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\begin{document}

\title{\bf \large Supplemental Materials for ``Quantifying Immune
Response to Influenza Virus Infection via Multivariate Nonlinear
ODE Models with Partially Observed State Variables and Time-Varying
Parameters"}

\maketitle

In general, if an estimating equation with state variables and their derivatives at different orders can be obtained after eliminating latent (unobserved) variables
\begin{equation}\label{implicit_func}
g \left( t, x,x',x'',\cdots ,{x^{\left( m-1 \right)}},{x^{\left( m \right)}},\bm{\beta}  \right) =0,
\end{equation}
we can substitute the smoothing estimates of these state variables and their derivatives into the estimating equation to get
$$\hat{\mathbb{F}}(t;\bm{\beta}) = g \left( t, \hat{x},\hat{{x}}',\hat{{x}}'',\cdots ,{{{\hat{x}}}^{\left( m-1 \right)}},{{{\hat{x}}}^{\left( m \right)}}, \bm{\beta}  \right).$$
Then the estimator of $\bm{\beta}$ can be defined as
\begin{eqnarray*}
{{\hat{\bm{\beta} }}_{n}} &=& \arg \underset{\bm{\beta} }{\mathop{\min }}\,\sum\limits_{i=1}^{n}{w({{t}_{i}}){{[\hat{\mathbb{F}}({{t}_{i}};\bm{\beta} )]}^{2}}} \\
 &=& \arg \underset{\bm{\beta} }{\mathop{\min }}\,\sum\limits_{i=1}^{n}{w({{t}_{i}}){{\left[ g \left( {{t}_{i}},\hat{x}\left( {{t}_{i}} \right),\hat{{x}}' \left( {{t}_{i}} \right),\hat{{x}}'' \left( {{t}_{i}} \right),\cdots ,{{{\hat{x}}}^{\left( m-1 \right)}}\left( {{t}_{i}} \right),{{{\hat{x}}}^{\left( m \right)}}\left( {{t}_{i}} \right), \bm{\beta}  \right) \right]}^{2}}}.
\end{eqnarray*}
We can establish the asymptotic properties of the above generalized estimating-equation based estimator. For this purpose, we need some mild assumptions as follows.
\begin{itemize}
\item[(A1)] Let $\kappa=
\max\limits_{0\leq j \leq d} (\tau_{j+1}-\tau_{j})$. There exists a
constant $M>0$, such that $\frac{\kappa}{\min\limits_{0\leq j\leq
d}(\tau_{j+1}-\tau_{j})}$\\ $\leq M$ and $\max\limits_{0\leq j\leq
d}|\frac{\tau_{j+1}-\tau_{j}}{\tau_{j}-\tau_{j-1}}-1|=o(1)$.
\item[(A2)] For fixed design points, $t_i\in [a,b]$ for $i=1,\cdots,n$, assume that there exists a distribution function
$Q(t)$ with a corresponding positive continuous design density $\rho(t)\in C[a,b]$, such that for the empirical distribution of $(t_1,\cdots,t_n)$, $Q_n(t)$, we have
$\sup_t|Q_n(t)-Q(t)|=o(d^{-1})$.
For random design points, $t_i\in [a,b]$ ($i=1,\cdots,n$) are i.i.d. with a cumulative distribution function $Q(t)$ and a positive continuous density $\rho(t)\in C[a,b]$.
Moreover, there exist two constants $0<c_1<c_2<\infty$ such that $c_1\leq \rho(t)\leq c_2$ for all $t\in [a,b]$.
\item[(A3)] The number of knots $d=o(n)$.
\item[(A4)] $X_k(t)\in C^{k+1}[a,b]$ with $k\geq m$.
\item[(A5)] $K=(d+k-1)(\lambda c)^{1/4}n^{-1/4}<1$ for some
constant $c$.
\item[(A6)]The function $g$ in Eq. (10) is continuous for $\beta\in \Omega_{\beta}$, where $\Omega_{\beta}$ is a compact subset of $R^q$ and $q$ is the dimension of $\beta$.
\item[(A7)] The weight function $w(.)$ is bounded and non-negative on $[a,b]$.
\item[(A8)] $E\{w(t)[g(t,x(t),x'(t),x''(t),\cdots,x^{(m-1)}(t),\beta)-g(t,x(t),x'(t),x''(t),\cdots,x^{(m-1)}(t),\beta_0)]^2\}=0$ if and only if $\beta=\beta_0$, where $E[h(t)]$ is the integral
$\int_a^b h(t)d Q(t)$ for function $h(t)$ in the case of fixed design.
\item[(A9)] The first order partial derivatives of $g(t,x_0,\cdots,x_m,\beta)$ with respect to $x_0,\cdots,x_m$ are continuous and uniformly bounded.
\item[(A10)] The first and second order partial derivatives, $\frac{\partial g(t,x_0,\cdots,x_m,\beta)}{\partial\beta}$, $\frac{\partial^2 g(t,x_0,\cdots,x_m,\beta)}{\partial x_0 \partial\beta}$,
$\cdots$, $\frac{\partial^2 g(t,x_0,\cdots,x_m,\beta)}{\partial x_m\partial\beta}$, $\frac{\partial^2 g(t,x_0,\cdots,x_m,\beta)}{\partial\beta\partial\beta^T}$,
 exist and are continuous for all $x_0,\cdots,x_m$ and $\beta$.
\item[(A11)] $w(.)$ has bounded derivatives up to the $m$ order, and $w^{(i)}(a)=w^{(i)}(b)=0$ for $i=0,1,\cdots,m-1$.
\item[(A12)] $\rho(t)\in C^m[a,b]$.
\item[(A13)] $\frac{\partial^m }{\partial t^m}g(t,x_0,\cdots,x_m,\beta)$ exists and is continuous for $t$.
\end{itemize}

