A Framework for Inferring Unobserved Multistrain Epidemic Subpopulations Using Synchronization Dynamics

Abstract

A new method is proposed to infer unobserved epidemic subpopulations by exploiting the synchronization properties of multistrain epidemic models. A model for dengue fever is driven by simulated data from secondary infective populations. Primary infective populations in the driven system synchronize to the correct values from the driver system. Most hospital cases of dengue are secondary infections, so this method provides a way to deduce unobserved primary infection levels. We derive center manifold equations that relate the driven system to the driver system and thus motivate the use of synchronization to predict unobserved primary infectives. Synchronization stability between primary and secondary infections is demonstrated through numerical measurements of conditional Lyapunov exponents and through time series simulations.

Appendices

Appendix 1: Shifted, Rescaled and Augmented System of Equations

The governing equations for the two-serotype multistrain disease model and the subsystem driven by secondary infectious individuals are given by Eqs. (4a)–(5c). We define a new set of variables, $\bar{s}$, $\bar{x}_i$, $\bar{r}_i$, $\bar{x}_{ij}$, $\bar{s}_d$ and $\bar{x}_{id}$ for all i, j as $\bar{s}(t)=s(t)-s_0$, $\bar{x}_i(t)=x_i(t)-x_{i,0}$, $\bar{r}_i(t)=r(t)-r_{i,0}$, $\bar{x}_{ij}(t)=x_{ij}(t)-x_{ij,0}$, $\bar{s}_d(t)=s_d(t)-s_{d,0}$, $\bar{x}_{id}(t)=x_{id}(t)-x_{id,0}$, and these new variables are substituted into Eqs. (4a)–(5c).

Then, treating $\mu$ as a small parameter, we rescale time by letting $t=\mu \tau$. We may then introduce the following rescaled parameters: $\beta =\beta _0/\mu$ and $\sigma =\sigma _0/\mu$, where $\beta _0$ and $\sigma _0$ are $\mathcal {O}(1)$. The inclusion of the parameter $\mu$ as a new state variable means that the terms in our rescaled system which contain $\mu$ are now nonlinear terms. Furthermore, the system is augmented with the auxiliary equation $\frac{d\mu }{d\tau }=0$. The addition of this auxiliary equation contributes an extra simple zero eigenvalue to the system and adds one new center direction that has trivial dynamics. The shifted, rescaled, and augmented system of equations is given as

$$\begin{aligned} \frac{d\bar{s}}{d\tau }= & {} \mu ^2 - \beta _0 \left( \bar{s}+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_1+\bar{x}_2+\phi \left( \bar{x}_{21}+\bar{x}_{12}\right) +\frac{\mu ^2(1+\phi )}{\sigma _0}\right) , \nonumber \\ \end{aligned}$$

(21a)

$$\begin{aligned} \frac{d\bar{x}_1}{d\tau }= & {} \beta _0\left( \bar{s}+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_1+\phi \bar{x}_{21}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) -\sigma _0\left( \bar{x}_1+\frac{\mu ^2}{2\sigma _0}\right) , \nonumber \\\end{aligned}$$

(21b)

$$\begin{aligned} \frac{d\bar{x}_2}{d\tau }= & {} \beta _0\left( \bar{s}+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_2+\phi \bar{x}_{12}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) -\sigma _0\left( \bar{x}_2+\frac{\mu ^2}{2\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21c)

$$\begin{aligned} \frac{d\bar{r}_1}{d\tau }= & {} \sigma _0\left( \bar{x}_1+\frac{\mu ^2}{2\sigma _0}\right) -\beta _0\left( \bar{r}_1+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_2+\phi \bar{x}_{12}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21d)

$$\begin{aligned} \frac{d\bar{r}_2}{d\tau }= & {} \sigma _0\left( \bar{x}_2+\frac{\mu ^2}{2\sigma _0}\right) -\beta _0\left( \bar{r}_2+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_1+\phi \bar{x}_{21}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21e)

