Appendix
In this appendix, we briefly show the derivation of the equations that constitute a preparatory step before the optimization procedure for the other seven parameters that were not included in the main text (three were already shown starting from Sect. 3.3.1). For more details the interested reader is referred to Sect. 3.3.1).
1.1 Parameter d
The derivative of the general function f (the vector [T, V]) with respect to d is:
$$\begin{aligned} \begin{aligned} \dfrac{\partial {^{}}f}{\partial {{d}^{}}} (t, y^d)=\Big [&- T - {d} \dfrac{\partial {^{}}T}{\partial {{d}^{}}} - {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{d}^{}}} T + V \dfrac{\partial {^{}}T}{\partial {{d}^{}}} \right) \\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }R(a, t) \dfrac{\partial {^{}}I}{\partial {{d}^{}}} (a, t) {\mathrm {d}}a - {c} \dfrac{\partial {^{}}V}{\partial {{d}^{}}} \Big ] \end{aligned} \end{aligned}$$
where \( {y^d} = {\left\{ \begin{array}{ll} T\\ V \\ { \dfrac{\partial {^{}}T}{\partial {d^{}}} } \\ { \dfrac{\partial {^{}}V}{\partial {d^{}}} } \end{array}\right. }\).
Furthermore:
$$\begin{aligned} { \dfrac{\partial {^{}}R(a,t)}{\partial {d^{}}} }= & {} 0, \\ { \dfrac{\partial {^{}}\bar{T}}{\partial {d^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}\bar{V}}{\partial {d^{}}} }= & {} -1/{\beta },\\ { \dfrac{\partial {^{}}I(a,t)}{\partial {d^{}}} }= & {} \left\{ \begin{array}{lr} {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{d}^{}}} (t-a) T(t-a) + V(t-a) \dfrac{\partial {^{}}T}{\partial {{d}^{}}} (t-a) \right) {\mathrm {e}}^{-{\delta }a} &{} a < t\\ - c / ({\beta }N) {\mathrm {e}}^{-{\delta }a} &{} a > t\\ \end{array} \right. , \end{aligned}$$
The upper right block matrix of the Jacobian is:
$$\begin{aligned} f'_{d, 2\times 2}= \left( \begin{array}{l@{\quad }l} 0 &{} 0 \\ \begin{array}{l@{}} (1 - \varepsilon _s)\int _0^t\rho R(a, t) \\ \quad \times \dfrac{\partial {^{}}I(a, t)}{\partial { \dfrac{\partial {^{}}T}{\partial {d^{}}} ^{}}} {\mathrm {d}}a\end{array} &{}\quad \begin{array}{l@{}}(1 - \varepsilon _s)\int _0^t \rho R(a, t)\\ \quad \times \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {d^{}}} ^{}}} {\mathrm {d}}a\end{array} \\ \end{array}\right) \end{aligned}$$
and the transposed last two rows of the Jacobian are:
$$\begin{aligned} {\left( \left( f'_{d, 3}\right) ^{\text {tr}}, \left( f'_{d, 4}\right) ^{\text {tr}}\right) }:= & {} \left( \begin{array}{ll} -1-{\beta } \dfrac{\partial {^{}}V}{\partial {{d}^{}}} &{}\quad (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a, t)}{\partial {d^{}}} }{\partial {T^{}}} }\hbox {d}a \\ -{\beta } \dfrac{\partial {^{}}T}{\partial {{d}^{}}} &{}\quad (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a, t)}{\partial {d^{}}} }{\partial {V^{}}} }\hbox {d}a \\ -{d}-{\beta }V &{}\quad (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a, t)}{\partial {d^{}}} }{\partial { \dfrac{\partial {^{}}T}{\partial {d^{}}} ^{}}} }\hbox {d}a \\ -{\beta }T &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a, t)}{\partial {d^{}}} }{\partial { \dfrac{\partial {^{}}V}{\partial {d^{}}} ^{}}} }\hbox {d}a -{c}\\ \end{array}\right) ,\\ \dfrac{\partial {^{}} \dfrac{\partial {^{}}f}{\partial {{d}^{}}} }{\partial {t^{}}}= & {} \begin{aligned} \Bigg [&0,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }\left( \dfrac{\partial {^{}}R(a, t)}{\partial {t^{}}} \dfrac{\partial {^{}}I}{\partial {{d}^{}}} (a, t) + R(a, t) \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a, t)}{\partial {d^{}}} }{\partial {t^{}}} \right) {\mathrm {d}}a\Bigg ]. \end{aligned} \end{aligned}$$
1.2 Parameter \({\beta }\)
The derivative of the general function f (the vector [T, V]) with respect to \(\beta \) is:
$$\begin{aligned} \begin{aligned} \dfrac{\partial {^{}}f}{\partial {{\beta }^{}}} (t, y^\beta ) = \Big [&- {d} \dfrac{\partial {^{}}T}{\partial {{\beta }^{}}} - (V T + {\beta } \dfrac{\partial {^{}}V}{\partial {{\beta }^{}}} T + {\beta }V \dfrac{\partial {^{}}T}{\partial {{\beta }^{}}} )\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }R(a, t) { \dfrac{\partial {^{}}I(a,t)}{\partial {{\beta }^{}}} } {\mathrm {d}}a - {c} \dfrac{\partial {^{}}V}{\partial {{\beta }^{}}} \Big ] \end{aligned} \end{aligned}$$
where \({y^\beta } = {\left\{ \begin{array}{ll} T\\ V \\ { \dfrac{\partial {^{}}T}{\partial {\beta ^{}}} }\\ { \dfrac{\partial {^{}}V}{\partial {\beta ^{}}} } \end{array}\right. }\).