\vspace{0.3mm}
\noindent\textbf{Proposition 1}: Under Assumptions A1-A9, if $d \sim cn^{\zeta}$ and $\lambda=O(n^{\gamma})$ with some constant $c$, $\zeta \ge \frac{1}{2k+3}$ and $\gamma \le \frac{k}{2k+3}$, then $\hat{\bm{\beta}}_{n}$ is a strong consistent estimator of $\bm{\beta}_0$.

\noindent\textbf{Proof}: Denote $S_n(\bm{\beta})
=\sum_{i=1}^nw(t_i)[g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm\beta)]^2$.
Then under Assumption A9 in the Appendix A of the paper, by the
mid-value theorem and
$g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta_0)=0$, we have
\begin{eqnarray*}
\frac{1}{n}S_n(\bm{\beta})&=&\frac{1}{n} \sum_{i=1}^n
w(t_i)[g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm\beta)-g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta) \\
& &+g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta)-g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta_0)]^2\\
&=&\frac{1}{n} \sum_{i=1}^n w(t_i)\big\{\frac{\partial
g(t_i,x,x'(t_i),\cdots,x^{(m)}(t_i),\bm\beta)}{\partial
x}|_{x=\tilde{x}_0}[\hat{x}(t_i)-x(t_i)]+\cdots\\
& &+\frac{\partial
g(t_i,x(t_i),\cdots,x^{(m-1)}(t_i),x,\bm\beta)}{\partial
x}|_{x=\tilde{x}_m}[\hat{x}^{(m)}(t_i)-x^{(m)}(t_i)]\\
& &+g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta)-g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta_0)\big\}^2\\
&=&\frac{1}{n} \sum_{i=1}^n
w(t_i)[g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta)-g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta_0)]^2\\
& &+\frac{c_0}{n}\sum_{i=1}^nw(t_i)[\hat{x}(t_i)-x(t_i)]^2+\cdots
 +\frac{c_m}{n}\sum_{i=1}^nw(t_i)[\hat{x}^{(m)}(t_i)-x^{(m)}(t_i)]^2+R,
\end{eqnarray*}
where $\tilde{x}_0$, $\cdots$, $\tilde{x}_m$ lie between
$\hat{x}(t_i)$ and $x(t_i)$, $\cdots$, $\hat{x}^{(m)}(t_i)$ and
$x^{(m)}(t_i)$, respectively, $c_0,\cdots,c_m$ are constants, and
$R$ is a term with a higher rate than other terms and can by
ignored. Under Assumptions A1-A7, if $d\sim cn^{\zeta}$ and
$\lambda=O(n^{\gamma})$ with some constant $c$, $\zeta\geq 1/(2k+3)$
and $\gamma\leq k/(2k+3)$, by the law of large numbers, then it
follows that
$\frac1n\sum_{i=1}^nw(t_i)[\hat{x}(t_i)-x(t_i)]^2\rightarrow
\|w(t)[\hat{x}(t)-x(t)]\|_2^2\rightarrow 0$, $\cdots$ and
$\frac1n\sum_{i=1}^nw(t_i)[\hat{x}^{(m)}(t_i)-x^{(m)}(t_i)]^2\rightarrow
\|w(t)[\hat{x}^{(m)}(t)-x^{(m)}(t)]\|_2^2\rightarrow 0$, a.s.
respectively. So for $\bm\beta\neq\bm\beta_0$, we have
\begin{eqnarray*}
\frac{1}{n}S_n(\bm{\beta})&=&\frac{1}{n} \sum_{i=1}^n
w(t_i)[g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta)-g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm\beta_0)]^2\\
&=&\mbox{E}\big\{w(t)[g(t,x(t),\cdots,x^{(m)}(t),\bm\beta)-g(t,x(t),\cdots,x^{(m)}(t),\bm\beta_0)]^2\big\}.
\end{eqnarray*}
Denote $S(\bm{\beta})=\mbox{E}\big\{w(t)[g(t,x(t),\cdots,x^{(m)}(t),\bm\beta)-g(t,x(t),\cdots,x^{(m)}(t),\bm\beta_0)]^2\big\}$. Then we have
$\lim\limits_{n \rightarrow \infty} \frac{S_n(\bm{\beta})}{n}-S(\bm{\beta})=0$, a.s.