$$\begin{aligned} \frac{d\bar{x}_{21}}{d\tau }= & {} \beta _0\left( \bar{r}_2+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_1+\phi \bar{x}_{21}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) -\sigma _0\left( \bar{x}_{21}+\frac{\mu ^2}{2\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21f)

$$\begin{aligned} \frac{d\bar{x}_{12}}{d\tau }= & {} \beta _0\left( \bar{r}_1+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_2+\phi \bar{x}_{12}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) -\sigma _0\left( \bar{x}_{12}+\frac{\mu ^2}{2\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21g)

$$\begin{aligned} \frac{d\bar{s}_d}{d\tau }= & {} \mu ^2-\beta _0 \left( \bar{s}_d +\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_{1d}+\bar{x}_{2d}+\phi \left( \bar{x}_{21}+\bar{x}_{12}\right) +\frac{\mu ^2(1+\phi )}{\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21h)

$$\begin{aligned} \frac{d\bar{x}_{1d}}{d\tau }= & {} \beta _0\left( \bar{s}_d+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_{1d}+\phi \bar{x}_{21}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) -\sigma _0\left( \bar{x}_{1d}+\frac{\mu ^2}{2\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21i)

$$\begin{aligned} \frac{d\bar{x}_{2d}}{d\tau }= & {} \beta _0\left( \bar{s}_d+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( \bar{x}_{2d}+\phi \bar{x}_{12}+\frac{\mu ^2(1+\phi )}{2\sigma _0}\right) -\sigma _0\left( \bar{x}_{2d}+\frac{\mu ^2}{2\sigma _0}\right) ,\nonumber \\ \end{aligned}$$

(21j)

$$\begin{aligned} \frac{d\mu }{d\tau }= & {} 0, \end{aligned}$$

(21k)

where the endemic fixed point is now located at the origin.

Appendix 2: Definition of New Variables

Using the fact that $(\bar{s},\bar{x}_1,\bar{x}_2,\bar{r}_1,\bar{r}_2,\bar{x}_{21},\bar{x}_{12},\bar{s}_d,\bar{x}_{1d},\bar{x}_{2d})^T = \mathbf{P}\cdot \mathbf{W}^T$, where $\mathbf{P}$ is given by Eq. (7) and $\mathbf{W}=(W_1,W_2,W_3,W_4,W_5,W_6,W_7,W_8,W_9,W_{10})$, then the transformation matrix leads to the following definition of new variables, $W_i$, $i=1\ldots 10$:

$$\begin{aligned} W_1= & {} \frac{\bar{x}_{21}-\bar{x}_1}{1+\phi }, \quad W_2 = \frac{\bar{x}_{12}-\bar{x}_2}{1+\phi }, \quad W_3 = \frac{\bar{x}_{1d}+\bar{x}_{2d}-\bar{x}_1-\bar{x}_2}{\phi }, \quad W_4 = \bar{x}_{2d}-\bar{x}_2, \nonumber \\ W_5= & {} \bar{s}, \quad W_6 =\frac{\bar{x}_{1}+\phi \bar{x}_{21}}{1+\phi }, \quad W_7 = \frac{\bar{x}_{2}+\phi \bar{x}_{12}}{1+\phi }, \quad W_8=\frac{\phi \bar{r}_{1}+\phi \bar{x}_{1}+\bar{r}_1-\phi \bar{x}_{21}}{1+\phi },\nonumber \\ W_9= & {} \frac{\phi \bar{r}_{2}+\phi \bar{x}_{2}+\bar{r}_2-\phi \bar{x}_{12}}{1+\phi },\quad W_{10} = \frac{\phi \bar{s}_{d}-\bar{x}_{1d}-\bar{x}_{2d}+\bar{x}_{1}+\bar{x}_2}{\phi }. \end{aligned}$$

(22)

Appendix 3: Transformed Evolution Equations

The application of the transformation matrix $\mathbf{P}$ given by Eqs. (7) to (21a)–(21j) leads to the following set of transformed evolution equations:

$$\begin{aligned} \frac{dW_1}{d\tau }= & {} \beta _0\left( W_6+\frac{\mu ^2}{2\sigma _0}\right) \left( W_9+\phi W_2-W_5\right) -\sigma _0 W_1 , \end{aligned}$$

