
Thus if (A < B,,then,frac{dL}{{dt}} < 0).
Noting that: (frac{dL}{{dt}} = 0) if and only if (S_{h}^{1*} = S_{h}^{1} ,,I_{h}^{1*} = I_{h}^{1} ,,,C_{h}^{1*} = C_{h}^{1} ,,S_{c}^{1*} = S_{c}^{1} ,,I_{c}^{1*} = I_{c}^{1} .) Therefore, the largest compact invariant set in (left{ {left( {S_{h}^{1*} ,I_{h}^{1*} ,C_{h}^{1*} ,S_{c}^{1*} ,I_{c}^{1*} } right) in R^{5} :frac{dL}{{dt}} = 0} right}) is the singleton (E^{*}), by invariance principle, it implies that (E^{*}) is GAS in (R^{5}) if (A < B). This implies that (R_{0} > 1) 38.
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- Source: https://www.nature.com/articles/s41598-024-63263-w