Structural identifiability analysis of epidemic models based on differential equations: a tutorial-based primer

Abstract

The successful application of epidemic models hinges on our ability to estimate model parameters from limited observations reliably. An often-overlooked step before estimating model parameters consists of ensuring that the model parameters are structurally identifiable from the observed states of the system. In this tutorial-based primer, intended for a diverse audience, including students training in dynamic systems, we review and provide detailed guidance for conducting structural identifiability analysis of differential equation epidemic models based on a differential algebra approach using differential algebra for identifiability of systems (DAISY) and Mathematica (Wolfram Research). This approach aims to uncover any existing parameter correlations that preclude their estimation from the observed variables. We demonstrate this approach through examples, including tutorial videos of compartmental epidemic models previously employed to study transmission dynamics and control. We show that the lack of structural identifiability may be remedied by incorporating additional observations from different model states, assuming that the system’s initial conditions are known, using prior information to fix some parameters involved in parameter correlations, or modifying the model based on existing parameter correlations. We also underscore how the results of structural identifiability analysis can help enrich compartmental diagrams of differential-equation models by indicating the observed state variables and the results of the structural identifiability analysis.

Appendix

1.1 Input–output equations of epidemic models

$\underline{\hbox {Input-output equation of the model }{\hbox {M}}_2}$

$$\begin{aligned}{} & {} (y_1^{'''})^2 y_1^{'} y_1 + (y_1^{'''})^2 y_1^2 k - y_1^{'''} (y_1^{''})^2 y_1 - y_1^{'''} y_1^{''} (y_1^{'})^2 +y_1^{'''} y_1^{''} y_1^{'} y_1 (2 \gamma - k) \nonumber \\{} & {} \quad +2 y_1^{'''} y_1^{''} y_1^2 k ( \gamma + k) - y_1^{'''} (y_1^{'})^3 ( \gamma + k) + 4 y_1^{'''} (y_1^{'})^2 y_1^2 \frac{\beta }{N} + y_1^{'''} (y_1^{'})^2 y_1 ( \gamma ^2 - 2 k^2) \nonumber \\{} & {} \quad + 8 y_1^{'''} y_1^{'} y_1^3 \frac{\beta k}{N} + y_1^{'''} y_1^{'} y_1^2 \gamma k (2 \gamma + k) + 4 y_1^{'''} y_1^4 \frac{\beta k^2}{N} \nonumber \\{} & {} \quad + y_1^{'''} y_1^3 \gamma ^2 k^2 + (y_1^{''})^3 y_1^{'} - (y_1^{''})^3 y_1 ( \gamma + k) \nonumber \\{} & {} \quad + 2 (y_1^{''})^2 (y_1^{'})^2 k - 3 (y_1^{''})^2 y_1^{'} y_1^2 \frac{\beta }{N} + (y_1^{''})^2 y_1^{'} y_1 ( \gamma ^2 - \gamma k - 2 k^2) - 3 (y_1^{''})^2 y_1^3 \frac{\beta k}{N} \nonumber \\{} & {} \quad + (y_1^{''})^2 y_1^2 k ( \gamma ^2 + 2 \gamma k + k^2) + y_1^{''} (y_1^{'})^3 ( - 2 \gamma ^2 - \gamma k + 2 k^2) + 2 y_1^{''} (y_1^{'})^2 y_1^2 \frac{\beta (3 \gamma - k)}{N} \nonumber \\{} & {} \quad + y_1^{''} (y_1^{'})^2 y_1 \frac{( \gamma ^3 - \gamma ^2 k - 3 \gamma k^2 - 2 k^3)}{N^2} + 2 y_1^{''} y_1^{'} y_1^3 \frac{\beta k (6 \gamma + k)}{N} \nonumber \\{} & {} \quad + y_1^{''} y_1^{'} y_1^2 \frac{\gamma k (2 \gamma ^2 + 2 \gamma k + k^2)}{N^2}+ 2 y_1^{''} y_1^4 \frac{\beta k^2 (3 \gamma + 2 k)}{N} \nonumber \\{} & {} \quad + y_1^{''} y_1^3 \gamma ^2 k^2 ( \gamma + k) - (y_1^{'})^5 \frac{\beta }{N} + (y_1^{'})^4 y_1 \frac{\beta ( - 4 \gamma - 3 k)}{N} \nonumber \\{} & {} \quad + (y_1^{'})^4 ( - \gamma ^3 - 2 \gamma ^2 k + k^3) + 4 (y_1^{'})^3 y_1^3 \frac{\beta ^2}{N^2} + (y_1^{'})^3 y_1^2 \frac{\beta ( \gamma ^2 - 6 \gamma k - 6 k^2)}{N} \nonumber \\{} & {} \quad + (y_1^{'})^3 y_1 \gamma k ( - 2 \gamma ^2 - 3 \gamma k - k^2) + 12 (y_1^{'})^2 y_1^4 \frac{\beta ^2 k}{N^2} + (y_1^{'})^2 y_1^3 \frac{\beta k (3 \gamma ^2 - 4 k^2)}{N} \nonumber \\{} & {} \quad -(y_1^{'})^2 y_1^2 \gamma ^2 k^2 ( \gamma + k) + 12 y_1^{'} y_1^5 \frac{\beta ^2 k^2}{N^2} \nonumber \\{} & {} \quad +y_1^{'} y_1^4 \frac{\beta \gamma k^2 (3 \gamma + 2 k)}{N} + 4 y_1^6 \frac{\beta ^2 k^3}{N^2} + y_1^5 \frac{\beta \gamma ^2 k^3}{N}=0 \end{aligned}$$

