The infection-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_0$$\end{document}E0 is locally asymptotically stable if\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \max \{\mathcal {R}_0^{\text {1D}},\mathcal {R}_{\text {eff}}^{\text {1D}}\}<1, \end{aligned}$$\end{document}max{R01D,Reff1D}<1,and is unstable otherwise.

We linearized Eq. 3 at the disease-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_0$$\end{document}E0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_0)=\begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & -\delta & \frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & p & -c-\phi & 0 & 0 & 0\\ 0 & 0 & 0 & -d & 0 & -\frac{\beta s_2}{d}\\ 0 & 0 & 0 & 0 & -\delta & \frac{\beta s_2}{d}\\ 0 & 0 & \phi & 0 & p & -c \end{bmatrix}\begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_0). \end{aligned}$$\end{document}ddtT1I1V1T2I2V2(E0)=-d0-βs1d0000-δβs1d0000p-c-ϕ000000-d0-βs2d0000-δβs2d00ϕ0p-cT1I1V1T2I2V2(E0).The corresponding Jacobian matrix at the disease-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_0$$\end{document}E0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J(E_0) = \begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & -\delta & \frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & p & -c-\phi & 0 & 0 & 0\\ 0 & 0 & 0 & -d & 0 & -\frac{\beta s_2}{d}\\ 0 & 0 & 0 & 0 & -\delta & \frac{\beta s_2}{d}\\ 0 & 0 & \phi & 0 & p & -c \end{bmatrix}, \end{aligned}$$\end{document}J(E0)=-d0-βs1d0000-δβs1d0000p-c-ϕ000000-d0-βs2d0000-δβs2d00ϕ0p-c,can be written as:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J(E_0) = \begin{bmatrix} A_{01} & \textbf{0}\\ \Phi & A_{02} \end{bmatrix}, \end{aligned}$$\end{document}J(E0)=A10ΦA2,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{0}$$\end{document}0 is the 3× 33×3 zero matrix,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{01}&= \begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d}\\ 0 & -\delta & \frac{\beta s_1}{d}\\ 0 & p & -c-\phi \end{bmatrix}, A_{02} =\begin{bmatrix} -d & 0 & -\frac{\beta s_2}{d}\\ 0 & -\delta & \frac{\beta s_2}{d}\\ 0 & p & -c \end{bmatrix}, \Phi = \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & \phi \end{bmatrix}. \end{aligned}$$\end{document}A1=-d0-βs1d0-δβs1d0p-c-ϕ,A2=-d0-βs2d0-δβs2d0p-c,Φ=00000000ϕ.The characteristic equation of model Eq. 3 at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_0$$\end{document}E0 is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0 = P_0(\lambda )&=\det (J(E_0)-\lambda 1\!\!1_6)=\det (A_{01}-\lambda 1\!\!1_3)\det (A_{02}-\lambda 1\!\!1_3) \\ &= (\lambda +d)^2P_{01}(\lambda )P_{02}(\lambda ). \end{aligned}$$\end{document}0=P0(λ)=det(J(E0)-λ16)=det(A1-λ13)det(A2-λ13)=(λ+d)2P1(λ)P2(λ).where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P_{01}(\lambda )&= \begin{vmatrix} -\delta -\lambda&\frac{\beta s_1}{d}\\ p&-c-\phi -\lambda \end{vmatrix} = \lambda ^2+(c+\delta +\phi )\lambda +\delta (c+\phi )(1-R_{\text {eff}}^{\text {1D}}),\\ P_{02}(\lambda )&= \begin{vmatrix} -\delta -\lambda&\frac{\beta s_2}{d}\\ p&-c-\lambda \end{vmatrix} = \lambda ^2+(c+\delta )\lambda + c\delta (1-\mathcal {R}_0^{\text {1D}}), \end{aligned}$$\end{document}P1(λ)=-δ-λβs1dp-c-ϕ-λ=λ2+(c+δ+ϕ)λ+δ(c+ϕ)(1-Reff1D),P2(λ)=-δ-λβs2dp-c-λ=λ2+(c+δ)λ+cδ(1-R01D),and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\!\!1_k$$\end{document}1k is the k× kk×k identity matrix. Two eigenvalues, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{1,2}=-d$$\end{document}λ1,2=-d are always negative. It is easy to see that the solutions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{01}(\lambda )$$\end{document}P1(λ) are negative or have negative real parts if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}<1$$\end{document}Reff1D<1 and the solutions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{02}(\lambda )$$\end{document}P2(λ) are negative or have negative real parts if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{\text {1D}}<1$$\end{document}R01D<1. Thus, when\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\max \{\mathcal {R}_0^{\text {1D}},R_{\text {eff}}^{\text {1D}}\}<1,\end{aligned}$$\end{document}max{R01D,Reff1D}<1,the infection-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_0$$\end{document}E0 is locally asymptotically stable. If, by contrast, either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{\text {1D}}>1$$\end{document}R01D>1 or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1, the infection-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_0$$\end{document}E0 is unstable. This completes the proof. □ □

The one-patch chronic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}E1 exists when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{\text {1D}}>1$$\end{document}R01D>1, is locally asymptotically stable if\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} R_{\text {eff}}^{\text {1D}}< 1 < \mathcal {R}_0^{\text {1D}}, \end{aligned}$$\end{document}Reff1D<1<R01D,and is unstable otherwise.

