## 4 Proof of Theorem 1  
### A Change of Variable  
**Lemma 5**. : _Suppose that the ordinary differential equations system (13)-(17) has a solution. Then, \((S,I,U,W)\) satisfies (13)-(17) if and only if \((S^{\star},I^{\star},I^{\star},W^{\star})\), defined as_  
\[(S,I,U,W)(t)=\Big{(}\exp(S^{\star}(t)),\exp(I^{\star}(t)),\exp(U^{\star}(t)), \exp(W^{\star}(t))\Big{)}, \tag{61}\]  
_is a solution of the following ordinary differential equations system:_  
\[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]\[-\beta(t)\exp(W^{\star})-h(t)+g(t)\exp(I^{\star}-S^{\star}), \tag{62a}\] \[\frac{dI^{\star}}{dt} =b_{I}(\exp(S^{\star}),\exp(I^{\star}))\bigg{(}1-\frac{\exp(S^{ \star})+\exp(I^{\star})}{\theta}\bigg{)}\] \[\quad+\beta(t)\exp(S^{\star}+W^{\star}-I^{\star})-(\alpha+h+g)(t),\] (62b) \[\frac{dU^{\star}}{dt} =a(t)(1+\exp(W^{\star}-S^{\star}))\bigg{(}1-\frac{\exp(U^{\star}) +\exp(W^{\star})}{K(\exp(S^{\star})+\exp(I^{\star}))}\bigg{)}\] \[\quad-\gamma(t)\exp(I^{\star})-(\mu+c)(t),\] (62c) \[\frac{dW^{\star}}{dt} =\gamma(t)\exp(I^{\star}+U^{\star}-W^{\star})-(\mu+c)(t),\] (62d) \[(S^{\star},I^{\star},U^{\star},W^{\star})(0)=(\ln(S_{0}),\ln(I_{0}),\ln(I_ {0}),\ln(W_{0})). \tag{62e}\]  
_In particular, it follows that the following two statements are valid:_  
1. _If_ \((S^{\star},I^{\star},U^{\star},W^{\star})\)_, is_ \(\omega\)_-periodic, then_ \((S,I,U,W)\) _is_ \(\omega\)_-periodic._
2. _If_ (62) _has a solution, then_ (13)_-_(17) _has a positive solution._  
Proof. : We prove that systems (13)-(17) and (62) are equivalent, in the sense that \((S,I,U,W)\) is a solution of system (13)-(17) if and only if \((S^{\star},I^{\star},U^{\star},W^{\star})\) is a solution of system (62). Firstly, let us consider that \((S,I,U,W)\) is a solution of system (13)-(17) and prove that \((S^{\star},I^{\star},U^{\star},W^{\star})\) is a solution of system (62). We observe that multiplying (13) by \(\exp(-S^{\star}(t))\) and using the change of variable (61), we deduce that  
\[\frac{dS^{\star}}{dt} =\frac{dS}{dt}\exp(-S^{\star}(t))\] \[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]0 \[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]1 \[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]2 \[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]3  
Then, \(S^{\star}\) satisfies (62a). Analogously, by using the change of variable defined in (61) and multiplying (14), (15), and (16) by \(\exp(-I^{\star}(t))\), \(\exp(-U^{\star}(t))\), and \(\exp(-W^{\star}(t))\), respectively, we deduce that \(I^{\star},U^{\star}\), and \(W^{\star}\) satisfy (62b), (62d), and (62d). Consequently, \((S^{\star},I^{\star},U^{\star},W^{\star})\) is a solution of system (62). Conversely, if we assume that \((S^{\star},I^{\star},U^{\star},W^{\star})\) is a solution of (62), we deduce that \((S,I,U,W)\) is a solution of (13)-(17) by (61) and differentiation. For instance, for the first equation, we have that  
\[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]4 \[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]5 \[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]6 \[\frac{dS^{\star}}{dt}=b_{S}(\exp(S^{\star}),\exp(I^{\star}),t)\bigg{(}1-\frac{ \exp(S^{\star})+\exp(I^{\star})}{\theta}\bigg{)}\]7  
To prove _(a)_ and _(b)_, we proceed as follows.