Furthermore:
$$\begin{aligned} { \dfrac{\partial {^{}}R(a,t)}{\partial {{\beta }^{}}} }= & {} 0,\\ \bar{ \dfrac{\partial {^{}}T}{\partial {{\beta }^{}}} }= & {} -{c}/({\beta }^2 N),\\ \bar{ \dfrac{\partial {^{}}V}{\partial {{\beta }^{}}} }= & {} {d}/ {\beta }^2, \end{aligned}$$
$$\begin{aligned} { \dfrac{\partial {^{}}I(a,t)}{\partial {{\beta }^{}}} }= & {} \left\{ \begin{array}{lr} \left( V(t-a) T(t-a) + {\beta } \dfrac{\partial {^{}}V}{\partial {{\beta }^{}}} (t-a) T(t-a) + {\beta }V(t-a) \dfrac{\partial {^{}}T}{\partial {{\beta }^{}}} (t-a) \right) {\mathrm {e}}^{-{\delta }a} &{} a < t\\ {d}c / ({{\beta }^2} N) {\mathrm {e}}^{-{\delta }a} &{} a > t\\ \end{array} \right. , \end{aligned}$$
The upper right block matrix of the Jacobian is:
$$\begin{aligned} f'_{\beta , 2\times 2}= \left( \begin{array}{l@{\quad }l} 0 &{} \quad 0 \\ \begin{array}{l@{}} (1 - \varepsilon _s)\int _0^t\rho R(a, t) \\ \quad \times \dfrac{\partial {^{}}I(a, t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\beta ^{}}} ^{}}} {\mathrm {d}}a\end{array} &{}\quad \begin{array}{l@{}}(1 - \varepsilon _s)\int _0^t \rho R(a, t)\\ \quad \times \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\beta ^{}}} ^{}}} {\mathrm {d}}a\end{array} \\ \end{array}\right) \end{aligned}$$
and the transposed last two rows of the Jacobian are:
$$\begin{aligned} {\left( \left( f'_{\beta , 3}\right) ^{\text {tr}}, \left( f'_{\beta , 4}\right) ^{\text {tr}}\right) }:= & {} \left( \begin{array}{ll} -\left( V + {\beta } \dfrac{\partial {^{}}V}{\partial {{\beta }^{}}} \right) &{}\quad (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\beta ^{}}} }{\partial {T^{}}} }\hbox {d}a \\ -\left( T + {\beta } \dfrac{\partial {^{}}T}{\partial {{\beta }^{}}} \right) &{}\quad (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\beta ^{}}} }{\partial {V^{}}} }\hbox {d}a \\ -{d}- {\beta }V &{}\quad (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a,t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\beta ^{}}} }{\partial { \dfrac{\partial {^{}}T}{\partial {\beta ^{}}} ^{}}} }\hbox {d}a \\ -{\beta }T &{}\quad (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a,t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\beta ^{}}} }{\partial { \dfrac{\partial {^{}}V}{\partial {\beta ^{}}} ^{}}} }\hbox {d}a - {c}\\ \end{array}\right) , \\ \dfrac{\partial {^{}} \dfrac{\partial {^{}}f}{\partial {{\beta }^{}}} }{\partial {t^{}}}= & {} \begin{aligned} \big [&0,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }\left( \dfrac{\partial {^{}}R(a, t)}{\partial {t^{}}} \dfrac{\partial {^{}}I}{\partial {{\beta }^{}}} (a, t) + R(a,t) \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\beta ^{}}} }{\partial {t^{}}} \right) {\mathrm {d}}a\Bigg ]. \end{aligned} \end{aligned}$$
1.3 Parameter \({\varepsilon _{s}}\)
The derivative of the general function f (the vector [T, V]) with respect to \(\varepsilon _{s}\) is:
$$\begin{aligned} \begin{aligned} \dfrac{\partial {^{}}f}{\partial {{\varepsilon _{s}}^{}}} (t, y^{\varepsilon _{s}}) = \Big [&- {d} \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{s}}^{}}} - {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{s}}^{}}} T + V \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{s}}^{}}} \right) ,\\&\begin{array}{l@{}} -\int _0^\infty {\rho }R(a, t) I (a, t) {\mathrm {d}}a + (1 - {\varepsilon _{s}})\int _0^\infty {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } I (a, t) {\mathrm {d}}a \\ \qquad +\, (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) \dfrac{\partial {^{}}I}{\partial {{\varepsilon _{s}}^{}}} (a, t) {\mathrm {d}}a - {c} \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{s}}^{}}} \Big ] \end{array} \end{aligned} \end{aligned}$$
where \({y^{\varepsilon _s}} = {\left\{ \begin{array}{ll} T\\ V \\ { \dfrac{\partial {^{}}T}{\partial {\varepsilon _s^{}}} } \\ { \dfrac{\partial {^{}}V}{\partial {\varepsilon _s^{}}} } \end{array}\right. }\).