We linearize model Eq. 3 at equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}E1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_1)=\begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & -\delta & \frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & p & -c-\phi & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{\beta p s_2}{c\delta } & 0 & -\frac{c\delta }{p}\\ 0 & 0 & 0 & -d + \frac{\beta p s_2}{c\delta } & -\delta & \frac{c\delta }{p}\\ 0 & 0 & \phi & 0 & p & -c \end{bmatrix}\begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_1). \end{aligned}$$\end{document}ddtT1I1V1T2I2V2(E1)=-d0-βs1d0000-δβs1d0000p-c-ϕ000000-βps2cδ0-cδp000-d+βps2cδ-δcδp00ϕ0p-cT1I1V1T2I2V2(E1).The corresponding Jacobian matrix at equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}E1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J(E_1) = \begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & -\delta & \frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & p & -c-\phi & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{\beta p s_2}{c\delta } & 0 & -\frac{c\delta }{p}\\ 0 & 0 & 0 & -d + \frac{\beta p s_2}{c\delta } & -\delta & \frac{c\delta }{p}\\ 0 & 0 & \phi & 0 & p & -c \end{bmatrix}, \end{aligned}$$\end{document}J(E1)=-d0-βs1d0000-δβs1d0000p-c-ϕ000000-βps2cδ0-cδp000-d+βps2cδ-δcδp00ϕ0p-c,can be written as:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J(E_1) = \begin{bmatrix} A_{11} & \textbf{0}\\ \Phi & A_{12} \end{bmatrix}, \end{aligned}$$\end{document}J(E1)=A110ΦA12,where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{11} = \begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d}\\ 0 & -\delta & \frac{\beta s_1}{d}\\ 0 & p & -c-\phi \end{bmatrix}, A_{12} =\begin{bmatrix} -\frac{\beta p s_2}{c\delta }& 0 & -\frac{c\delta }{p}\\ -d+\frac{\beta p s_2}{c\delta } & -\delta & \frac{c\delta }{p}\\ 0 & p & -c \end{bmatrix}, \end{aligned}$$\end{document}A11=-d0-βs1d0-δβs1d0p-c-ϕ,A12=-βps2cδ0-cδp-d+βps2cδ-δcδp0p-c,and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{0}$$\end{document}0, Φ Φ are as before (see Proposition 1.). The characteristic equation of model Eq. 3 at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}E1 is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0&= P_1(\lambda ) =\det (J(E_1)-\lambda 1\!\!1_6) = \det (A_{11}-\lambda 1\!\!1_3)\det (A_{12}-\lambda 1\!\!1_{3})\\ &= (\lambda +d)P_{11}(\lambda )P_{12}(\lambda ), \end{aligned}$$\end{document}0=P1(λ)=det(J(E1)-λ16)=det(A11-λ13)det(A12-λ13)=(λ+d)P11(λ)P12(λ),where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P_{11}(\lambda )&= \begin{vmatrix} -\delta - \lambda&\frac{\beta s_1}{d}\\ p&-c-\phi -\lambda \end{vmatrix} = \lambda ^2+(c+\delta )\lambda + c\delta (1-\mathcal {R}_{\text {eff}}^{\text {1D}}),\\ P_{12}(\lambda )&= \begin{vmatrix} -\frac{\beta p s_2}{c\delta } - \lambda&0&-\frac{c\delta }{p}\\ -d+\frac{\beta p s_2}{c\delta }&-\delta -\lambda&\frac{c\delta }{p}\\ 0&p&-c-\lambda \end{vmatrix}\\ &= \lambda ^3 + \left( c+\delta + \frac{\beta p s_2}{c\delta }\right) \lambda ^2+\left( \frac{\beta p s_2}{\delta }+\frac{\beta p s_2}{c}\right) \lambda \\&\quad +cd\delta (\mathcal {R}_0^{\text {1D}}-1). \end{aligned}$$\end{document}P11(λ)=-δ-λβs1dp-c-ϕ-λ=λ2+(c+δ)λ+cδ(1-Reff1D),P12(λ)=-βps2cδ-λ0-cδp-d+βps2cδ-δ-λcδp0p-c-λ=λ3+c+δ+βps2cδλ2+βps2δ+βps2cλ+cdδ(R01D-1).One eigenvalue λ =-dλ=-d is always negative. It is easy to show that the eigenvalues of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{11}(\lambda )$$\end{document}P11(λ) are negative or have negative real parts if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}<1$$\end{document}Reff1D<1. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{12}(\lambda )$$\end{document}P12(λ), we apply the Routh-Hurwitz conditions which state that the solutions are negative or have negative real part if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i>0$$\end{document}bi>0 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=\{1,2,3\}$$\end{document}i={1,2,3} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1 b_2>b_3$$\end{document}b1b2>b3, where:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P_{12}(\lambda ) = \lambda ^3 + b_1\lambda ^2 + b_2\lambda + b_3, \end{aligned}$$\end{document}P12(λ)=λ3+b1λ2+b2λ+b3,and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b_1&= c+\delta + \frac{\beta p s_2}{c\delta },\\ b_2&= \frac{\beta p s_2}{\delta } + \frac{\beta p s_2}{c},\\ b_3&= cd\delta (\mathcal {R}_0^{\text {1D}}-1). \end{aligned}$$\end{document}b1=c+δ+βps2cδ,b2=βps2δ+βps2c,b3=cdδ(R01D-1).Observe that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1,b_2>0$$\end{document}b1,b2>0 always, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_3 > 0$$\end{document}b3>0 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{\text {1D}}>1$$\end{document}R01D>1. Moreover,8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} b_1b_2 - b_3&= \left( c+\delta + \frac{\beta p s_2}{c\delta }\right) \left( \frac{\beta p s_2}{\delta } + \frac{\beta p s_2}{c}\right) \\&\quad - cd\delta (\mathcal {R}_0^{\text {1D}}-1) \\&\quad \left( c+ \frac{\beta p s_2}{c\delta }\right) \left( \frac{\beta p s_2}{\delta } + \frac{\beta p s_2}{c}\right) \\&\quad + \frac{\delta \beta p s_2}{c} + cd\delta > 0. \end{aligned} \end{aligned}$$\end{document}b1b2-b3=c+δ+βps2cδβps2δ+βps2c-cdδ(R01D-1)c+βps2cδβps2δ+βps2c+δβps2c+cdδ>0.Therefore, when\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}R_{\text {eff}}^{\text {1D}}< 1 < \mathcal {R}_0^{\text {1D}},\end{aligned}$$\end{document}Reff1D<1<R01D,the one-patch chronic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}E1 is locally asymptotically stable. Conversely, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}} > 1$$\end{document}Reff1D>1 or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal {R}_0^{\text {1D}}<1$$\end{document}R01D<1, the one-patch chronic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}E1 is unstable. This completes the proof. □ □

The biologically realistic two-patch chronic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document}E2 exists and is locally asymptotically stable if\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}R_{\text {eff}}^{\text {1D}}>1.\end{aligned}$$\end{document}Reff1D>1.

We linearize model Eq. 3 at equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document}E2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\frac{d}{dt} \begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_2) \\ &= \begin{bmatrix} -d-d(R_{\text {eff}}^{\text {1D}}-1) & 0 & -\frac{\delta (c+\phi )}{p} & 0 & 0 & 0\\ d(R_{\text {eff}}^{\text {1D}}-1) & -\delta & \frac{\delta (c+\phi )}{p} & 0 & 0 & 0\\ 0 & p & -c-\phi & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{d(R_0^{\text {1D}}+1)}{2}- \frac{\phi \delta d(R_{\text {eff}}^{\text {1D}}-1) + \sqrt{D}}{2c\delta } & 0 & -\frac{cd\delta (R_0^{\text {1D}}+1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) - \sqrt{D}}{2dp}\\ 0 & 0 & 0 & \frac{d(R_0^{\text {1D}}-1)}{2}+ \frac{\phi \delta d(R_{\text {eff}}^{\text {1D}}-1) + \sqrt{D}}{2c\delta } & -\delta & \frac{cd\delta (R_0^{\text {1D}}+1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) - \sqrt{D}}{2dp}\\ 0 & 0 & \phi & 0 & p & -c \end{bmatrix}\\&\quad \times \begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_2). \end{aligned} \end{aligned}$$\end{document}ddtT1I1V1T2I2V2(E2)=-d-d(Reff1D-1)0-δ(c+ϕ)p000d(Reff1D-1)-δδ(c+ϕ)p0000p-c-ϕ000000-d(R01D+1)2-ϕδd(Reff1D-1)+D2cδ0-cdδ(R01D+1)+ϕδd(Reff1D-1)-D2dp000d(R01D-1)2+ϕδd(Reff1D-1)+D2cδ-δcdδ(R01D+1)+ϕδd(Reff1D-1)-D2dp00ϕ0p-c×T1I1V1T2I2V2(E2).The corresponding Jacobian matrix at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document}E2 is,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} J(E_2)&= \begin{bmatrix} A_{11} & 0\\ \Phi & A_{22} \end{bmatrix}. \end{aligned} \end{aligned}$$\end{document}J(E2)=A110ΦA22.where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{11}&= \begin{bmatrix} -d-d(R_{\text {eff}}^{\text {1D}}-1) & 0 & -\frac{\delta (c+\phi )}{p}\\ d(R_{\text {eff}}^{\text {1D}}-1) & -\delta & \frac{\delta (c+\phi )}{p}\\ 0 & p & -c-\phi \end{bmatrix},\\ A_{22}&= \begin{bmatrix} -\frac{d(R_0^{\text {1D}}+1)}{2}- \frac{\phi \delta d(R_{\text {eff}}^{\text {1D}}-1) + \sqrt{D}}{2c\delta } & 0 & -\frac{cd\delta (R_0^{\text {1D}}+1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) - \sqrt{D}}{2dp}\\ \frac{d(R_0^{\text {1D}}-1)}{2}+ \frac{\phi \delta d(R_{\text {eff}}^{\text {1D}}-1) + \sqrt{D}}{2c\delta } & -\delta & \frac{cd\delta (R_0^{\text {1D}}+1) + \phi \delta d(R_{\text {eff}}^{\text {1D}}-1) - \sqrt{D}}{2dp}\\ 0 & p & -c \end{bmatrix}, \end{aligned}$$\end{document}A11=-d-d(Reff1D-1)0-δ(c+ϕ)pd(Reff1D-1)-δδ(c+ϕ)p0p-c-ϕ,A22=-d(R01D+1)2-ϕδd(Reff1D-1)+D2cδ0-cdδ(R01D+1)+ϕδd(Reff1D-1)-D2dpd(R01D-1)2+ϕδd(Reff1D-1)+D2cδ-δcdδ(R01D+1)+ϕδd(Reff1D-1)-D2dp0p-c,and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{0}$$\end{document}0 and Φ Φ are as before (see Proposition 1). The characteristic equation of model Eq. 3 at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document}E2 is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0=P_2(\lambda )=\det (J(E_2) - \lambda 1\!\!1_6) = \det (A_{11} - \lambda 1\!\!1_3)\det (A_{22} - \lambda 1\!\!1_3)= P_{21}(\lambda ) P_{22}(\lambda ). \end{aligned}$$\end{document}0=P2(λ)=det(J(E2)-λ16)=det(A11-λ13)det(A22-λ13)=P21(λ)P22(λ).For both polynomials, we apply the Routh-Hurwitz condition. Let\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P_{21}(\lambda ) = \lambda ^3 + a_1\lambda ^2 + a_2\lambda + a_3, \end{aligned}$$\end{document}P21(λ)=λ3+a1λ2+a2λ+a3,where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} a_1&= c+d+\delta + \phi + d(R_{\text {eff}}^{\text {1D}}-1),\\ a_2&= d(c+\delta +\phi ) +d(c+\delta +\phi )(R_{\text {eff}}^{\text {1D}}-1),\\ a_3&= \delta d(c+\phi )(R_{\text {eff}}^{\text {1D}}-1). \end{aligned}$$\end{document}a1=c+d+δ+ϕ+d(Reff1D-1),a2=d(c+δ+ϕ)+d(c+δ+ϕ)(Reff1D-1),a3=δd(c+ϕ)(Reff1D-1).Polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{21}(\lambda )$$\end{document}P21(λ) has negative roots or roots with negative real parts when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i>0$$\end{document}ai>0 for all i and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1 a_2>a_3$$\end{document}a1a2>a3. Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i>0$$\end{document}ai>0 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1. Moreover, it is easy to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1 a_2- a_3>0$$\end{document}a1a2-a3>0 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1.