For any probability measure $Q$, let ${\cal F}_n$ be the set $\{\frac{S_n(\bm{\beta})}{n}-S(\bm{\beta}):\bm{\beta}\in\Omega_{\bm\beta}\}$ and
$N_1(\epsilon,Q,{\cal F}_n)$ be the covering number of the class ${\cal F}_n$ in the probability measure $Q$, as given in Pollard(1984,page 25).
Under Assumption A6, using the similar arguments to those in the proof of Theorem 3.1 of Xue, Miao and Wu(2010), we have
$\sup_Q N_1(\epsilon,Q,{\cal F}_n)\leq C(\frac{1}{\epsilon})^q$ for $0<\epsilon<1$, where $C$ is some constant.
Then by Theorem II.37 in Pollard(1984),
$\sup_{\bm\beta}|\frac{S_n(\bm{\beta})}{n}-S(\bm{\beta})|\rightarrow 0$, a.s.,
under $P_{\bm{\beta}_0}$.

Next, from Assumption A8, we know that $\bm{\beta}_0$ is the unique minimum point of $S(\bm{\beta})$. Since $\bm{\beta}_0$ is an interior point of $\Omega_{\beta}$, if follows that the first-order derivative $\frac{\partial S(\bm{\beta})}{\partial\bm{\beta}}$ of $S(\bm{\beta})$ at $\bm{\beta}_0$ equals to zero and the second-order derivative $\frac{\partial^2 S(\bm{\beta})}{\partial\bm{\beta}\partial\bm{\beta}^T}$ of $S(\bm{\beta})$ at $\bm{\beta}_0$ is positive definite. By Assumptions A2 and A10, the second-order derivative of
$S(\bm{\beta})$ in a small neighborhood of $\bm{\beta}_0$ is bounded away from 0 and $\infty$. Then the second-order Taylor expansion of $S(\bm{\beta})$ gives that there exists a constant $0<C<\infty$ such that $|S(\hat{\bm\beta}_n)-S(\bm{\beta}_0)|\geq C\|\hat{\bm\beta}_n-\bm{\beta}_0\|^2$. Moreover,
\begin{eqnarray*}
0\leq S(\hat{\bm\beta}_n)-S(\bm{\beta}_0)&= & S(\hat{\bm\beta}_n)-\frac{S_n(\hat{\bm\beta}_n)}{n}+\frac{S_n(\hat{\bm\beta}_n)}{n}-S(\bm{\beta}_0)\\
&\leq &S(\hat{\bm\beta}_n)-\frac{S_n(\hat{\bm\beta}_n)}{n}+\frac{S_n({\bm\beta}_0)}{n}-S(\bm{\beta}_0)\\
&\leq &2\sup_{\bm\beta}|\frac{S_n(\bm{\beta})}{n}-S(\bm{\beta})|\rightarrow 0, \hbox{ a.s. }
\end{eqnarray*}
Thus $\|\hat{\bm\beta}_n-\bm{\beta}_0\|\rightarrow 0$, a.s., i.e.
$\hat{\bm\beta}_n$ strongly converges to $\bm{\beta}_0$ when $n$ is large. $\blacksquare$

\noindent\textbf{Proposition 2}: Under Assumptions A1-A13, if if $d \sim cn^{\zeta}$ and $\lambda=O(n^{\gamma})$ with some constant $c$, $\zeta \ge \frac{1}{2k+3}$ and $\gamma \le \frac{k}{2k+3}$, then $\sqrt{n} \left( \hat{\bm{\beta}}_n - \bm{\beta}_0 \right) \rightarrow N \left( 0, \Sigma_{\bm{\beta}} \right)$, where $\Sigma_{\bm{\beta}}$ is the covariance matrix defined in the proof below.