(23a)

$$\begin{aligned} \frac{dW_2}{d\tau }= & {} \beta _0\left( W_7+\frac{\mu ^2}{2\sigma _0}\right) \left( W_8+\phi W_1-W_5\right) -\sigma _0 W_2 , \end{aligned}$$

(23b)

$$\begin{aligned} \frac{dW_3}{d\tau }= & {} W_3\left( \beta _0\left( W_3+W_{10}\right) -\frac{\sigma _0\phi }{1+\phi }\right) \nonumber \\&+\,\frac{\beta _0\left( 1+\phi \right) }{\phi }\left( W_6+W_7+\frac{\mu ^2}{\sigma _0}\right) \left( W_3+W_{10}-W_5\right) , \end{aligned}$$

(23c)

$$\begin{aligned} \frac{dW_4}{d\tau }= & {} \left( \beta _0\left( W_3+W_{10}\right) -\frac{\sigma _0\phi }{1+\phi }\right) W_4 \nonumber \\&+\,\beta _0\left( W_3+W_{10}-W_5\right) \left( \left( 1+\phi \right) W_7+\frac{\mu ^2\left( 1+\phi \right) }{2\sigma _0}\right) , \end{aligned}$$

(23d)

$$\begin{aligned} \frac{dW_5}{d\tau }= & {} \mu ^2-\beta _0\left( W_5+\frac{\sigma _0}{\beta _0 (1+\phi )}\right) \left( (1+\phi )\left( W_6+W_7+\frac{\mu ^2}{\sigma _0}\right) \right) , \end{aligned}$$

(23e)

$$\begin{aligned} \frac{dW_6}{d\tau }= & {} \beta _0\left( W_6+\frac{\mu ^2}{2\sigma _0}\right) \left( \phi ^2 W_2+\phi W_9+W_5\right) , \end{aligned}$$

(23f)

$$\begin{aligned} \frac{dW_7}{d\tau }= & {} \beta _0\left( W_7+\frac{\mu ^2}{2\sigma _0}\right) \left( \phi ^2 W_1+\phi W_8+W_5\right) , \end{aligned}$$

(23g)

$$\begin{aligned} \frac{dW_8}{d\tau }= & {} \left( W_6+\frac{\mu ^2}{2\sigma _0}\right) \left( \sigma _0+\beta _0\left( \phi W_5 -\phi W_9 -\phi ^2 W_2 \right) \right) \nonumber \\&-\,\left( 1+\phi \right) \beta _0\left( W_7 + \frac{\mu ^2}{2\sigma _0} \right) \left( \phi W_1 + W_8 + \frac{\sigma _0}{\beta _0\left( 1+\phi \right) } \right) , \end{aligned}$$

(23h)

$$\begin{aligned} \frac{dW_9}{d\tau }= & {} \left( W_7+\frac{\mu ^2}{2\sigma _0}\right) \left( \sigma _0+\beta _0\left( \phi W_5 -\phi W_8 -\phi ^2 W_1 \right) \right) \nonumber \\&-\,\left( 1+\phi \right) \beta _0\left( W_6 + \frac{\mu ^2}{2\sigma _0} \right) \left( \phi W_2 + W_9 + \frac{\sigma _0}{\beta _0\left( 1+\phi \right) } \right) , \end{aligned}$$

(23i)

$$\begin{aligned} \frac{dW_{10}}{d\tau }= & {} \mu ^2-\left( 1+\phi \right) \beta _0 W_3\left( W_3+W_{10}\right) \nonumber \\&+\, \frac{\left( 1+\phi \right) \beta _0}{\phi }\left( W_6+W_7+\frac{\mu ^2}{\sigma _0}\right) \nonumber \\&\left( W_5-\left( 1+\phi \right) \left( W_3+W_{10}\right) -\frac{\phi \sigma _0}{\beta _0\left( 1+\phi \right) }\right) , \end{aligned}$$