Similarly, let\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P_{22}(\lambda ) = \lambda ^3 + b_1\lambda ^2 + b_2\lambda + b_3, \end{aligned}$$\end{document}P22(λ)=λ3+b1λ2+b2λ+b3,where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b_1&= c + \delta + d + \beta (V_2^2)\\&= c + \delta + \frac{d(R_0^{\text {1D}} + 1)}{2} + \frac{\phi \delta d(R_{\text {eff}}^{\text {1D}} - 1) + \sqrt{D}}{2c \delta },\\ b_2&= (T_2^2) \beta p + (V_2^2) \beta c + (V_2^2) \beta \delta + cd + c \delta + d \delta \\&= \frac{c + \delta }{2c \delta }\left( cd \delta (R_0^{\text {1D}} - 1) + \phi \delta d(R_{\text {eff}}^{\text {1D}} - 1) + \sqrt{D}\right) + d \delta + cd + c \delta \\&\quad + \frac{1}{2d}\left( cd\delta (R_0^{\text {1D}} + 1) + \phi \delta d(R_{\text {eff}}^{\text {1D}} - 1) - \sqrt{D}\right) ,\\ b_3&= (T_2^2) \beta dp + (V_2^2) \beta c \delta + cd \delta \\&= cd \delta (R_0^{\text {1D}} + 1) + \phi \delta d(R_{\text {eff}}^{\text {1D}} - 1), \end{aligned}$$\end{document}b1=c+δ+d+β(V22)=c+δ+d(R01D+1)2+ϕδd(Reff1D-1)+D2cδ,b2=(T22)βp+(V22)βc+(V22)βδ+cd+cδ+dδ=c+δ2cδcdδ(R01D-1)+ϕδd(Reff1D-1)+D+dδ+cd+cδ+12dcdδ(R01D+1)+ϕδd(Reff1D-1)-D,b3=(T22)βdp+(V22)βcδ+cdδ=cdδ(R01D+1)+ϕδd(Reff1D-1),and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_2^2$$\end{document}V22 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2^2$$\end{document}T22 are the target cell and virus equilibrium values for the nodular structure 2 as given by Eq. 5. Polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{22}(\lambda )$$\end{document}P22(λ) has negative roots or roots with negative real parts when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i>0$$\end{document}bi>0 for all i and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1 b_2>b_3$$\end{document}b1b2>b3. Note that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i>0$$\end{document}bi>0 for all i. Moreover,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} b_1 b_2 - b_3&= \beta ^2 (c + \delta ) V_2^2 + T_2 V_2 \beta ^2 p \\&\quad + V_2 \beta (c + \delta )(c+\delta +2d) \\&\quad + T_2\beta p(c + \delta ) \\&\quad + (c+\delta )(c+d)(\delta +d), \end{aligned} \end{aligned}$$\end{document}b1b2-b3=β2(c+δ)V22+T2V2β2p+V2β(c+δ)(c+δ+2d)+T2βp(c+δ)+(c+δ)(c+d)(δ+d),is positive when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1. Then, by the Routh-Hurwitz condition, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{22}(\lambda )$$\end{document}P22(λ) has negative roots or roots with negative real part when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}>1$$\end{document}Reff1D>1. This guarantees the local stability analysis of the chronic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document}E2. Conversely, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{\text {eff}}^{\text {1D}}<1$$\end{document}Reff1D<1, equilibrum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document}E2 does not exist. This concludes the proof. □ □

When measurements for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1(t)$$\end{document}V1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_2(t)$$\end{document}V2(t) variables are given, system Eq. 3 is unidentifiable under unknown initial conditions. In particular, parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _1=\{\beta , c, d, \delta , \phi \}$$\end{document}π1={β,c,d,δ,ϕ} are globally identifiable and parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _2=\{s_1, s_2, p\}$$\end{document}π2={s1,s2,p} are unidentifiable. If, additionally, initial conditions are known, then all parameters are globally structurally identifiable.