\noindent\textbf{Proof}: For $\hat{\bm\beta}_n$, it
satisfies $\frac{\partial
S_n(\hat{\bm\beta}_n)}{\partial\bm\beta}=0$. From the proof of
Proposition 1, it follows that
\begin{eqnarray*}
& &\frac{1}{n}\frac{\partial
S_n(\hat{\bm\beta}_n)}{\partial\bm\beta}\\
&=&\frac{2}{n} \sum_{i=1}^n
w(t_i)g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\hat{\bm\beta}_n)\frac{\partial
g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta}|_{\bm{\beta}=\hat{\bm\beta}_n}\\
&=&\frac{2}{n} \sum_{i=1}^n w(t_i)\frac{\partial
g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta}|_{\bm{\beta}=\hat{\bm\beta}_n}
[g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\hat{\bm\beta}_n)-g(t_i,x(t_i),\cdots,x^{(m)}(t_i),{\bm\beta}_0)]\\
&=&\frac{2}{n} \sum_{i=1}^n w(t_i)\frac{\partial
g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta}|_{\bm{\beta}=\hat{\bm\beta}_n}
\big\{\frac{\partial
g(t_i,x,x'(t_i),\cdots,x^{(m)}(t_i),{\bm\beta}_0)}{\partial
x}|_{x=\tilde{x}_0}[\hat{x}(t_i)-x(t_i)]\\
& &+\cdots+\frac{\partial
g(t_i,x(t_i),\cdots,x^{(m-1)}(t_i),x,{\bm\beta}_0)}{\partial
x}|_{x=\tilde{x}_m}[\hat{x}^{(m)}(t_i)-x^{(m)}(t_i)]\\
& &+\frac{\partial
g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta^T}|_{\bm{\beta}=\tilde{\bm\beta}}
(\hat{\bm\beta}_n-\bm\beta_0)\big\},
\end{eqnarray*}
with $\tilde{x}_0$, $\cdots$, and $\tilde{x}_m$ being some points
between $x(t_i)$ and $\hat{x}(t_i)$, $\cdots$, and $x^{(m)}(t_i)$
and $\hat{x}^{(m)}(t_i)$, respectively, and $\tilde{\bm\beta}$ being
some point between $\bm\beta_0$ and $\hat{\bm\beta}_n$. So we obtain
an asymptotic expression for $\hat{\bm\beta}_n-\bm\beta_0$ as
follows
\begin{eqnarray}\label{asym_expre}
& &\frac{1}{n} \sum_{i=1}^n w(t_i)\frac{\partial
g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta}|_{\bm{\beta}=\hat{\bm\beta}_n}\frac{\partial
g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta^T}|_{\bm{\beta}=\tilde{\bm\beta}}(\hat{\bm\beta}_n-\bm\beta_0)\nonumber\\
&=&-\frac{1}{n} \sum_{i=1}^n w(t_i)\frac{\partial
g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta}|_{\bm{\beta}=\hat{\bm\beta}_n}
\big\{\frac{\partial
g(t_i,x,\cdots,x^{(m)}(t_i),{\bm\beta}_0)}{\partial
x}|_{x=\tilde{x}_0}[\hat{x}(t_i)-x(t_i)]\nonumber\\
& &+\cdots+\frac{\partial
g(t_i,x(t_i),\cdots,x,{\bm\beta}_0)}{\partial
x}|_{x=\tilde{x}_m}[\hat{x}^{(m)}(t_i)-x^{(m)}(t_i)]\big\}.
\end{eqnarray}
Denote $J_*=\mbox{E}\big\{w(t)[\frac{\partial
g(t,x(t),\cdots,x^{(m)}(t),\bm\beta_0)}{\partial\bm\beta_0}]^{\otimes
2}\big\}$, $G_n$ being the right side expression in
(\ref{asym_expre}), and
\begin{eqnarray*}
M_n&=&-\int_a^b w(t)\rho(t)\frac{\partial
g(t,x(t),\cdots,x^{(m)}(t),{\bm\beta}_0)}{\partial\bm\beta_0}
\big\{\frac{\partial
g(t,x,\cdots,x^{(m)}(t),{\bm\beta}_0)}{\partial
x}|_{x=x(t)}[\hat{x}(t)-x(t)]\\
& &+\cdots+\frac{\partial
g(t,x(t),\cdots,x,{\bm\beta}_0)}{\partial
x}|_{x=x^{(m)}(t)}[\hat{x}^{(m)}(t)-x^{(m)}(t)]\big\}dt.
\end{eqnarray*}
Under Assumption A10, from the law of large numbers and Proposition
1, we have
\begin{equation*}
\frac{1}{n} \sum_{i=1}^n w(t_i)\frac{\partial
g(t_i,\hat{x}(t_i),\cdots,\hat{x}^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta}|_{\bm{\beta}=\hat{\bm\beta}_n}\frac{\partial
g(t_i,x(t_i),\cdots,x^{(m)}(t_i),\bm{\beta})}{\partial\bm\beta^T}|_{\bm{\beta}=\tilde{\bm\beta}}\rightarrow
J_*
\end{equation*}
and $G_n-M_n\rightarrow 0$ in probability. So
$\hat{\bm\beta}_n-\bm\beta_0=J_*^{-1}M_n+o_p(n^{-1/2})$.