Furthermore:
$$\begin{aligned} { \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} }= & {} \left\{ \begin{array}{@{}lr@{}} \begin{array}{l@{}} {\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ -{\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} \\ + {\rho }a\left( 1 - \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\right) {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} \end{array} &{} a< t\\ \\ \begin{array}{l@{}} {\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ - {\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha }{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t} \\ +{\rho }t\bigg (\frac{\alpha }{{\rho }+ \mu } + \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} - \frac{(1 - {\varepsilon _{\alpha }})\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\bigg ) {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t} \end{array} &{} a> t\\ \end{array} \right. ,\\ { \dfrac{\partial {^{}}\bar{T}}{\partial {{\varepsilon _{s}}^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}\bar{V}}{\partial {{\varepsilon _{s}}^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}I(a,t)}{\partial {{\varepsilon _{s}}^{}}} }= & {} \left\{ \begin{array}{lr} {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{s}}^{}}} (t-a) T(t-a) + V(t-a) \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{s}}^{}}} (t-a)\right) {\mathrm {e}}^{-{\delta }a} &{} a < t\\ 0 &{} a > t\\ \end{array} \right. , \end{aligned}$$
The upper right block matrix of the Jacobian is:
$$\begin{aligned} f'_{\varepsilon _{s}, 2\times 2}= \left( \begin{array}{l@{\quad }l} 0 &{} \quad 0 \\ \begin{array}{l@{}} (1 - \varepsilon _s)\int _0^t\rho R(a, t) \\ \quad \times \dfrac{\partial {^{}}I(a, t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\varepsilon _{s}^{}}} ^{}}} {\mathrm {d}}a\end{array} &{}\quad \begin{array}{l@{}}(1 - \varepsilon _s)\int _0^t \rho R(a, t)\\ \quad \times \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\varepsilon _{s}^{}}} ^{}}} {\mathrm {d}}a\end{array} \\ \end{array}\right) \end{aligned}$$
and the transposed last two rows of the Jacobian are:
$$\begin{aligned} {\left( \left( f'_{\varepsilon _s, 3}\right) ^{\text {tr}}, \left( f'_{\varepsilon _s, 4}\right) ^{\text {tr}}\right) }:= & {} \left( \begin{array}{ll} -{\beta } \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{s}}^{}}} &{}\begin{array}{l@{}}\displaystyle -\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}}I(a,t)}{\partial {T^{}}} }{\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial {T^{}}} }\hbox {d}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _s^{}}} }{\partial {T^{}}} }\hbox {d}a\\ \end{array}\\ \\ - {\beta } \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{s}}^{}}} &{}\quad \begin{array}{l@{}}\displaystyle -\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}}I(a,t)}{\partial {V^{}}} }\hbox {d}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial {V^{}}} }\hbox {d}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _s^{}}} }{\partial {V^{}}} }\hbox {d}a\\ \end{array}\\ \\ -{d}- {\beta }V &{}\quad \begin{array}{l@{}}\displaystyle -\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\varepsilon _s^{}}} ^{}}} }\hbox {d}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\varepsilon _s^{}}} ^{}}} }\hbox {d}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _s^{}}} }{\partial { \dfrac{\partial {^{}}T}{\partial {\varepsilon _s^{}}} ^{}}} }\hbox {d}a\\ \end{array}\\ \\ -{\beta }T&{}\begin{array}{l@{}}\displaystyle -\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\varepsilon _s^{}}} ^{}}} }\hbox {d}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\varepsilon _s^{}}} ^{}}} }\hbox {d}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _s^{}}} }{\partial { \dfrac{\partial {^{}}V}{\partial {\varepsilon _s^{}}} ^{}}} }\hbox {d}a\\ \qquad \qquad - {c}\end{array}\\ \\ \end{array}\right) ,\\ \dfrac{\partial {^{}} \dfrac{\partial {^{}}f}{\partial {{\varepsilon _{s}}^{}}} }{\partial {t^{}}}= & {} \begin{aligned}&\left[ 0,-\int _0^\infty {\rho }\left( \dfrac{\partial {^{}}R(a,t)}{\partial {t^{}}} I (a, t) +R(a,t) \dfrac{\partial {^{}}I(a,t)}{\partial {t^{}}} \right) {\mathrm {d}}a \right. \\&\qquad + (1 - {\varepsilon _{s}})\int _0^\infty {\rho }\left( \dfrac{\partial {^{}} \dfrac{\partial {^{}}R(a, t)}{\partial {{\varepsilon _{s}}^{}}} }{\partial {t^{}}} I (a, t) + \dfrac{\partial {^{}}R(a, t)}{\partial {\varepsilon _s^{}}} \dfrac{\partial {^{}}I (a, t)}{\partial {t^{}}} \right) {\mathrm {d}}a \\&\qquad \left. + (1 - {\varepsilon _{s}})\int _0^t {\rho }\left( \dfrac{\partial {^{}}R(a, t)}{\partial {t^{}}} \dfrac{\partial {^{}}I(a, t)}{\partial {{\varepsilon _{s}}^{}}} + R(a,t) \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a, t)}{\partial {\varepsilon _s^{}}} }{\partial {t^{}}} \right) {\mathrm {d}}a \right] \end{aligned} \end{aligned}$$