The input-output equation for model Eq. 3, given data for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1=V_{1}(t)$$\end{document}y1=V1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2=V_{2}(t)$$\end{document}y2=V2(t), is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f(y_1, y_2, \pi _1, \pi _2)=\frac{d^3 y_1}{dt^3} y_1 - \frac{d^2 y_1}{dt^2} \frac{d y_1}{dt} + \frac{d^2 y_1}{dt^2} y_1^2 \beta \\&\quad + \frac{d^2 y_1}{dt^2} y_1 (c + d + \delta + \phi ) - \left( \frac{d y_1}{dt} \right) ^2 (c + \delta + \phi ) \\&\quad + \frac{d y_1}{dt} y_1^2 \beta (c + \delta + \phi ) + \frac{d y_1}{dt} y_1 d (c + \delta + \phi ) \\&\quad + y_1^3 \beta \delta (c + \phi ) + y_1^2 \left( - \beta p s_1 + c d \delta + d \delta \phi \right) =0,\\&g(y_1, y_2, \pi _1, \pi _2)=\frac{d^2 y_1}{dt^2} y_2 \phi + \frac{d y_1}{dt} \frac{d y_2}{dt} \phi - \frac{d y_1}{dt} y_2^2 \beta \phi \\&\quad - \frac{d y_1}{dt} y_2 \phi (d + \delta ) + \frac{d^3 y_2}{dt^3} y_2 - \frac{d^2 y_2}{dt^2} \frac{d y_2}{dt} \\&\quad + \frac{d^2 y_2}{dt^2} y_2^2 \beta + \frac{d^2 y_2}{dt^2} y_2 (c + d + \delta ) - \left( \frac{d y_2}{dt} \right) ^2 (c + \delta ) \\&\quad + \frac{d y_2}{dt} y_1 \delta \phi + \frac{d y_2}{dt} y_2^2 \beta (c + \delta ) \\&\quad + \frac{d y_2}{dt} y_2 d (c + \delta ) - y_1 y_2^2 \beta \delta \phi - y_1 y_2 d \delta \phi \\&\quad + y_2^3 \beta c \delta + y_2^2 \left( - \beta p s_2 + c d \delta \right) =0. \end{aligned}$$\end{document}f(y1,y2,π1,π2)=d3y1dt3y1-d2y1dt2dy1dt+d2y1dt2y12β+d2y1dt2y1(c+d+δ+ϕ)-dy1dt2(c+δ+ϕ)+dy1dty12β(c+δ+ϕ)+dy1dty1d(c+δ+ϕ)+y13βδ(c+ϕ)+y12-βps1+cdδ+dδϕ=0,g(y1,y2,π1,π2)=d2y1dt2y2ϕ+dy1dtdy2dtϕ-dy1dty22βϕ-dy1dty2ϕ(d+δ)+d3y2dt3y2-d2y2dt2dy2dt+d2y2dt2y22β+d2y2dt2y2(c+d+δ)-dy2dt2(c+δ)+dy2dty1δϕ+dy2dty22β(c+δ)+dy2dty2d(c+δ)-y1y22βδϕ-y1y2dδϕ+y23βcδ+y22-βps2+cdδ=0.Parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\pi _1,\pi _2\}$$\end{document}{π1,π2} are globally structurally identifiable if:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}f(y_1, y_2, \pi _1, \pi _2)=f(y_1, y_2, \hat{\pi }_1, \hat{\pi }_2) \text { implies } \pi _1=\hat{\pi }_1 \text { and } \pi _2=\hat{\pi }_2.\end{aligned}$$\end{document}f(y1,y2,π1,π2)=f(y1,y2,π^1,π^2)impliesπ1=π^1andπ2=π^2.It is easy to see that:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\{\beta =\hat{\beta }\} \text { , } \{d=\hat{d}\} \text { , } \{\delta =\hat{\delta }\} \text { , } \{c=\hat{c}\} \text { , } \{\phi =\hat{\phi }\} \text { , } \{p \times s_1=\hat{p}\times \hat{s}_1\} \text { , } \{p \times s_2=\hat{p}\times \hat{s}_2\}.\end{aligned}$$\end{document}{β=β^},{d=d^},{δ=δ^},{c=c^},{ϕ=ϕ^},{p×s1=p^×s^1},{p×s2=p^×s^2}.Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\beta , c, d, \delta , \phi \}$$\end{document}{β,c,d,δ,ϕ} are globally structurally identifiable and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{s_1, s_2, p\}$$\end{document}{s1,s2,p} are unidentifiable. When initial conditions are known, we have:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\frac{dV_{1}}{dt}(0)=p I_1(0) -cV_1(0)-\phi V_1(0)+\phi V_2(0).\end{aligned}$$\end{document}dV1dt(0)=pI1(0)-cV1(0)-ϕV1(0)+ϕV2(0).As long as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1(0)\ne 0$$\end{document}I1(0)≠0, p (and by default \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1$$\end{document}s1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_2$$\end{document}s2) are globally structurally identifiable. Hence, model Eq. 3 is globally structurally identifiable for known initial conditions. A summary of the results is given in Table 1. □ □

When measurements for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1(t)$$\end{document}T1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2(t)$$\end{document}T2(t) variables are given, system Eq. 3 is unidentifiable when initial conditions are unknown. In particular, parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _1=\{ c, d, \delta , \phi , s_{1}, s_{2}\}$$\end{document}π1={c,d,δ,ϕ,s1,s2} are globally structurally identifiable, and parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _2=\{\beta , p\}$$\end{document}π2={β,p} are unidentifiable. If, additionally, initial conditions are known, then all parameters are globally structurally identifiable.

The input-output equations for model Eq. 3, given data for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1=T_{1}(t)$$\end{document}y1=T1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2=T_{2}(t)$$\end{document}y2=T2(t), are:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f(y_1,y_2, \pi _1, \pi _2)= - \frac{d^3 y_1}{dt^3} y_1^2 y_2^2 + 3 \frac{d^2 y_1}{dt^2} \frac{d y_1}{dt} y_1 y_2^2 - \frac{d^2 y_1}{dt^2} y_1^2 y_2^2 (c + \delta + \phi )\\&\quad - \frac{d^2 y_1}{dt^2} y_1 y_2^2 s_1 - 2 \left( \frac{d y_1}{dt} \right) ^3 y_2^2 \\&+ \left( \frac{d y_1}{dt} \right) ^2 y_1 y_2^2 (c + \delta + \phi ) + 2 \left( \frac{d y_1}{dt} \right) ^2 y_2^2 s_1 + \frac{d y_1}{dt} y_1^3 y_2^2 \beta p\\&\quad - \frac{d y_1}{dt} y_1^2 y_2^2 \delta (c + \phi ) - \frac{d y_1}{dt} y_1 y_2^2 s_1 (c + \delta + \phi ) \\&+ \frac{d^2 y_2}{dt^2} y_1^3 y_2 \phi - \left( \frac{d y_2}{dt} \right) ^2 y_1^3 \phi + \frac{d y_2}{dt} y_1^3 y_2 \delta \phi + \frac{d y_2}{dt} y_1^3 \phi s_2 + y_1^4 y_2^2 \beta d p \\&\quad - y_1^3 y_2^2 \left( \beta p s_1 + c d \delta \right) - y_1^3 y_2 \delta \phi s_2 \\&+ y_1^2 y_2^2 \delta s_1 (c + \phi )=0,\\&g(y_1,y_2, \pi _1, \pi _2) = \frac{d^2 y_1}{dt^2} y_1 y_2^3 \phi - \left( \frac{d y_1}{dt} \right) ^2 y_2^3 \phi + \frac{d y_1}{dt} y_1 y_2^3 \delta \phi + \frac{d y_1}{dt} y_2^3 \phi s_1 \\&\quad - \frac{d^3 y_2}{dt^3} y_1^2 y_2^2 + 3 \frac{d^2 y_2}{dt^2} \frac{d y_2}{dt} y_1^2 y_2 \\&- \frac{d^2 y_2}{dt^2} y_1^2 y_2^2 (c + \delta ) - \frac{d^2 y_2}{dt^2} y_1^2 y_2 s_2 - 2 \left( \frac{d y_2}{dt} \right) ^3 y_1^2 + \left( \frac{d y_2}{dt} \right) ^2 y_1^2 y_2 (c + \delta ) \\&\quad + 2 \left( \frac{d y_2}{dt} \right) ^2 y_1^2 s_2 \\&+ \frac{d y_2}{dt} y_1^2 y_2^3 \beta p - \frac{d y_2}{dt} y_1^2 y_2^2 c \delta - \frac{d y_2}{dt} y_1^2 y_2 s_2 (c + \delta )+ y_1^2 y_2^4 \beta d p \\&\quad + y_1^2 y_2^3 \left( - \beta p s_2 - c d \delta + d \delta \phi \right) + y_1^2 y_2^2 c \delta s_2 - y_1 y_2^3 \delta \phi s_1=0. \end{aligned}$$\end{document}f(y1,y2,π1,π2)=-d3y1dt3y12y22+3d2y1dt2dy1dty1y22-d2y1dt2y12y22(c+δ+ϕ)-d2y1dt2y1y22s1-2dy1dt3y22+dy1dt2y1y22(c+δ+ϕ)+2dy1dt2y22s1+dy1dty13y22βp-dy1dty12y22δ(c+ϕ)-dy1dty1y22s1(c+δ+ϕ)+d2y2dt2y13y2ϕ-dy2dt2y13ϕ+dy2dty13y2δϕ+dy2dty13ϕs2+y14y22βdp-y13y22βps1+cdδ-y13y2δϕs2+y12y22δs1(c+ϕ)=0,g(y1,y2,π1,π2)=d2y1dt2y1y23ϕ-dy1dt2y23ϕ+dy1dty1y23δϕ+dy1dty23ϕs1-d3y2dt3y12y22+3d2y2dt2dy2dty12y2-d2y2dt2y12y22(c+δ)-d2y2dt2y12y2s2-2dy2dt3y12+dy2dt2y12y2(c+δ)+2dy2dt2y12s2+dy2dty12y23βp-dy2dty12y22cδ-dy2dty12y2s2(c+δ)+y12y24βdp+y12y23-βps2-cdδ+dδϕ+y12y22cδs2-y1y23δϕs1=0.Parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\pi _1,\pi _2\}$$\end{document}{π1,π2} are globally structurally identifiable if:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} f(y_1, y_2, \pi _1, \pi _2)&= f(y_1, y_2, \hat{\pi }_1, \hat{\pi }_2) \text { and }\\ g(y_1, y_2, \pi _1, \pi _2)&= g(y_1, y_2, \hat{\pi }_1, \hat{\pi }_2) \\&\text { imply } \pi _1 = \hat{\pi }_1 \text { and } \pi _2 = \hat{\pi }_2. \end{aligned} \end{aligned}$$\end{document}f(y1,y2,π1,π2)=f(y1,y2,π^1,π^2)andg(y1,y2,π1,π2)=g(y1,y2,π^1,π^2)implyπ1=π^1andπ2=π^2.It is easy to see that:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\{s_1=\hat{s}_1\} \text { , } \{s_2=\hat{s}_2\} \text { , } \{\delta =\hat{\delta }\} \text { , } \{d=\hat{d}\} \text { , } \{c=\hat{c}\} \text { , } \{\phi =\hat{\phi }\} \text { , } \{\beta \times p=\hat{\beta }\times \hat{p}\}.\end{aligned}$$\end{document}{s1=s^1},{s2=s^2},{δ=δ^},{d=d^},{c=c^},{ϕ=ϕ^},{β×p=β^×p^}.Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{s_1, s_2, d, \delta , c, \phi \}$$\end{document}{s1,s2,d,δ,c,ϕ} are globally structurally identifiable and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\beta , p\}$$\end{document}{β,p} are unidentifiable. When initial conditions are known, we have:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\frac{dT_{1}}{dt}(0)=s_1-d T_1(0)-\beta T_1(0) V_1(0),\end{aligned}$$\end{document}dT1dt(0)=s1-dT1(0)-βT1(0)V1(0),making β β (and by default p) globally structurally identifiable. Hence, model Eq. 3 is globally structurally identifiable for known initial conditions. A summary of the results is given in Table 1. □ □

