Structural and Practical Identifiability Analysis of Zika Epidemiological Models

Abstract

The Zika virus (ZIKV) epidemic has caused an ongoing threat to global health security and spurred new investigations of the virus. Use of epidemiological models for arbovirus diseases can be a powerful tool to assist in prevention and control of the emerging disease. In this article, we introduce six models of ZIKV, beginning with a general vector-borne model and gradually including different transmission routes of ZIKV. These epidemiological models use various combinations of disease transmission (vector and direct) and infectious classes (asymptomatic and pregnant), with addition to loss of immunity being included. The disease-induced death rate is omitted from the models. We test the structural and practical identifiability of the models to find whether unknown model parameters can uniquely be determined. The models were fit to obtain time-series data of cumulative incidences and pregnant infections from the Florida Department of Health Daily Zika Update Reports. The average relative estimation errors (AREs) were computed from the Monte Carlo simulations to further analyze the identifiability of the models. We show that direct transmission rates are not practically identifiable; however, fixed recovery rates improve identifiability overall. We found ARE is low for each model (only slightly higher for those that account for a pregnant class) and help to confirm a reproduction number greater than one at the start of the Florida epidemic. Basic reproduction number, $\mathcal {R}_0$, is an epidemiologically important threshold value which gives the number of secondary cases generated by one infected individual in a totally susceptible population in duration of infectiousness. Elasticity of the reproduction numbers suggests that the mosquito-to-human ratio, mosquito life span and biting rate have the greatest potential for reducing the reproduction number of Zika, and therefore, corresponding control measures need to be focused on.

Appendix

Here, we prove Theorem 1.

Proof

Let $\xi _p = 0$. We assume that $\gamma \ge \gamma _p$. The case $\gamma <\gamma _p$ is similar. We express all variables in terms of $i+i_p$ where $i=I/N$ and $i_p=I_p/N$

$$\begin{aligned} I_v=(i+i_p)\frac{\beta N_v}{\beta (i+i_p)+\mu _v} =: (i+i_p) f_v(i+i_p) N_v. \end{aligned}$$

We solve for $s = S/N$:

$$\begin{aligned} \displaystyle s = \frac{\mu +\frac{\omega \gamma _p}{\mu +\omega }(i+i_p)+\frac{\omega }{\mu +\omega }(\gamma -\gamma _p) i}{\beta _v I_v/N +\beta _d(i+i_p) +\mu +\xi }. \end{aligned}$$

Substituting in the equation for i and solving for i, we have

$$\begin{aligned} i= & {} (i+i_p) \frac{(\beta _v f_v m+\beta _d)(\mu +\frac{\omega \gamma _p}{\mu +\omega }(i+i_p))}{(\mu +\gamma )(\beta _vI_v/N+\beta _d(i+i_p)+\mu +\xi )(1-g(i+i_p))}\\=: & {} (i+i_p)G(i+i_p), \end{aligned}$$

where

$$\begin{aligned} g(i+i_p) = \frac{\frac{\omega }{\mu +\omega }(\gamma -\gamma _p)(\beta _v I_v/N+\beta _d(i+i_p))}{(\beta _v I_v/N+\beta _d(i+i_p)+\mu +\xi )(\mu +\gamma )}. \end{aligned}$$

It is not hard to show that $i>0$. Replacing i from the above expression in s, we can obtain s in terms of $i+i_p$.

$$\begin{aligned} \displaystyle s = \frac{\mu +\frac{\omega \gamma _p}{\mu +\omega }(i+i_p)+\frac{\omega }{\mu +\omega }(\gamma -\gamma _p) (i+i_p)G(i+i_p)}{\beta _v I_v/N +\beta _d(i+i_p) +\mu +\xi }. \end{aligned}$$

We express $s_p$ in terms of $i+i_p$:

$$\begin{aligned} s_p =\frac{\xi s}{\beta _{v_p}I_v/N+\beta _{d_p}(i+i_p) G(i+i_p) +\mu }. \end{aligned}$$

From here, we obtain $i_p$ in terms of $i+i_p$:

$$\begin{aligned} i_p = (i+i_p)\frac{\beta _{v_p}f_vm +\beta _{d_p}G(i+i_p)}{\mu +\gamma _p} s_p =:(i+i_p)\mathcal K(i+i_p). \end{aligned}$$