(23j)

$$\begin{aligned} \frac{d\mu }{d\tau }= & {} 0. \end{aligned}$$

(23k)

Appendix 4: Center Manifold Condition

Substitution of the center manifold functions $W_i=h_i$ given by Eq. (12) into the transformed evolution equations given in “Appendix 3” leads to the following center manifold condition:

$$\begin{aligned}&\frac{\partial h_1}{\partial W_5}\frac{dW_5}{d\tau } + \frac{\partial h_1}{\partial W_6}\frac{dW_6}{d\tau } + \frac{\partial h_1}{\partial W_7}\frac{dW_7}{d\tau } + \frac{\partial h_1}{\partial W_8}\frac{dW_8}{d\tau } + \frac{\partial h_1}{\partial W_9}\frac{dW_9}{d\tau } + \frac{\partial h_1}{\partial W_{10}}\frac{dW_{10}}{d\tau } \nonumber \\&\quad =\beta _0\left( W_6+\frac{\mu ^2}{2\sigma _0}\right) \left( W_9+\phi h_2-W_5\right) -\sigma _0 h_1, \end{aligned}$$

(24a)

$$\begin{aligned}&\frac{\partial h_2}{\partial W_5}\frac{dW_5}{d\tau } + \frac{\partial h_2}{\partial W_6}\frac{dW_6}{d\tau } + \frac{\partial h_2}{\partial W_7}\frac{dW_7}{d\tau } + \frac{\partial h_2}{\partial W_8}\frac{dW_8}{d\tau } + \frac{\partial h_2}{\partial W_9}\frac{dW_9}{d\tau } + \frac{\partial h_2}{\partial W_{10}}\frac{dW_{10}}{d\tau } \nonumber \\&\quad =\beta _0\left( W_7+\frac{\mu ^2}{2\sigma _0}\right) \left( W_8+\phi h_1-W_5\right) -\sigma _0 h_2, \end{aligned}$$

(24b)

$$\begin{aligned}&\frac{\partial h_3}{\partial W_5}\frac{dW_5}{d\tau } + \frac{\partial h_3}{\partial W_6}\frac{dW_6}{d\tau } + \frac{\partial h_3}{\partial W_7}\frac{dW_7}{d\tau } + \frac{\partial h_3}{\partial W_8}\frac{dW_8}{d\tau } + \frac{\partial h_3}{\partial W_9}\frac{dW_9}{d\tau } + \frac{\partial h_3}{\partial W_{10}}\frac{dW_{10}}{d\tau } \nonumber \\&\quad =h_3\left( \beta _0\left( h_3+W_{10}\right) -\frac{\sigma _0\phi }{1+\phi }\right) \nonumber \\&\quad \quad +\,\frac{\beta _0\left( 1+\phi \right) }{\phi }\left( W_6+W_7+\frac{\mu ^2}{\sigma _0}\right) \left( h_3+W_{10}-W_5\right) , \end{aligned}$$

(24c)

$$\begin{aligned}&\frac{\partial h_4}{\partial W_5}\frac{dW_5}{d\tau } + \frac{\partial h_4}{\partial W_6}\frac{dW_6}{d\tau } + \frac{\partial h_4}{\partial W_7}\frac{dW_7}{d\tau } + \frac{\partial h_4}{\partial W_8}\frac{dW_8}{d\tau } + \frac{\partial h_4}{\partial W_9}\frac{dW_9}{d\tau } + \frac{\partial h_4}{\partial W_{10}}\frac{dW_{10}}{d\tau } \nonumber \\&\quad =\left( \beta _0\left( h_3+W_{10}\right) -\frac{\sigma _0\phi }{1+\phi }\right) h_4 \nonumber \\&\quad \quad +\,\beta _0\left( h_3+W_{10}-W_5\right) \left( \left( 1+\phi \right) W_7+\frac{\mu ^2\left( 1+\phi \right) }{2\sigma _0}\right) . \end{aligned}$$

(24d)