Table 1. Identifiability analysis for model Eq. 3, performed using the DAISY software (Bellu et al. 2007). For the simulations with initial conditions, all initial conditions are known

The infection-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4$$\end{document}E4 is locally asymptotically stable if\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {R}_0^{2D} = \frac{\beta p s_2}{d\delta }\frac{p s_1\beta + d \delta (c+\phi )}{ps_1\beta (c+\phi ) + d c\delta (c+2\phi )}<1, \end{aligned}$$\end{document}R02D=βps2dδps1β+dδ(c+ϕ)ps1β(c+ϕ)+dcδ(c+2ϕ)<1,and unstable otherwise.

We linearize Eq. 9 at the infection-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4$$\end{document}E4:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_4)=\begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & -\delta & \frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & p & -c-\phi & 0 & 0 & \phi \\ 0 & 0 & 0 & -d & 0 & -\frac{\beta s_2}{d}\\ 0 & 0 & 0 & 0 & -\delta & \frac{\beta s_2}{d}\\ 0 & 0 & \phi & 0 & p & -c-\phi \end{bmatrix}\begin{bmatrix} T_1\\ I_1\\ V_1\\ T_2\\ I_2\\ V_2 \end{bmatrix}(E_4), \end{aligned}$$\end{document}ddtT1I1V1T2I2V2(E4)=-d0-βs1d0000-δβs1d0000p-c-ϕ00ϕ000-d0-βs2d0000-δβs2d00ϕ0p-c-ϕT1I1V1T2I2V2(E4),The corresponding Jacobian matrix at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4$$\end{document}E4 is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J(E_4) = \begin{bmatrix} -d & 0 & -\frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & -\delta & \frac{\beta s_1}{d} & 0 & 0 & 0\\ 0 & p & -c-\phi & 0 & 0 & \phi \\ 0 & 0 & 0 & -d & 0 & -\frac{\beta s_2}{d}\\ 0 & 0 & 0 & 0 & -\delta & \frac{\beta s_2}{d}\\ 0 & 0 & \phi & 0 & p & -c-\phi \end{bmatrix}, \end{aligned}$$\end{document}J(E4)=-d0-βs1d0000-δβs1d0000p-c-ϕ00ϕ000-d0-βs2d0000-δβs2d00ϕ0p-c-ϕ,and the characteristic equation for model Eq. 9 at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4$$\end{document}E4 is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0=P_{4}(E_4)=\det (J(E_4)-\lambda 1\!\!1_6) = (\lambda + d)^2\begin{vmatrix} -\delta -\lambda&-\frac{\beta s_1}{d}&0&0\\ p&-c-\phi -\lambda&0&\phi \\ 0&0&-\delta - \lambda&\frac{\beta s_2}{d}\\ 0&\phi&p&-c-\phi -\lambda \end{vmatrix}. \end{aligned}$$\end{document}0=P4(E4)=det(J(E4)-λ16)=(λ+d)2-δ-λ-βs1d00p-c-ϕ-λ0ϕ00-δ-λβs2d0ϕp-c-ϕ-λ.Two eigenvalues, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{1,2}=-d$$\end{document}λ1,2=-d are always negative. The remaining polynomial is:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P_{41}(\lambda ) = \lambda ^4 + a_1\lambda ^3 + a_2\lambda ^2 + a_3\lambda + a_4, \end{aligned}$$\end{document}P41(λ)=λ4+a1λ3+a2λ2+a3λ+a4,where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} a_1&= 2 c+2 \delta +2 \phi ,\\ a_2&= (c+\delta )^2+2\phi (c+2\delta )+\frac{c^2\delta }{c+\phi }+\frac{\beta ^2 p^2 s1 s2}{d^2\delta (c+\phi )}\\ &\quad +\frac{\beta p s_1(c+\phi )+dc\delta (c+2\phi )}{d(c+\phi )}(1-\mathcal {R}_0^{2D}),\\ a_3&= 2(c+\delta )(c+\phi )\delta +2c\phi \delta +\frac{(c+\phi +\delta )\beta ^2 p^2 s_1 s_2}{d^2 \delta (c+\phi )}\\&\quad +\frac{\beta ps_1 (c+\phi )+dc\delta (c+2\phi )}{d(c+\phi )}(c+\phi +\delta )(1-\mathcal {R}_0^{2D}),\\ a_4&=\frac{\beta ps_1 \delta (c+\phi )+dc\delta ^2(c+2\phi )}{d}(1-\mathcal {R}_0^{2D}). \end{aligned}$$\end{document}a1=2c+2δ+2ϕ,a2=(c+δ)2+2ϕ(c+2δ)+c2δc+ϕ+β2p2s1s2d2δ(c+ϕ)+βps1(c+ϕ)+dcδ(c+2ϕ)d(c+ϕ)(1-R02D),a3=2(c+δ)(c+ϕ)δ+2cϕδ+(c+ϕ+δ)β2p2s1s2d2δ(c+ϕ)+βps1(c+ϕ)+dcδ(c+2ϕ)d(c+ϕ)(c+ϕ+δ)(1-R02D),a4=βps1δ(c+ϕ)+dcδ2(c+2ϕ)d(1-R02D).By the Routh-Hurwitz criteria, polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{41}(\lambda )$$\end{document}P41(λ) has negative roots or roots with negative real parts when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i>0$$\end{document}ai>0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1 a_2 a_3 > a_3^2+ a_1^2 a_4$$\end{document}a1a2a3>a32+a12a4. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{2D}<1$$\end{document}R02D<1, then all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i>0$$\end{document}ai>0. Moreover, using Maple, we can show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1a_2a_3 > a_3^2 + a_1^2a_4$$\end{document}a1a2a3>a32+a12a4 when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{2D} < 1$$\end{document}R02D<1 (see Appendix A). Thus, by the Routh-Hurwitz criteria, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{2D} < 1$$\end{document}R02D<1, the infection-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4$$\end{document}E4 is locally asymptotically stable. In contrast, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0^{2D} > 1$$\end{document}R02D>1, the infection-free equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_4$$\end{document}E4 is unstable. This concludes the proof. □ □