where
$$\begin{aligned} { \dfrac{\partial {^{}} \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} }{\partial {t^{}}} } = \left\{ \begin{array}{@{}lr@{}} \begin{array}{l@{}} -{\gamma }{\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ +{\gamma }{\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} \\ + {\gamma }{\rho }a \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} \end{array} &{} a < t\\ \\ \begin{array}{l@{}} -{\gamma }{\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ +((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu ) {\rho }\frac{(1 - {\varepsilon _{\alpha }})\alpha }{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t} \\ +{\rho }\bigg (\frac{\alpha }{{\rho }+ \mu } + \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} - \frac{(1 - {\varepsilon _{\alpha }})\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\bigg ) {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\\ +({\rho }+ \mu ){\rho }t \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\\ -((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu ){\rho }t\bigg (\frac{\alpha }{{\rho }+ \mu } + \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} - \frac{(1 - {\varepsilon _{\alpha }})\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\bigg )\\ \times {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t} \end{array} &{} a > t\\ \end{array} \right. . \end{aligned}$$
1.4 Parameter \({\varepsilon _{\alpha }}\)
The derivative of the general function f (the vector [T, V]) with respect to \({\varepsilon _{\alpha }}\) is:
$$\begin{aligned} \begin{aligned} \dfrac{\partial {^{}}f}{\partial {{\varepsilon _{\alpha }}^{}}} (t, y^{\varepsilon _{\alpha }}) =&\left[ - {d} \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{\alpha }}^{}}} - {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{\alpha }}^{}}} T + V \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{\alpha }}^{}}} \right) ,\right. \\&\quad (1 - {\varepsilon _{s}})\int _0^\infty {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{\alpha }}^{}}} } I (a, t) {\mathrm {d}}a \\&\left. \qquad +(1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) \dfrac{\partial {^{}}I}{\partial {{\varepsilon _{\alpha }}^{}}} (a, t) {\mathrm {d}}a - {c} \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{\alpha }}^{}}} \right] \end{aligned} \end{aligned}$$
where \({y^{\varepsilon _{\alpha }}} = {\left\{ \begin{array}{ll} T\\ V \\ { \dfrac{\partial {^{}}T}{\partial {\varepsilon _\alpha ^{}}} } \\ { \dfrac{\partial {^{}}V}{\partial {\varepsilon _\alpha ^{}}} } \end{array}\right. }\).
Furthermore:
$$\begin{aligned} { \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{\alpha }}^{}}} }= & {} \left\{ \begin{array}{@{}lr@{}} -\frac{\alpha {\mathrm {e}}^{-{\gamma }t}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }} + \frac{\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} &{} a< t\\ \\ \begin{array}{l@{}} -\frac{\alpha {\mathrm {e}}^{-{\gamma }t}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }} + \frac{\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\end{array} &{} a> t\\ \end{array} \right. ,\\ { \dfrac{\partial {^{}}\bar{T}}{\partial {{\varepsilon _{\alpha }}^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}\bar{V}}{\partial {{\varepsilon _{\alpha }}^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}I(a,t)}{\partial {{\varepsilon _{\alpha }}^{}}} }= & {} \left\{ \begin{array}{lr} {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{\alpha }}^{}}} (t-a) T(t-a) + V(t-a) \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{\alpha }}^{}}} (t-a)\right) {\mathrm {e}}^{-{\delta }a} &{} a < t\\ 0 &{} a > t\\ \end{array} \right. , \end{aligned}$$
The upper right block matrix of the Jacobian is:
$$\begin{aligned} f'_{\varepsilon _{\alpha }, 2\times 2}= \left( \begin{array}{l@{\quad }l} 0 &{} 0 \\ \begin{array}{l@{}} (1 - \varepsilon _s)\int _0^t\rho R(a, t) \\ \quad \times \dfrac{\partial {^{}}I(a, t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\varepsilon _{\alpha }^{}}} ^{}}} {\mathrm {d}}a\end{array} &{}\quad \begin{array}{l@{}}(1 - \varepsilon _s)\int _0^t \rho R(a, t)\\ \quad \times \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\varepsilon _{\alpha }^{}}} ^{}}} {\mathrm {d}}a\end{array} \\ \end{array}\right) \end{aligned}$$
and the transposed last two rows of the Jacobian are:
$$\begin{aligned} {\left( \left( f'_{\varepsilon _\alpha , 3}\right) ^{\text {tr}}, \left( f'_{\varepsilon _\alpha , 4}\right) ^{\text {tr}}\right) }:= & {} \left( \begin{array}{ll} -{\beta } \dfrac{\partial {^{}}V}{\partial {{\varepsilon _{\alpha }}^{}}} &{}\begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{\alpha }}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial {T^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _\alpha ^{}}} }{\partial {T^{}}} } {\mathrm {d}}a\\ \end{array}\\ \\ - {\beta } \dfrac{\partial {^{}}T}{\partial {{\varepsilon _{\alpha }}^{}}} &{}\quad \begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial {V^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _\alpha ^{}}} }{\partial {V^{}}} } {\mathrm {d}}a\\ \end{array}\\ \\ -{d}- {\beta }V &{}\quad \begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\varepsilon _\alpha ^{}}} ^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _\alpha ^{}}} }{\partial { \dfrac{\partial {^{}}T}{\partial {\varepsilon _\alpha ^{}}} ^{}}} } {\mathrm {d}}a\\ \end{array}\\ \\ -{\beta }T&{}\quad \begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{s}}^{}}} } { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\varepsilon _\alpha ^{}}} ^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _\alpha ^{}}} }{\partial { \dfrac{\partial {^{}}V}{\partial {\varepsilon _\alpha ^{}}} ^{}}} } {\mathrm {d}}a - {c}\end{array}\\ \\ \end{array}\right) ,\\ \dfrac{\partial {^{}} \dfrac{\partial {^{}}f}{\partial {{\varepsilon _{s}}^{}}} }{\partial {t^{}}}= & {} \begin{aligned} \Bigg [&0,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }\left( \dfrac{\partial {^{}} \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{\alpha }}^{}}} }{\partial {t^{}}} I (a, t) + \dfrac{\partial {^{}}R(a, t)}{\partial {\varepsilon _\alpha ^{}}} \dfrac{\partial {^{}}I (a, t)}{\partial {t^{}}} \right) {\mathrm {d}}a \\&\qquad + (1 - {\varepsilon _{s}})\int _0^t {\rho }\left( \dfrac{\partial {^{}}R(a, t)}{\partial {t^{}}} \dfrac{\partial {^{}}I}{\partial {{\varepsilon _{\alpha }}^{}}} (a, t) + R(a, t) \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\varepsilon _\alpha ^{}}} }{\partial {t^{}}} \right) {\mathrm {d}}a \Bigg ]\end{aligned} \end{aligned}$$
where:
$$\begin{aligned} { \dfrac{\partial {^{}} \dfrac{\partial {^{}}R(a,t)}{\partial {{\varepsilon _{\alpha }}^{}}} }{\partial {t^{}}} } = \left\{ \begin{array}{@{}lr@{}} {\gamma }\frac{\alpha {\mathrm {e}}^{-{\gamma }t}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }} -{\gamma }\frac{\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} &{} a < t\\ \\ \begin{array}{l@{}} {\gamma }\frac{\alpha {\mathrm {e}}^{-{\gamma }t}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu ) \frac{\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\end{array} &{} a > t\\ \end{array} \right. . \end{aligned}$$
1.5 Parameter \({\kappa }\)
The derivative of the general function f (the vector [T, V]) with respect to \(\kappa \) is:
$$\begin{aligned} \begin{aligned} \dfrac{\partial {^{}}f}{\partial {{\kappa }^{}}} (t, y^\kappa ) =\Big [&- {d} \dfrac{\partial {^{}}T}{\partial {{\kappa }^{}}} - {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\kappa }^{}}} T + V \dfrac{\partial {^{}}T}{\partial {{\kappa }^{}}} \right) ,\\&\begin{array}{l@{}} (1 - {\varepsilon _{s}})\int _0^\infty {\rho }{ \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} } I (a, t) {\mathrm {d}}a \\ \quad +\, (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) \dfrac{\partial {^{}}I}{\partial {{\kappa }^{}}} (a, t) {\mathrm {d}}a - {c} \dfrac{\partial {^{}}V}{\partial {{\kappa }^{}}} \end{array}\Big ] \end{aligned} \end{aligned}$$
where \({y^{\kappa }} = {\left\{ \begin{array}{ll} T\\ V \\ { \dfrac{\partial {^{}}T}{\partial {\kappa ^{}}} } \\ { \dfrac{\partial {^{}}V}{\partial {\kappa ^{}}} } \end{array}\right. }\).