Substituting in $i+i_p$ and canceling $i+i_p$, we obtain the following equation for $i+i_p$:

$$\begin{aligned} 1 = G(i+i_p)+\mathcal K(i+i_p). \end{aligned}$$

It is not hard to see that if $\omega =0$, the right-hand side of the above equality is a decreasing function of $i+i_p$ that does not depend of $\gamma -\gamma _p$. Hence, in the case $\omega =0$, if the above equation has a solution, it must be unique. In the general case to see that the above equation has a solution, notice that $G(0)+\mathcal K(0)=\mathcal R_{05}>1$. Next, we show that $G(1)+\mathcal K(1)<1$. Some computation shows that

$$\begin{aligned} G(1) \le \frac{(\mu +\gamma _p)(\beta _v f_v m+\beta _d)}{(\mu +\gamma _p)(\beta _v f_v m+\beta _d)+(\mu +\gamma )(\mu +\xi )}. \end{aligned}$$

Furthermore,

$$\begin{aligned} s(1) \le \frac{(\mu +\gamma _p)(\mu +\gamma )}{(\mu +\gamma _p)(\beta _v f_v m+\beta _d)+(\mu +\gamma )(\mu +\xi )}. \end{aligned}$$

This gives the estimate

$$\begin{aligned} \mathcal K(1) \le \frac{\xi s}{\mu +\gamma _p} =\frac{\xi (\mu +\gamma )}{(\mu +\gamma _p)(\beta _v f_v m+\beta _d)+(\mu +\gamma )(\mu +\xi )}. \end{aligned}$$

Hence,

$$\begin{aligned} G(1) +\mathcal K(1) \le \frac{(\mu +\gamma _p)(\beta _v f_v m+\beta _d) +\xi (\mu +\gamma )}{(\mu +\gamma _p)(\beta _v f_v m+\beta _d)+(\mu +\gamma )(\mu +\xi )}<1. \end{aligned}$$

We conclude that in the case $\mathcal R_{05}>1$ there must be at least one solution. $\square$

Now we prove Theorem 2.

Proof

We apply Theorem 4.1 in Castillo-Chavez and Song (2004). We denote by $f_1,\dots , f_6$ the right-hand sides of Eq. (7) with variables ordered the same way as the equations. We use $\beta$ as a bifurcation parameter in place of $\phi$ in Theorem 4.1. It is not hard to show that if $\beta =\beta ^*$ where $\beta ^*$ is the value that makes $\mathcal R_{05}=1$, the Jacobian of the system has a unique zero eigenvalue and all other eigenvalues have negative real parts. We compute the right eigenvector to obtain:

$$\begin{aligned} \begin{array}{l l l } \displaystyle w_6=1 &{} \displaystyle w_3=\frac{\beta _v s^0}{\mu +\gamma } &{} \displaystyle w_4 = \frac{\beta _{v_p} s_p^0}{\mu +\gamma _p}\\ \displaystyle w_5 = \frac{\gamma w_3+\gamma _p w_4}{\mu =\omega } &{}\quad \displaystyle w_1 = \frac{\omega w_5 - \beta _v s^0}{\mu +\xi } &{} \quad \displaystyle w_2 = \frac{\xi w_1 -\beta _{v_p} s_p^0}{\mu }. \end{array} \end{aligned}$$

(24)

Next, we compute the left eigenvector which has the following nonzero components:

$$\begin{aligned} v_6=1\qquad v_3 = \frac{\beta ^* m}{\mu +\gamma } \qquad v_4 = \frac{\beta ^* m}{\mu +\gamma _p}. \end{aligned}$$

(25)

Since only $v_2, v_3, v_6$ are nonzero, we need the partial derivatives of $f_3, f_4$ and $f_6$. They are not hard to compute:

$$\begin{aligned} \frac{\partial ^2f_3}{\partial S\partial I_v} = \frac{\beta _v}{N} \qquad \frac{\partial ^2f_4}{\partial S_p\partial I_v} = \frac{\beta _{v_p}}{N}\qquad \frac{\partial ^2f_6}{\partial I\partial I_v} = -\frac{\beta ^*}{N}\qquad \frac{\partial ^2f_6}{\partial I_p\partial I_v} = -\frac{\beta ^*}{N}. \end{aligned}$$

(26)

In addition, the only right-hand side in (21) that has nonzero derivative with respect to $\beta$ is $f_6$. Hence, we have

$$\begin{aligned} \frac{\partial ^2f_6}{\partial I\partial \beta } = m \qquad \frac{\partial ^2f_6}{\partial I_p\partial \beta } = m. \end{aligned}$$

(27)

From here, $b= v_6 m(w_3+w_4)>0$. Thus, the direction of the bifurcation is controlled by a and is a backward bifurcation if and only if $a>0$ where

$$\begin{aligned} a = 2 \beta ^*/N w_6\left( \frac{\beta _{v} m}{\mu +\gamma } w_1 + \frac{\beta _{v_p} m}{\mu +\gamma _p} w_2 -w_2-w_3\right) , \end{aligned}$$

which gives the condition in the statement of the theorem. This concludes the proof. $\square$