Lastly, since explicit forms of chronic equilibria \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_j$$\end{document}Ej (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j={5,6,7}$$\end{document}j=5,6,7) are hard to determine, we examined their existence and asymptotic stability numerically.

Given measurements for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1(t)$$\end{document}V1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_2(t)$$\end{document}V2(t) variables, system Eq. 9 is unidentifiable when initial conditions are unknown. In particular, parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _1=\{ \beta , c, d, \delta , \}$$\end{document}π1={β,c,d,δ,} are globally structurally identifiable, and parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _2=\{s_{1}, s_{2}, p\}$$\end{document}π2={s1,s2,p} are unidentifiable. If, additionally, initial conditions are known, then all parameters are globally structurally identifiable.

The input-output equations for model Eq. 9, given data for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1=V_{1}(t)$$\end{document}y1=V1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2=V_{2}(t)$$\end{document}y2=V2(t), are:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f(y_1, y_2, \pi _1,\pi _2) = \frac{d^3 y_1}{dt^3} y_1 - \frac{d^2 y_1}{dt^2} \frac{d y_1}{dt} + \frac{d^2 y_1}{dt^2} y_1^2 \beta + \frac{d^2 y_1}{dt^2} y_1 (c + d + \delta + \phi ) \\&\quad - \left( \frac{d y_1}{dt} \right) ^2 (c + \delta + \phi ) + \frac{d y_1}{dt} \frac{d y_2}{dt} \phi + \frac{d y_1}{dt} y_1^2 \beta (c + \delta + \phi ) \\&\quad + \frac{d y_1}{dt} y_1 d (c + \delta + \phi ) + \frac{d y_1}{dt} y_2 \delta \phi - \frac{d^2 y_2}{dt^2} y_1 \phi \\&\quad - \frac{d y_2}{dt} y_1^2 \beta \phi - \frac{d y_2}{dt} y_1 \phi (d + \delta ) + y_1^3 \beta \delta (c + \phi ) \\&\quad - y_1^2 y_2 \beta \delta \phi + y_1^2 \left( - \beta p s_1 + c d \delta + d \delta \phi \right) \\&\quad - y_1 y_2 d \delta \phi = 0, \\&g(y_1, y_2, \pi _1, \pi _2) = - \frac{d^2 y_1}{dt^2} y_2 \phi + \frac{d y_1}{dt} \frac{d y_2}{dt} \phi \\&\quad - \frac{d y_1}{dt} y_2^2 \beta \phi - \frac{d y_1}{dt} y_2 \phi (d + \delta ) + \frac{d^3 y_2}{dt^3} y_2 \\&\quad - \frac{d^2 y_2}{dt^2} \frac{d y_2}{dt} + \frac{d^2 y_2}{dt^2} y_2^2 \beta + \frac{d^2 y_2}{dt^2} y_2 (c + d + \delta + \phi ) \\&\quad - \left( \frac{d y_2}{dt} \right) ^2 (c + \delta + \phi ) + \frac{d y_2}{dt} y_1 \delta \phi \\&\quad + \frac{d y_2}{dt} y_2^2 \beta (c + \delta + \phi ) + \frac{d y_2}{dt} y_2 d (c + \delta + \phi ) \\&\quad - y_1 y_2^2 \beta \delta \phi - y_1 y_2 d \delta \phi + y_2^3 \beta \delta (c + \phi ) \\&\quad + y_2^2 \left( - \beta p s_2 + c d \delta + d \delta \phi \right) = 0. \end{aligned}$$\end{document}f(y1,y2,π1,π2)=d3y1dt3y1-d2y1dt2dy1dt+d2y1dt2y12β+d2y1dt2y1(c+d+δ+ϕ)-dy1dt2(c+δ+ϕ)+dy1dtdy2dtϕ+dy1dty12β(c+δ+ϕ)+dy1dty1d(c+δ+ϕ)+dy1dty2δϕ-d2y2dt2y1ϕ-dy2dty12βϕ-dy2dty1ϕ(d+δ)+y13βδ(c+ϕ)-y12y2βδϕ+y12-βps1+cdδ+dδϕ-y1y2dδϕ=0,g(y1,y2,π1,π2)=-d2y1dt2y2ϕ+dy1dtdy2dtϕ-dy1dty22βϕ-dy1dty2ϕ(d+δ)+d3y2dt3y2-d2y2dt2dy2dt+d2y2dt2y22β+d2y2dt2y2(c+d+δ+ϕ)-dy2dt2(c+δ+ϕ)+dy2dty1δϕ+dy2dty22β(c+δ+ϕ)+dy2dty2d(c+δ+ϕ)-y1y22βδϕ-y1y2dδϕ+y23βδ(c+ϕ)+y22-βps2+cdδ+dδϕ=0.Parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\pi _1,\pi _2\}$$\end{document}{π1,π2} are globally structurally identifiable if:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} f(y_1, y_2, \pi _1, \pi _2)&= f(y_1, y_2, \hat{\pi }_1, \hat{\pi }_2) \text { and }\\ g(y_1, y_2, \pi _1, \pi _2)&= g(y_1, y_2, \hat{\pi }_1, \hat{\pi }_2) \\&\text { imply } \pi _1 = \hat{\pi }_1 \text { and } \pi _2 = \hat{\pi }_2. \end{aligned} \end{aligned}$$\end{document}f(y1,y2,π1,π2)=f(y1,y2,π^1,π^2)andg(y1,y2,π1,π2)=g(y1,y2,π^1,π^2)implyπ1=π^1andπ2=π^2.It is easy to see that:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\{\beta =\hat{\beta }\} \text { , } \{d=\hat{d}\} \text { , } \{\delta =\hat{\delta }\} \text { , } \{c=\hat{c}\} \text { , } \{\phi =\hat{\phi }\} \text { , } \{p \times s_1=\hat{p}\times \hat{s}_1\} \text { , } \{p \times s_2=\hat{p}\times \hat{s}_2\}.\end{aligned}$$\end{document}{β=β^},{d=d^},{δ=δ^},{c=c^},{ϕ=ϕ^},{p×s1=p^×s^1},{p×s2=p^×s^2}.Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\beta , c, d, \delta , \phi \}$$\end{document}{β,c,d,δ,ϕ} are globally structurally identifiable and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{s_1, s_2, p\}$$\end{document}{s1,s2,p} are unidentifiable. When initial conditions are known, we have\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\frac{dV_{1}}{dt}(0)=p I_1(0) -cV_1(0)-\phi V_1(0)+\phi V_2(0).\end{aligned}$$\end{document}dV1dt(0)=pI1(0)-cV1(0)-ϕV1(0)+ϕV2(0).As long as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1(0)\ne 0$$\end{document}I1(0)≠0, p (and by default \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1$$\end{document}s1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_2$$\end{document}s2) are globally structurally identifiable. Hence, model Eq. 9 is globally structurally identifiable for known initial conditions. A summary of the results is given in Table 3. □ □

Given measurements for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1(t)$$\end{document}T1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2(t)$$\end{document}T2(t) variables, system Eq. 9 is unidentifiable when initial conditions are unknown. In particular, parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _1=\{c, d, \delta , \phi , s_{1}, s_{2}\}$$\end{document}π1={c,d,δ,ϕ,s1,s2} are globally structurally identifiable, and parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _2=\{\beta , p\}$$\end{document}π2={β,p} are unidentifiable. If, additionally, initial conditions are known, then all parameters are globally structurally identifiable.