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$\underline{\hbox {Input-output equation of the model }{\hbox {M}}_3}$

$$\begin{aligned}{} & {} (y_1^{'''})^2 y_1^{'} y_1 + (y_1^{'''})^2 y_1^2 k - y_1^{'''} (y_1^{''})^2 y_1 - y_1^{'''} y_1^{''} (y_1^{'})^2 + y_1^{'''} y_1^{''} y_1^{'} y_1 (2 \gamma - k) \nonumber \\{} & {} \quad +2 y_1^{'''} y_1^{''} y_1^2 k (\gamma + k) - y_1^{'''} (y_1^{'})^3 (\gamma + k) + 4 y_1^{'''} (y_1^{'})^2 y_1^2 \frac{( - \beta _A \rho + \beta _A + \beta _I \rho )}{N \rho } \nonumber \\{} & {} \quad +y_1^{'''} (y_1^{'})^2 y_1 (\gamma ^2 - 2 k^2) +8 y_1^{'''} y_1^{'} y_1^3 \frac{k ( - \beta _A \rho + \beta _A + \beta _I \rho )}{N \rho }+ y_1^{'''} y_1^{'} y_1^2 \gamma k (2 \gamma + k) \nonumber \\{} & {} \quad +4 y_1^{'''} y_1^4 \frac{k^2 ( - \beta _A \rho + \beta _A + \beta _I \rho )}{N \rho } +y_1^{'''} y_1^3 \gamma ^2 k^2 +(y_1^{''})^3 y_1^{'} \nonumber \\{} & {} \quad - (y_1^{''})^3 y_1 (\gamma + k) + 2 (y_1^{''})^2 (y_1^{'})^2 k \nonumber \\{} & {} \quad +3 (y_1^{''})^2 y_1^{'} y_1^2 \frac{(\beta _A \rho - \beta _A - \beta _I \rho )}{N \rho } +(y_1^{''})^2 y_1^{'} y_1 (\gamma ^2 - \gamma k - 2 k^2) \nonumber \\{} & {} \quad +3 (y_1^{''})^2 y_1^3 \frac{k (\beta _A \rho - \beta _A - \beta _I \rho )}{ N \rho } \nonumber \\{} & {} \quad +(y_1^{''})^2 y_1^2 k (\gamma ^2 + 2 \gamma k + k^2) + y_1^{''} (y_1^{'})^3( - 2 \gamma ^2 - \gamma k + 2 k^2) \end{aligned}$$