Furthermore:
$$\begin{aligned} { \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} }= & {} \left\{ \begin{array}{@{}lr@{}} \begin{array}{l@{}} {-}\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ {+}\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a}\\ -\mu a \left( 1 - \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\right) {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} \end{array} &{} a< t\\ \\ \begin{array}{l@{}} {-}\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ {+}\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha }{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\\ -\mu t\bigg (\frac{\alpha }{{\rho }+ \mu } + \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} - \frac{(1 - {\varepsilon _{\alpha }})\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\bigg ) {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t} \end{array} &{} a> t\\ \end{array} \right. ,\\ { \dfrac{\partial {^{}}\bar{T}}{\partial {{\kappa }^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}\bar{V}}{\partial {{\kappa }^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}I(a,t)}{\partial {{\kappa }^{}}} }= & {} \left\{ \begin{array}{lr} {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\kappa }^{}}} (t-a) T(t-a) + V(t-a) \dfrac{\partial {^{}}T}{\partial {{\kappa }^{}}} \right) {\mathrm {e}}^{-{\delta }a} &{} a < t\\ 0 &{} a > t\\ \end{array} \right. , \end{aligned}$$
The upper right block matrix of the Jacobian is:
$$\begin{aligned} f'_{\kappa , 2\times 2}= \left( \begin{array}{l@{\quad }l} 0 &{} 0 \\ \begin{array}{l@{}} (1 - \varepsilon _s)\int _0^t\rho R(a, t) \\ \quad \times \dfrac{\partial {^{}}I(a, t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\kappa ^{}}} ^{}}} {\mathrm {d}}a\end{array} &{} \begin{array}{l@{}}(1 - \varepsilon _s)\int _0^t \rho R(a, t)\\ \quad \times \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\kappa ^{}}} ^{}}} {\mathrm {d}}a\end{array} \\ \end{array}\right) \end{aligned}$$
and the transposed last two rows of the Jacobian are:
$$\begin{aligned} {\left( \left( f'_{\kappa , 3}\right) ^{\text {tr}}, \left( f'_{\kappa , 4}\right) ^{\text {tr}}\right) }:= & {} \left( \begin{array}{ll} -{\beta } \dfrac{\partial {^{}}V}{\partial {{\kappa }^{}}} &{}\begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho } \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} { \dfrac{\partial {^{}}I(a,t)}{\partial {T^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\kappa ^{}}} }{\partial {T^{}}} } {\mathrm {d}}a\\ \end{array}\\ \\ - {\beta } \dfrac{\partial {^{}}T}{\partial {{\kappa }^{}}} &{} \begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho } \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} { \dfrac{\partial {^{}}I(a,t)}{\partial {V^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\kappa ^{}}} }{\partial {V^{}}} } {\mathrm {d}}a\\ \end{array}\\ \\ -{d}- {\beta }V &{}\begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho } \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\kappa ^{}}} ^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\kappa ^{}}} }{\partial { \dfrac{\partial {^{}}T}{\partial {\kappa ^{}}} ^{}}} } {\mathrm {d}}a\\ \end{array}\\ \\ -{\beta }T&{}\begin{array}{l@{}}\displaystyle (1 - {\varepsilon _{s}})\int _0^t {\rho } \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} { \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\kappa ^{}}} ^{}}} } {\mathrm {d}}a \\ \qquad \displaystyle + (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t) { \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\kappa ^{}}} }{\partial { \dfrac{\partial {^{}}V}{\partial {\kappa ^{}}} ^{}}} } {\mathrm {d}}a - {c}\end{array}\\ \\ \end{array}\right) ,\\ \dfrac{\partial {^{}} \dfrac{\partial {^{}}f}{\partial {{\kappa }^{}}} }{\partial {t^{}}}= & {} \begin{aligned} \Big [&0,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }\left( \dfrac{\partial {^{}} \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} }{\partial {t^{}}} I (a, t) + \dfrac{\partial {^{}}R(a,t)}{\partial {\kappa ^{}}} \dfrac{\partial {^{}}I(a,t)}{\partial {t^{}}} \right) {\mathrm {d}}a \\&\qquad + (1 - {\varepsilon _{s}})\int _0^t {\rho }\left( \dfrac{\partial {^{}}R(a,t)}{\partial {t^{}}} \dfrac{\partial {^{}}I(a, t)}{\partial {{\kappa }^{}}} + R(a,t) \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a, t)}{\partial {\kappa ^{}}} }{\partial {t^{}}} \right) {\mathrm {d}}a\Big ] \end{aligned} \end{aligned}$$
where:
$$\begin{aligned} { \dfrac{\partial {^{}} \dfrac{\partial {^{}}R(a,t)}{\partial {{\kappa }^{}}} }{\partial {t^{}}} } = \left\{ \begin{array}{@{}lr@{}} \begin{array}{l@{}} {+}{\gamma }\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ {-}{\gamma }\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a}\\ -{\gamma }\mu a \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }(t - a)}}{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}{\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )a} \end{array} &{} a < t\\ \\ \begin{array}{l@{}} {+}{\gamma }\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha {\mathrm {e}}^{-{\gamma }t}}{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} \\ {-}((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )\mu \frac{(1 - {\varepsilon _{\alpha }})\alpha }{\left( (1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }\right) ^2} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\\ -\mu \bigg (\frac{\alpha }{{\rho }+ \mu } + \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} - \frac{(1 - {\varepsilon _{\alpha }})\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\bigg ) {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\\ {-}({\rho }+ \mu )\mu t \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t}\\ + ((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )\mu t\bigg (\frac{\alpha }{{\rho }+ \mu } + \left( 1 - \frac{\alpha }{{\rho }+ \mu }\right) {\mathrm {e}}^{-({\rho }+ \mu )(a-t)} - \frac{(1 - {\varepsilon _{\alpha }})\alpha }{(1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu - {\gamma }}\bigg )\\ \times {\mathrm {e}}^{-((1-{\varepsilon _{s}}){\rho }+ {\kappa }\mu )t} \end{array} &{} a > t\\ \end{array} \right. . \end{aligned}$$
1.6 Parameter \({c}\)
The derivative of the general function f (the vector [T, V]) with respect to c is:
$$\begin{aligned} \begin{aligned} \dfrac{\partial {^{}}f}{\partial {{c}^{}}} (t, y^c) =\Big [&- {d} \dfrac{\partial {^{}}T}{\partial {{c}^{}}} - {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{c}^{}}} T + V \dfrac{\partial {^{}}T}{\partial {{c}^{}}} \right) ,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }R(a, t) \dfrac{\partial {^{}}I}{\partial {{c}^{}}} (a, t) {\mathrm {d}}a - \left( V + {c} \dfrac{\partial {^{}}V}{\partial {{c}^{}}} \right) \Big ] \end{aligned} \end{aligned}$$
where \({y^c} = {\left\{ \begin{array}{ll} T\\ V\\ { \dfrac{\partial {^{}}T}{\partial {c^{}}} } \\ { \dfrac{\partial {^{}}V}{\partial {c^{}}} } \end{array}\right. }\).