Denote $A_0(t)=\frac{\partial
g(t,x,x'(t),\cdots,x^{(m)}(t),\bm\beta_0)}{\partial x}|_{x=x(t)}$, $\cdots$,
$A_m(t)=\frac{\partial
g(t,x(t),\cdots,x^{(m-1)}(t),x,\bm\beta_0)}{\partial x}|_{x=x^{(m)}(t)}$, and
$D(t)=\frac{\partial
g(t,x(t),\cdots,x^{(m)}(t),\bm{\beta}_0)}{\partial\bm\beta_0}$.
Continuing to apply integration by parts to $M_n$ under Assumptions A11-A13, we
have $M_n=\int_a^bH(t)[\hat{X}(t)-X(t)]dt$ with
$H(t)=-w(t)\rho(t)D(t)A_0(t)+\cdots
+(-1)^{m+1}\frac{d^m}{dt^m}[w(t)\rho(t)D(t)A_m(t)]$. Then
\begin{equation}\label{main_term}
\hat{\bm\beta}_n-\bm\beta_0=J_*^{-1}\int_a^bH(t)[\hat{X}(t)-X(t)]dt+o_p(n^{-1/2}).
\end{equation}
By the Delta-method(van der Vaart and Wellner, 1996, p.377), the
right side of (\ref{main_term}) is asymptotically normal with mean
as
\begin{equation*}
\mu_1=J_*^{-1}\int_a^b H(t)\mbox{E}[\hat{X}(t)-X(t)]dt,
\end{equation*}
and variance as
\begin{equation*}
\Sigma_{\beta}^*=J_*^{-1}\big\{\int_a^b\int_a^b
H(s)\mbox{Cov}[\hat{X}(s)-X(s),\hat{X}(t)-X(t)]H^T(t)dsdt\big\}
J_*^{-1}.
\end{equation*}
From Theorem 2(a) of Claeskens, Krivobokova and Opsomer(2009) and Lemma 6 of Wu, Xue and Kumar(2011), we have
$\mu_1=O(\kappa^{k+1})=o(n^{-1/2})$ and
\begin{eqnarray*}
\Sigma_{\beta}^*&=&\frac{\sigma^2}{n}J_*^{-1}\Big[\int_a^b\int_a^b
H(s)\bm{B}^T_{k+1}(s)(\bm{B}+\frac{\lambda}{n}\bm{D}_2)^{-1}\\
& &\bm{B}(\bm{B}+\frac{\lambda}{n}\bm{D}_2)^{-1} \bm{B}_{k+1}(t)
H^T(t)dsdt\Big] J_*^{-1}\\
&=&O(1/n)
\end{eqnarray*}
with $\bm{B}=\int_a^b\bm{B}_{k+1}(t)\bm{B}_{k+1}^T(t)\rho(t)dt$.
Thus if $K<1$, $d\sim c n^{\varsigma}$ with $\varsigma\geq
1/(2k+3)$, and $\lambda=O(n^{\gamma})$ with $\gamma\leq k/(2k+3)$,
then
\begin{equation*}
\sqrt{n}(\hat{\bm{\beta}}_n-\bm{\beta}_0)\rightarrow
N(0,\Sigma_{\beta})~~~~~~\mbox{with}~~~~~~\Sigma_{\beta}=n\Sigma_{\beta}^*.
\end{equation*}
Obviously, $\Sigma_{\beta}=O(1)$, since $\Sigma_{\beta}^*=O(1/n)$. $\blacksquare$


\vspace{0.5cm}
\renewenvironment{thebibliography}{\centerline{\large{\sc Additional References}}
\begin{description}}{\end{description}}
\renewcommand{\endthebibliography}{\end{description}}
\begin{thebibliography}
\item
Pollard, D. (1984). {\it Convergence of Stochastic Processes}. Springer-Verlag, New York.

\item
van der Vaart, A. W. and Wellner, J. A. (1996). {\it Weak
Convergence and Empirical Processes}. Springer-Verlag, New York.
\end{thebibliography}

\end{document}