The input-output equations for model Eq. 9, given data for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1=T_{1}(t)$$\end{document}y1=T1(t) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2=T_{2}(t)$$\end{document}y2=T2(t), are:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f(y_1,y_2, \pi _1, \pi _2)= - \frac{d^3 y_1}{dt^3} y_1^2 y_2^2 + 3 \frac{d^2 y_1}{dt^2} \frac{d y_1}{dt} y_1 y_2^2 - \frac{d^2 y_1}{dt^2} y_1^2 y_2^2 (c + \delta + \phi ) \\&\quad - \frac{d^2 y_1}{dt^2} y_1 y_2^2 s_1 - 2 \left( \frac{d y_1}{dt} \right) ^3 y_2^2 \\&\quad + \frac{d y_1}{dt}^2 y_1 y_2^2 (c + \delta + \phi ) + 2 \frac{d y_1}{dt}^2 y_2^2 s_1 + \frac{d y_1}{dt} y_1^3 y_2^2 \beta p \\&\quad - \frac{d y_1}{dt} y_1^2 y_2^2 \delta (c + \phi ) - \frac{d y_1}{dt} y_1 y_2^2 s_1 (c + \delta + \phi ) \\&\quad + \frac{d^2 y_2}{dt^2} y_1^3 y_2 \phi - \left( \frac{d y_2}{dt} \right) ^2 y_1^3 \phi + \frac{d y_2}{dt} y_1^3 y_2 \delta \phi \\&\quad + \frac{d y_2}{dt} y_1^3 \phi s_2 + y_1^4 y_2^2 \beta d p - y_1^3 y_2^2 \left( \beta p s_1 + c d \delta \right) \\&\quad - y_1^3 y_2 \delta \phi s_2 + y_1^2 y_2^2 \delta s_1 (c + \phi )=0,\\&g(y_1,y_2, \pi _1, \pi _2)= \frac{d^2 y_1}{dt^2} y_1 y_2^3 \phi - \left( \frac{d y_1}{dt} \right) ^2 y_2^3 \phi \\&\quad + \frac{d y_1}{dt} y_1 y_2^3 \delta \phi + \frac{d y_1}{dt} y_2^3 \phi s_1 \\&\quad - \frac{d^3 y_2}{dt^3} y_1^2 y_2^2 + 3 \frac{d^2 y_2}{dt^2} \frac{d y_2}{dt} y_1^2 y_2 \\&\quad - \frac{d^2 y_2}{dt^2} y_1^2 y_2^2 (c + \delta + \phi ) - \frac{d^2 y_2}{dt^2} y_1^2 y_2 s_2 \\&\quad - 2 \left( \frac{d y_2}{dt} \right) ^3 y_1^2 + \left( \frac{d y_2}{dt} \right) ^2 y_1^2 y_2 (c + \delta + \phi ) \\&\quad + 2 \left( \frac{d y_2}{dt} \right) ^2 y_1^2 s_2 + \frac{d y_2}{dt} y_1^2 y_2^3 \beta p \\&\quad - \frac{d y_2}{dt} y_1^2 y_2^2 \delta (c + \phi ) \\&\quad - \frac{d y_2}{dt} y_1^2 y_2 s_2 (c + \delta + \phi ) \\&\quad + y_1^2 y_2^4 \beta d p - y_1^2 y_2^3 (\beta p s_2 + c d \delta ) \\&\quad + y_1^2 y_2^2 \delta s_2 (c + \phi ) - y_1 y_2^3 \delta \phi s_1=0. \end{aligned}$$\end{document}f(y1,y2,π1,π2)=-d3y1dt3y12y22+3d2y1dt2dy1dty1y22-d2y1dt2y12y22(c+δ+ϕ)-d2y1dt2y1y22s1-2dy1dt3y22+dy1dt2y1y22(c+δ+ϕ)+2dy1dt2y22s1+dy1dty13y22βp-dy1dty12y22δ(c+ϕ)-dy1dty1y22s1(c+δ+ϕ)+d2y2dt2y13y2ϕ-dy2dt2y13ϕ+dy2dty13y2δϕ+dy2dty13ϕs2+y14y22βdp-y13y22βps1+cdδ-y13y2δϕs2+y12y22δs1(c+ϕ)=0,g(y1,y2,π1,π2)=d2y1dt2y1y23ϕ-dy1dt2y23ϕ+dy1dty1y23δϕ+dy1dty23ϕs1-d3y2dt3y12y22+3d2y2dt2dy2dty12y2-d2y2dt2y12y22(c+δ+ϕ)-d2y2dt2y12y2s2-2dy2dt3y12+dy2dt2y12y2(c+δ+ϕ)+2dy2dt2y12s2+dy2dty12y23βp-dy2dty12y22δ(c+ϕ)-dy2dty12y2s2(c+δ+ϕ)+y12y24βdp-y12y23(βps2+cdδ)+y12y22δs2(c+ϕ)-y1y23δϕs1=0.Parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\pi _1,\pi _2\}$$\end{document}{π1,π2} are globally structurally identifiable if:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} f(y_1, y_2, \pi _1, \pi _2)&= f(y_1, y_2, \hat{\pi }_1, \hat{\pi }_2) \text { and }\\ g(y_1, y_2, \pi _1, \pi _2)&= g(y_1, y_2, \hat{\pi }_1, \hat{\pi }_2) \\&\text { imply } \pi _1 = \hat{\pi }_1 \text { and } \pi _2 = \hat{\pi }_2. \end{aligned} \end{aligned}$$\end{document}f(y1,y2,π1,π2)=f(y1,y2,π^1,π^2)andg(y1,y2,π1,π2)=g(y1,y2,π^1,π^2)implyπ1=π^1andπ2=π^2.It is easy to see that:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\{s_1=\hat{s}_1\} \text { , } \{s_2=\hat{s}_2\} \text { , } \{\delta =\hat{\delta }\} \text { , } \{d=\hat{d}\} \text { , } \{c=\hat{c}\} \text { , } \{\phi =\hat{\phi }\} \text { , } \{\beta \times p=\hat{\beta }\times \hat{p}\}.\end{aligned}$$\end{document}{s1=s^1},{s2=s^2},{δ=δ^},{d=d^},{c=c^},{ϕ=ϕ^},{β×p=β^×p^}.Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{s_1, s_2, d, \delta , c, \phi \}$$\end{document}{s1,s2,d,δ,c,ϕ} are globally structurally identifiable and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\beta , p\}$$\end{document}{β,p} are unidentifiable. When initial conditions are known, we have:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\frac{dT_{1}}{dt}(0)=s_1-d T_1(0)-\beta T_1(0) V_1(0),\end{aligned}$$\end{document}dT1dt(0)=s1-dT1(0)-βT1(0)V1(0),making β β (and by default p) globally structurally identifiable. Hence, model Eq. 9 is globally structurally identifiable for known initial conditions. A summary of the results is given in Table 3. □ □