Furthermore:
$$\begin{aligned} { \dfrac{\partial {^{}}R(a,t)}{\partial {{c}^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}\bar{T}}{\partial {{c}^{}}} }= & {} 1/({\beta }N),\\ { \dfrac{\partial {^{}}\bar{V}}{\partial {{c}^{}}} }= & {} - N {s}/ {c}^2,\\ { \dfrac{\partial {^{}}I(a,t)}{\partial {{c}^{}}} }= & {} \left\{ \begin{array}{lr} {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{c}^{}}} (t-a) T(t-a) + V(t-a) \dfrac{\partial {^{}}T}{\partial {{c}^{}}} (t-a)\right) {\mathrm {e}}^{-{\delta }a} &{} a < t\\ - {d}/ ({\beta }N) {\mathrm {e}}^{-{\delta }a} &{} a > t\\ \end{array} \right. , \end{aligned}$$
The upper right block matrix of the Jacobian is:
$$\begin{aligned} f'_{c, 2\times 2}= \left( \begin{array}{l@{\quad }l} 0 &{} 0 \\ \begin{array}{l@{}} (1 - \varepsilon _s)\int _0^t\rho R(a, t) \\ \quad \times \dfrac{\partial {^{}}I(a, t)}{\partial { \dfrac{\partial {^{}}T}{\partial {c^{}}} ^{}}} {\mathrm {d}}a\end{array} &{} \begin{array}{l@{}}(1 - \varepsilon _s)\int _0^t \rho R(a, t)\\ \quad \times \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {c^{}}} ^{}}} {\mathrm {d}}a\end{array} \\ \end{array}\right) \end{aligned}$$
and the transposed last two rows of the Jacobian are:
$$\begin{aligned} {\left( \left( f'_{c, 3}\right) ^{\text {tr}}, \left( f'_{c, 4}\right) ^{\text {tr}}\right) }:= & {} \left( \begin{array}{ll} -{\beta } \dfrac{\partial {^{}}V}{\partial {{c}^{}}} &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {c^{}}} }{\partial {T^{}}} } \mathrm da \\ -{\beta } \dfrac{\partial {^{}}T}{\partial {{c}^{}}} &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {c^{}}} }{\partial {V^{}}} } \mathrm da - 1 \\ -{d}-{\beta }V &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {c^{}}} }{\partial { \dfrac{\partial {^{}}T}{\partial {c^{}}} ^{}}} } \mathrm da \\ -{\beta }T &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {c^{}}} }{\partial { \dfrac{\partial {^{}}V}{\partial {c^{}}} ^{}}} } \mathrm da - {c}\\ \end{array}\right) ,\\ \dfrac{\partial {^{}} \dfrac{\partial {^{}}f}{\partial {{c}^{}}} }{\partial {t^{}}}= & {} \begin{aligned} \Big [&0,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }\left( \dfrac{\partial {^{}}R(a,t)}{\partial {t^{}}} \dfrac{\partial {^{}}I}{\partial {{c}^{}}} (a, t)+ R(a,t) \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {c^{}}} }{\partial {t^{}}} \right) {\mathrm {d}}a\Big ] \end{aligned} \end{aligned}$$
1.7 Parameter \({\delta }\)
The derivative of the general function f (the vector [T, V]) with respect to \(\delta \) is:
$$\begin{aligned} \begin{aligned} \dfrac{\partial {^{}}f}{\partial {{\delta }^{}}} (t, y^\delta )=\Big [&- {d} \dfrac{\partial {^{}}T}{\partial {{\delta }^{}}} - {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\delta }^{}}} T + V \dfrac{\partial {^{}}T}{\partial {{\delta }^{}}} \right) ,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }R(a, t) \dfrac{\partial {^{}}I}{\partial {{\delta }^{}}} (a, t) {\mathrm {d}}a - {c} \dfrac{\partial {^{}}V}{\partial {{\delta }^{}}} \Big ] \end{aligned} \end{aligned}$$
where \({y^{\delta }} = {\left\{ \begin{array}{ll} T\\ V\\ { \dfrac{\partial {^{}}T}{\partial {\delta ^{}}} }\\ { \dfrac{\partial {^{}}V}{\partial {\delta ^{}}} } \end{array}\right. }\).