$$\begin{aligned}{} & {} \quad +2 y_1^{''} (y_1^{'})^2 y_1^2 \frac{( - 3 \beta _A \gamma \rho + 3 \beta _A \gamma + \beta _A k \rho - \beta _A k + 3 \beta _I \gamma \rho - \beta _I k \rho )}{N \rho } \nonumber \\{} & {} \quad +y_1^{''} (y_1^{'})^2 y_1 (\gamma ^3 - \gamma ^2 k - 3 \gamma k^2 - 2 k^3)\nonumber \\{} & {} \quad +2 y_1^{''} y_1^{'} y_1^3 \frac{k ( - 6 \beta _A \gamma \rho + 6 \beta _A \gamma - \beta _A k \rho + \beta _A k + 6 \beta _I \gamma \rho + \beta _I k \rho )}{N \rho } \nonumber \\{} & {} \quad +y_1^{''} y_1^{'} y_1^2 \gamma k (2 \gamma ^2 + 2 \gamma k + k^2) \nonumber \\{} & {} \quad + 2 y_1^{''} y_1^4 \frac{k^2 ( - 3 \beta _A \gamma \rho + 3 \beta _A \gamma - 2 \beta _A k \rho + 2 \beta _A k + 3 \beta _I \gamma \rho + 2 \beta _I k \rho )}{N \rho } \nonumber \\{} & {} \quad +y_1^{''} y_1^3 \gamma ^2 k^2(\gamma + k) + (y_1^{'})^5 \frac{ (\beta _A \rho - \beta _A - \beta _I \rho )}{N \rho } \nonumber \\{} & {} \quad +(y_1^{'})^4 y_1 \frac{ (4 \beta _A \gamma \rho - 4 \beta _A \gamma + 3 \beta _A k \rho - 3 \beta _A k - 4 \beta _I \gamma \rho - 3 \beta _I k \rho )}{N \rho } \nonumber \\{} & {} \quad +( y_1^{'})^4 ( - \gamma ^3 - 2 \gamma ^2 k + k^3) \nonumber \\{} & {} \quad + 4 (y_1^{'})^3 y_1^3 \frac{(\beta _A^2 \rho ^2 - 2 \beta _A^2 \rho + \beta _A^2 - 2 \beta _A \beta _I \rho ^2 + 2 \beta _A \beta _I \rho + \beta _I^2 \rho ^2)}{N^2 \rho ^2} \nonumber \\{} & {} \quad +(y_1^{'})^3 y_1^2 \frac{ ( - \beta _A \gamma ^2 \rho + \beta _A \gamma ^2 + 6 \beta _A \gamma k \rho - 6 \beta _A \gamma k + 6 \beta _A k^2 \rho - 6 \beta _A k^2 + \beta _I \gamma ^2 \rho - 6 \beta _I \gamma k \rho - 6 \beta _I k^2 \rho )}{N \rho } \nonumber \\{} & {} \quad +(y_1^{'})^3 y_1 \gamma k ( - 2 \gamma ^2 - 3 \gamma k - k^2) \nonumber \\{} & {} \quad + 12 (y_1^{'})^2 y_1^4 \frac{k (\beta _A^2 \rho ^2 - 2 \beta _A^2 \rho + \beta _A^2 - 2 \beta _A \beta _I \rho ^2 + 2 \beta _A \beta _I \rho + \beta _I^2 \rho ^2)}{N^2 \rho ^2} \nonumber \\{} & {} \quad +(y_1^{'})^2 y_1^3 \frac{k ( - 3 \beta _A \gamma ^2 \rho + 3 \beta _A \gamma ^2 + 4 \beta _A k^2 \rho - 4 \beta _A k^2 + 3 \beta _I \gamma ^2 \rho - 4 \beta _I k^2 \rho )}{N \rho }\nonumber \\{} & {} \quad - (y_1^{'})^2 y_1^2 \gamma ^2 k^2 (\gamma + k) \nonumber \\{} & {} \quad +12 y_1^{'} y_1^5 \frac{k^2 (\beta _A^2 \rho ^2 - 2 \beta _A^2 \rho + \beta _A^2 - 2 \beta _A \beta _I \rho ^2 + 2 \beta _A \beta _I \rho + \beta _I^2 \rho ^2)}{N^2 \rho ^2} \nonumber \\{} & {} \quad +y_1^{'} y_1^4 \frac{ \gamma k^2 ( - 3 \beta _A \gamma \rho + 3 \beta _A \gamma - 2 \beta _A k \rho + 2 \beta _A k + 3 \beta _I \gamma \rho + 2 \beta _I k \rho )}{N \rho } \nonumber \\{} & {} \quad +4 y_1^6 \frac{ k^3 (\beta _A^2 \rho ^2 - 2 \beta _A^2 \rho + \beta _A^2 - 2 \beta _A \beta _I \rho ^2 + 2 \beta _A \beta _I \rho + \beta _I^2 \rho ^2)}{N^2 \rho ^2} \nonumber \\{} & {} \quad + y_1^5 \frac{ \gamma ^2 k^3 ( - \beta _A \rho + \beta _A + \beta _I \rho )}{N \rho } \end{aligned}$$

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