## 2 Problem Statement
: _Let the parameters \(\mu\), \(\gamma_{1}\), \(\gamma_{2}\), \(\sigma\), \(\alpha\) and \(\rho\) fulfill the condition_  
\[0\leq\mu,\gamma_{1},\gamma_{2},\sigma,\alpha,\rho.\]  
**Assumption 2** (unknown variable boundedness). : _Let the unknown time-varying transmission rate \(\beta(t)\) fulfill the following conditions:_  
_(i)_ _It is bounded by the non-negative and known parameters_ \(\beta_{m}\) _and_ \(\beta_{M}\)_;_  
\[\beta(t)\in[\beta_{m},\beta_{M}],\ \forall t\in[t_{0},\infty),\] _where_ \(t_{0}\in[0,\infty).\)__ _(ii)_ _It is continuously differentiable with respect to time with a bounded time derivative;_  
\[|\dot{\beta}(t)|\leq\overline{\mu},\ \forall t\in[t_{0},\infty).\]  
Note that throughout this paper, \(t_{0}\) denotes a non-negative real number (i.e., \(t_{0}\in[0,\infty)\)). **Remark 1**. : _Since the transmission rate is proportional to the number of contact rates, as long as the contact rate is positive and bounded (i.e., in real life, it is not possible to have negative nor infinite contacts per day), and the probability of being infected is also positive and bounded, the transmission rate is also positive and bounded._  
**Remark 2**. : _The contact rate is given by the average number of people that a person meets during the day. Therefore, any self-driven population will not have abrupt changes in the contact rate, and consequently, the transmission rate can be considered continuously differentiable in \(t\). Even if abrupt and global policies affecting everybody equally are considered, which could cause a piecewise continuously differentiable transmission rate, people's self-awareness would have already changed the contact rate (e.g., before governments apply a lockdown, people's fear of disease would have decreased in value). So the assumption of a continuously differentiable transmission rate is still valid._  
**Assumption 3** (input variables). : _Regarding the input variables, the following conditions are fulfilled:_  
1. _The input variables_ \(\Lambda(t)\) _and_ \(v(t)\) _are continuous on_ \(t\in[t_{0},\infty)\)_._
2. _Vaccination_ \(v(t)\) _must be non-negative and it must be lesser than the sum of the susceptible subpopulation (_\(S(t)\)_) and the new susceptible population (_\(\Lambda(t)+\rho R(t)\)_);_ \(0\leq v(t)<S(t)+\Lambda(t)+\rho R(t)\)_._
3. _The number of newborns is bounded and has positive values:_ \(0<\Lambda(t)\leq\Lambda_{M}\)_._
4. _The normalized newborns per day are assumed to be approximately constant._ \[\frac{\Lambda(t)}{N(t)}\approx\frac{\Lambda}{N}=\lambda=\text{const}.\]  
**Assumption 4** (initial conditions). : _Let the initial conditions fulfill the following conditions:_  
\[S(t_{0})\in(0,N(t_{0})]\quad\text{and}\quad X(t_{0})\in[0,N(t_{0})]:\,X(t)\in\{ I(t),H(t),R(t)\},\,\forall t\in[t_{0},\infty), \tag{5}\]  
_where_  
\[N(t_{0})=S(t_{0})+I(t_{0})+H(t_{0})+R(t_{0}),\quad\text{and}\quad 0<N(t_{0}). \tag{6}\]  
**Proposition 1**. : _Under Assumptions 1-4, the following statements hold:_  
1. _The solution for the susceptible subpopulation will be positive:_ \(0<S(t),\,\,\,\forall\,t\in[t_{0},\infty),\,\,\forall\,S(t_{0})\in(0,N(t_{0})]\)_._
2. _The solution for all subpopulations but susceptible will be non-negative:_ \(0\leq X(t),\,\,\forall\,t\in[t_{0},\infty),\,\,\forall\,X(t_{0})\in[0,N(t_{0})]\)_, where_ \(X(t)\in\{I(t),H(t),R(t)\}\)_._  
**Proposition 2** (Positively Invariant Region). : _Under Assumptions 1-3, the set_  
\[\mathfrak{D}=\bigg{\{}[S,I,H,R]^{T}\in\mathbb{R}^{4}_{+}:N\leq\frac{\Lambda}{ \mu}\bigg{\}}, \tag{7}\]  
_where \(\mathbb{R}^{4}_{+}=\{x\in\mathbb{R}^{4}:x\geq 0\}\) is positively invariant with respect to the dynamical system (3); for any \([S(t_{0}),I(t_{0}),H(t_{0}),R(t_{0})]^{T}\in\mathfrak{D}\), then \([S(t),I(t),H(t),R(t)]^{T}\in\mathfrak{D}\) for all \(t\in[t_{0},\infty)\)._Considering Assumption 3, point _(iv)_, the normalized model (i.e., \(x_{1}(t)=S(t)/N(t)\), \(x_{2}(t)=I(t)/N(t)\), \(x_{3}(t)=H(t)/N(t)\), and \(x_{4}(t)=R(t)/N(t)\)) is given by the following set of equations:  
\[