Table 3. Identifiability analysis for model Eq. 9, performed using the DAISY software (Bellu et al. 2007). For the simulations with initial conditions, all initial conditions are known

System Eq. 12 is said to be globally structurally identifiable for parameter vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{q} $$\end{document}q if, for every other parameter vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{q}},$$\end{document}q^,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}y(t,\textbf{q})=y(t,\hat{\textbf{q}}) \text { implies } \textbf{q}=\hat{\textbf{q}};\end{aligned}$$\end{document}y(t,q)=y(t,q^)impliesq=q^;it is locally structurally identifiable for parameter vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{q}$$\end{document}q if, for every other parameter vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{q}}$$\end{document}q^,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}y(t,\textbf{q})=y(t,\hat{\textbf{q}}), \text { and } \hat{\textbf{q}}\in B(\textbf{q})\text { implies } \textbf{q}=\hat{\textbf{q}}, \text { where } B(\textbf{q}) \text { is a ball centered at } \textbf{q};\end{aligned}$$\end{document}y(t,q)=y(t,q^),andq^∈B(q)impliesq=q^,whereB(q)is a ball centered atq;and it is unidentifiable when at least one of its parameters fails the local identifiability test.

The methodology for determining the structural identifiability of systems of ordinary differential equations can range from the Taylor series approach (Pohjanpalo 1978), the differential algebra approach (Bellu et al. 2007; Ljung and Glad 1994), the generating series approach (Walter and Pronzato 1997), the implicit functions approach (Xia and Moog 2003), and several others. Additionally, there are several tools (platforms) that can assist in establishing whether a system of ordinary differential equations is identifiable, such as the COMBOS (Meshkat et al. 2014), the Differential Algebra for Identifiability of SYstems (DAISY, Bellu et al. (2007)), the Exact Arithmetic Rank (EAR, Anguelova et al. (2012)), the GenSSI2 (Ligon et al. 2018), the Structural Identifiability Analyser (SIAN, Hong et al. (2019)), the STRIKE-GOLDD (Villaverde et al. 2016), and the StructuralIdentifiability.jl (Dong et al. 2023). In this study, we used the differential algebra approach and the DAISY platform (Bellu et al. 2007) to determine the structural identifiability of systems Eqs. 3 and 9. The goal of the structural identifiability is to determine which parameters we can confidently estimated from unlimited observations.

Known parameters. We assumed that hepatocytes have a life-span of 100 days (M.-J. Wang et al. 2017), resulting in an uninfected hepatocyte death rate of d=0.01d=0.01 /day. Moreover, we assumed that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$6.8\times 10^{5}$$\end{document}6.8×105 hepatocytes/ml are susceptible to HBV infection (Ciupe et al. 2024), resulting in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(0)=T_1(0)+T_2(0)=6.8\times 10^{5}$$\end{document}T(0)=T1(0)+T2(0)=6.8×105 cells/ml and a total recruitment rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=d \times T(0)= 6.8\times 10^{3}$$\end{document}s=d×T(0)=6.8×103 cells/(ml× × day). HBV is cleared at rate c=4.4c=4.4 /day (Murray et al. 2005). Since we fit the model to data from immunosupressed mice, we assumed δ =d=0.01δ=d=0.01 /day.

We assumed that the HBV inoculum is seeded in patch 1, and set patch 1-specific initial virus and infected cells to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1(0)=10^4$$\end{document}V1(0)=104 HBV DNA/ml and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1(0)=I_0=1$$\end{document}I1(0)=I0=1 cells/ml (Zhang et al. 2023). Moreover, we assumed that there is no virus in patch 2 at the beginning of infection. Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_2(0)=0$$\end{document}I2(0)=0 cells/ml and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_2(0)=0$$\end{document}V2(0)=0 HBV DNA/ml. We split the number of uninfected hepatocytes between the two patches. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1(0)=T_2(0)=3.4\times 10^{5}$$\end{document}T1(0)=T2(0)=3.4×105 hepatocytes/ml, and assumed three cases for hepatocyte recruitment: (case 1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1=0.1\times s=6.8\times 10^2$$\end{document}s1=0.1×s=6.8×102 cells/(ml× × day), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_2=0.9\times s=6.12\times 10^3$$\end{document}s2=0.9×s=6.12×103 cells/(ml× × day); (case 2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1=s_2=0.5\times s =3.4 \times 10^3$$\end{document}s1=s2=0.5×s=3.4×103 cells/(ml× × day); and (case 3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1=0.9\times s=6.12\times 10^3$$\end{document}s1=0.9×s=6.12×103 cells/(ml× × day), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_2=0.1\times s=6.8\times 10^2$$\end{document}s2=0.1×s=6.8×102 cells/(ml× × day).

Data fitting algorithm. The remaining parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi =\{\beta , p, \phi \}$$\end{document}π={β,p,ϕ} are assumed unknown and are estimated by minimizing the functional:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}J(\pi )=\left( \sum _{t_{data}} \left( \log _{10} \left( V_1(t_{data}, \pi )+ V_2(t_{data}, \pi )\right) -\log _{10} V_{data} (t_{data}) \right) ^2\right) ^{1/2},\end{aligned}$$\end{document}J(π)=∑tdatalog10V1(tdata,π)+V2(tdata,π)-log10Vdata(tdata)21/2,over the parameter space π π using the built-in function fminsearchbnd in MATLAB R2021a. Parameter bounds are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-10}\le \beta \le 10^{-7}$$\end{document}10-10≤β≤10-7 ml/(virion× ×day), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.1\le \phi \le 5$$\end{document}0.1≤ϕ≤5 /day, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le p\le 1500$$\end{document}0≤p≤1500 virus/(ml× ×day) for (case 1 - case 3). The initial guesses are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =5\times 10^{-9}$$\end{document}β=5×10-9 ml/(virion× ×day), ϕ =0.5ϕ=0.5/day and p=100p=100 virus/(ml× ×day). The resulting values for model Eq. 3 are given in Table 2 and the dynamics of model Eq. 3 over time are given in Fig. 2. Similarly, the resulting values for model Eq. 9 are given in Table 4 and the dynamics of model Eq. 9 over time are given in Fig. 5.

A model is practically identifiable if a unique parameter set can be consistently obtained through the fitting procedure to noisy data.

There are several approaches and conditions for assessing practical identifiability (Wu et al. 2008; Eisenberg et al. 2013; Roosa and Chowell 2019; Saucedo et al. 2024; Tuncer and Le 2018). In this paper, we used the Monte Carlo (MC) approach to analyze the practical identifiability of the one-directional two-patch model Eq. 3 and two-directional two-patch model Eq. 9. The Monte Carlo approach is a sampling technique that uses random numbers and probability distributions to determine the practical identifiability of a model (Metropolis and Ulam 1949). We performed MC simulations by generating M=1,000M=1,0 data sets using the true parameter set π π and adding noise to the data in increasing amounts. The MC simulations are outlined in the following steps: The objective of this algorithm is to determine if the parameters are sensitive to gradual noise introduced into the given dataset. We used the AREs as a metric to determine the practical identifiability of model Eq. 3 (model Eq. 9, respectively) by applying the definition below (see (Heitzman-Breen et al. 2024) for details).

Let σ σ be the measurement error introduced to a dataset, and let the ARE be the average measurement error in the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^{(k)}$$\end{document}π(k). We state that a model is practically identifiable if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^{(k)}$$\end{document}π(k) is practically identifiable for all values of k.

The results for model Eq. 3 are given in Table 5.

The results for model Eq. 9 are given in Table 6.