Appendix 1: Formulating the governing equations of the model
The state variables are depicted by the boxes in the flow diagram, while the arrows illustrate the movement of people between different states in the population. The flows shown as arrows are calculated using the terms on the right-hand side of the equations: a flow pointing out of a box is taken away from a state variable and is a negative term, while a flow pointing into a box is added and is a positive term. Some constant parameters are introduced adjacent to the arrows to represent the proportionality rates of the flows. An illustration of an example of a flow diagram is shown in Fig.
2.
Formulating differential equations based on the flow diagram:
See Table 3.
Table 3 Description of the parameters in the model Appendix 2: The computation of equilibrium point
Consider a mathematical model governed by a system of differential equations:
$$\begin{aligned} & \frac{{{\text{d}}x_{1} }}{{{\text{d}}t}} = f_{1} \left( {x_{1,} x_{2, \ldots ,} x_{n} } \right) \\ & \frac{{{\text{d}}x_{2} }}{{{\text{d}}t}} = f_{2} \left( {x_{1,} x_{2, \ldots ,} x_{n} } \right) \\ & \quad \quad \quad \quad \vdots \\ & \frac{{{\text{d}}x_{n} }}{{{\text{d}}t}} = f_{n} \left( {x_{1,} x_{2, \ldots ,} x_{n} } \right) \\ & f_{1} \left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right) = 0 \\ & f_{2} \left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right) = 0 \\ \vdots \\ f_{n} \left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right) = 0 \\ \end{aligned}$$
The equilibrium point \(\left( {\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}} \right)\) is obtained by solving the equations above simultaneously or using any methods of solving the system of equations in algebra.
Appendix 3: The stability analysis using Routh–Hurwitz criteria
In this study, we will study a multiple variables model with continuous time. Therefore, the stability analysis is performed using the following steps:
$$J = \left( {\begin{array}{*{20}c} {\frac{{\partial f_{1} }}{{\partial x_{1} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & {\frac{{\partial f_{1} }}{{\partial x_{2} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & \cdots & {\frac{{\partial f_{1} }}{{\partial x_{n} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} \\ {\frac{{\partial f_{2} }}{{\partial x_{1} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & {\frac{{\partial f_{2} }}{{\partial x_{2} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & \cdots & {\frac{{\partial f_{2} }}{{\partial x_{n} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {\frac{{\partial f_{n} }}{{\partial x_{1} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & {\frac{{\partial f_{n} }}{{\partial x_{2} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} & \cdots & {\frac{{\partial f_{n} }}{{\partial x_{n} }}\left( {x_{1} , x_{2} , \ldots ,x_{n} } \right)} \\ \end{array} } \right)$$
where \(\frac{{\partial f_{i} }}{{\partial x_{j} }}\left( {x_{1} , x_{2} \ldots ,x_{n} } \right)\) is the partial derivative of \(f_{i}\) with respect to its variable, \(x_{j}\) (\(i,j = 1, 2, \ldots , n\)).
The Jacobian matrix is evaluated at the equilibrium values, \(\widehat{{x_{1} }}, \widehat{{x_{2} , }} \ldots , \widehat{{x_{n} }}\). A local stability matrix, \(\hat{J} = \left. J \right|_{{x_{1} = \widehat{{x_{1} }}, x_{2} = \widehat{{x_{2} }}, \ldots , x_{n} = \widehat{{x_{n} }}}}\) is obtained. Then, find the characteristic polynomial using \(\det \left( {\hat{J} - \lambda I} \right) = 0\), where \(I\) is the identity matrix, and rewrite in the following form:
$$P\left( \lambda \right) = \lambda^{n} + a_{1} \lambda { }^{n - 1} + \ldots + a_{n - 1} \lambda + a_{n}$$
with real coefficients \(a_{i}\) for \(i = 1, 2, \ldots , n.\)
The \(n\) Hurwitz matrices are defined as follows:
$$\begin{aligned} & H_{1} = \left( {a_{1} } \right),\;H_{2} = \left( {\begin{array}{*{20}c} {a_{1} } & 1 \\ {a_{3} } & {a_{2} } \\ \end{array} } \right),\;{\text{and}} \\ & H_{n} = \left( {\begin{array}{*{20}c} {a_{1} } & 1 & 0 & 0 & \cdots & 0 \\ {a_{3} } & {a_{2} } & {a_{1} } & 1 & \cdots & 0 \\ {a_{5} } & {a_{4} } & {a_{3} } & {a_{2} } & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & {a_{n} } \\ \end{array} } \right) \\ \end{aligned}$$
Note that if \(j > n\), then \(a_{j} = 0.\)
If and only if all \(\det H_{j} > 0\) with \(j = 1, 2, \ldots ,n\), then \(P\left( \lambda \right)\) has roots that are negative or have negative real part and hence the equilibrium point is said to be asymptotically stable.
The Routh–Hurwitz criteria for polynomials of degree, n = 4 are:
$$a_{1} > 0, a_{3} > 0,a_{4} > 0,\quad {\text{and}}\quad a_{1} a_{2} a_{3} > a_{3}^{2} + a_{1}^{2} a_{4} .$$
Appendix 4: Proof of Theorem 1
Given that the initial conditions \({A\left(0\right)=A}_{0}, { B\left(0\right)=B}_{0}, {C\left(0\right)=C}_{0}\) and \({D\left(0\right)=D}_{0}\) are nonnegative. It is clear from Eq. (1) that
$$\frac{{{\text{d}}A}}{{{\text{d}}t}} + \left[ {\xi + \mu + \varphi + \alpha } \right]A\left( t \right) \ge 0,$$
so that
$$\frac{{\text{d}}}{{{\text{d}}t}}\left[ {A\left( t \right)\exp \left( {\xi + \mu + \varphi + \alpha } \right)t} \right] \ge 0.$$
(13)
Integrating (13) with respect to \(t\) gives
$$A\left( t \right) \ge A\left( 0 \right)\exp \left[ { - \left( {\xi + \mu + \varphi + \alpha } \right)t} \right] > 0, \quad \forall \,t \ge 0.$$
Similarly, it can be shown that \(B\left( t \right) > 0,C\left( t \right) > 0\), \(D\left( t \right) > 0\) for all time \(t > 0\). This completes the proof.
It is crucial to note that Eqs. (1)–(4) will be analyzed in a feasible region \(D\) given by
$$D = \left\{ {\left( {A,B,C,D} \right) \in R_{ + }^{4} :A + B + C + D = N} \right\},$$
which can be easily verified to be positively invariant with respect to Eqs. (1)–(4), In what follows, the model is epidemiologically and mathematically well posed in D (see Hethcote 2000).