Furthermore:
$$\begin{aligned} { \dfrac{\partial {^{}}R(a,t)}{\partial {{\delta }^{}}} }= & {} 0,\\ { \dfrac{\partial {^{}}\bar{T}}{\partial {\delta ^{}}} }= & {} \dfrac{\partial {^{}}\frac{1}{N}}{\partial {{\delta }^{}}} {c}/{\beta },\\ { \dfrac{\partial {^{}}\bar{V}}{\partial {\delta ^{}}} }= & {} \dfrac{\partial {^{}}N}{\partial {{\delta }^{}}} {s}/ {c},\\ \displaystyle \dfrac{\partial {^{}}N}{\partial {{\delta }^{}}}= & {} \frac{{\rho }\left( {\delta }({\rho }+ \mu + {\delta }) - (\alpha + {\delta })({\rho }+ \mu + {\delta }) - {\delta }(\alpha + {\delta })\right) }{{\delta }^2({\rho }+ \mu + {\delta })^2}, \\ \dfrac{\partial {^{}}\frac{1}{N}}{\partial {{\delta }^{}}}= & {} \frac{({\rho }+ \mu + {\delta })(\alpha + {\delta }) + {\delta }(\alpha + {\delta }) - {\delta }({\rho }+ \mu + {\delta })}{{\rho }(\alpha + {\delta })^2}, \end{aligned}$$
$$\begin{aligned} { \dfrac{\partial {^{}}I(a,t)}{\partial {{\delta }^{}}} }= & {} \left\{ \begin{array}{lr} {\beta }\left( \dfrac{\partial {^{}}V}{\partial {{\delta }^{}}} (t-a) T(t-a) + V(t-a) \dfrac{\partial {^{}}T}{\partial {{\delta }^{}}} (t-a) - aV(t-a) T(t-a) \right) {\mathrm {e}}^{-{\delta }a} &{} a < t\\ \left( - \dfrac{\partial {^{}}\frac{1}{N}}{\partial {{\delta }^{}}} {d}{c}/ {\beta }- a({\beta }N{s}- {d}{c}) / ({\beta }N) \right) {\mathrm {e}}^{-{\delta }a} &{} a > t\\ \end{array} \right. , \end{aligned}$$
The upper right block matrix of the Jacobian is:
$$\begin{aligned} f'_{\delta , 2\times 2}= \left( \begin{array}{l@{\quad }l} 0 &{} 0 \\ \begin{array}{l@{}} (1 - \varepsilon _s)\int _0^t\rho R(a, t) \\ \quad \times \dfrac{\partial {^{}}I(a, t)}{\partial { \dfrac{\partial {^{}}T}{\partial {\delta ^{}}} ^{}}} {\mathrm {d}}a\end{array} &{} \begin{array}{l@{}}(1 - \varepsilon _s)\int _0^t \rho R(a, t)\\ \quad \times \dfrac{\partial {^{}}I(a,t)}{\partial { \dfrac{\partial {^{}}V}{\partial {\delta ^{}}} ^{}}} {\mathrm {d}}a\end{array} \\ \end{array}\right) \end{aligned}$$
and the transposed last two rows of the Jacobian are:
$$\begin{aligned} {\left( \left( f'_{\delta , 3}\right) ^{\text {tr}}, \left( f'_{\delta , 4}\right) ^{\text {tr}}\right) }:= & {} \left( \begin{array}{ll} -{\beta } \dfrac{\partial {^{}}V}{\partial {{\delta }^{}}} &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\delta ^{}}} }{\partial {T^{}}} } \mathrm da \\ -{\beta } \dfrac{\partial {^{}}T}{\partial {{\delta }^{}}} &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\delta ^{}}} }{\partial {V^{}}} } \mathrm da \\ -{d}-{\beta }V &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\delta ^{}}} }{\partial { \dfrac{\partial {^{}}T}{\partial {\delta ^{}}} ^{}}} } \mathrm da \\ -{\beta }T &{} (1 - {\varepsilon _{s}})\int _0^t {\rho }R(a, t){ \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\delta ^{}}} }{\partial { \dfrac{\partial {^{}}V}{\partial {\delta ^{}}} ^{}}} } \mathrm da - {c}\\ \end{array}\right) ,\\ \dfrac{\partial {^{}} \dfrac{\partial {^{}}f}{\partial {{\delta }^{}}} }{\partial {t^{}}}= & {} \begin{aligned} \Bigg [&0,\\&(1 - {\varepsilon _{s}})\int _0^\infty {\rho }\left( \dfrac{\partial {^{}}R(a,t)}{\partial {t^{}}} \dfrac{\partial {^{}}I}{\partial {{\delta }^{}}} (a, t)+ R(a,t) \dfrac{\partial {^{}} \dfrac{\partial {^{}}I(a,t)}{\partial {\delta ^{}}} }{\partial {t^{}}} \right) {\mathrm {d}}a\Bigg ] \end{aligned} \end{aligned}$$
