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\AtBeginDocument{{\noindent\small
Third International Conference on Applications of Mathematics to Nonlinear Sciences,\newline
\emph{Electronic Journal of Differential Equations},
Conference 27 (2024), pp. 1--12.\newline
ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu \newline
DOI: 10.58997/ejde.conf.27.g1}
\thanks{\copyright 2024 This work is licensed under a CC BY 4.0 license.}
\vspace{8mm}}

\begin{document} \setcounter{page}{1}
\title[\hfilneg EJDE-2024/conf/27\hfil  Clinical trial modeling]
{Respiratory Illness clinical trial modeling}

\author[D. Gerberry, H. Joshi, M. Peloquin, S. Vargas \hfil EJDE-2022/conf/26\hfilneg]
{David Gerberry, Hem Joshi,  Mac Peloquin, Sonia Vargas}

\address{David Gerberry \newline
Department of Mathematics,
Xavier University,
Cincinnati, OH 45207, USA}
\email{david.gerberry@xavier.edu}

\address{Hem Joshi \newline
Department of Mathematics,
Xavier University,
Cincinnati, OH 45207, USA}
\email{joshi@xavier.edu}

\address{Mac Peloquin \newline
Department of Mathematics, 
Xavier University,
Cincinnati, OH 45207, USA}
\email{peloquinj@xavier.edu}

\address{Sonia Vargas \newline
Department of Mathematics, 
Xavier University,
Cincinnati, OH 45207, USA}
\email{vargass2@xavier.edu}

\thanks{Published August 20, 2024}
\subjclass[2020]{34-11, 00A71, 92-10}
\keywords{Modeling; clinical trials; vaccine efficacy; reproductive number}

\begin{abstract}
Respiratory illnesses pose a significant burden on public health, 
causing numerous fatalities annually. Using the classical SEIR framework, 
we model a general respiratory infection to investigate the interaction 
between the disease's epidemiological dynamics and the dynamics of a 
vaccine clinical trial. We develop a system to describe clinical trials 
for a vaccine targeting a new respiratory infection. 
We model the interplay between epidemiological dynamics and clinical trials,
establishing the relationship between these two systems.

We also explore the impact of vaccines becoming available to adults and 
children at different times by dividing each epidemiological compartment into 
two age-based categories: adults and children. We calculate the basic 
reproductive number for the epidemiological model for each transmission 
category (adults and children) and then determine an overall reproductive 
number for the entire population. Additionally, we conduct numerical 
simulations under various assumptions and present our findings.
\end{abstract}


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\maketitle

\section{Introduction}\label{S:1}

Mathematical modeling is a powerful tool used for understanding complex 
systems including the spread of infectious diseases. 
Doing so involves creating mathematical equations and simulations that 
capture the dynamics of disease transmission and their impact on populations. 
By incorporating factors such as population size, contact rates, disease 
parameters, and intervention strategies, mathematical modeling can 
investigate potential scenarios to help inform public health decisions. 
These models can estimate the future course of an outbreak, evaluate the 
effectiveness of various interventions, and assess the potential impact of 
policy measures. They allow researchers and policymakers to explore 
different strategies and make evidence-based decisions to control and 
mitigate the spread of infectious diseases, ultimately helping to save 
lives and reduce the burden on health care systems~\cite{NIH3}.  
 
Respiratory illnesses are a significant burden to modern life and a large 
cause of human death and suffering each year.  The COVID-19 pandemic proved 
that these types of illnesses can change the way a society functions. 
To stop the spread and deaths caused by these diseases, efforts to quickly 
design vaccines for these respiratory illnesses are paramount. 
As shown during the response to the COVID-19 pandemic, the emergence of mRNA 
technology allows for vaccines to be formulated much faster than ever 
before \cite{ Seattle, Li, Racaniello, Tsang, NIH4}. With the new found 
ability to develop vaccines so quickly, the time it takes to test new 
vaccines (i.e.\ clinical trials) represents the major limitation to the 
speed with which society can begin to immunize its population and ultimately 
change the course of the next pandemic \cite{Pfizer, FDA, Maryland, Pfizer2}.

Of course, it is important to note that vaccine clinical trials do not occur 
in a vacuum but rather involve individuals (both in the control and treatment 
arms of the clinical trial) interacting with the general population under the 
current conditions of the epidemic. Therefore, the conditions of the epidemic 
will always impact the clinical trial. For example, a clinical trial will 
require a longer period to demonstrate a statistically significant protective 
effect if conditions are such that disease incidence is very low in the 
general population. If a clinical trial is conducted while disease incidence 
is very high, a shorter period of time will be needed to establish protection. 
Complicating matters further, vaccination is typically not tested and rolled 
out for the entire population at the same time. With COVID vaccines for 
example, clinical trials for adults were conducted first followed by children. 
Therefore, the clinical trial for vaccines in children occurred at the same
time that vaccination was available to adults in the general population 
\cite{Piechotta, Thompson}.

In this modeling work,  our goal is to examine the interaction between the 
epidemiological dynamics of a disease and the dynamics of a vaccine clinical 
trial. While motivated by the COVID-19 pandemic, our work generalizes to study 
a pandemic of a general respiratory disease; as respiratory infections 
(other corona viruses, avian influenza, H1N5, SARS, etc.) share many 
characteristics and all have the potential for causing future pandemics.


\section{Epidemiological model}\label{epimodel}

We use the classical SEIR framework to model a general respiratory infection, 
where individuals progress from being susceptible ($S$),  exposed ($E$) 
in which individuals have contracted the disease but are not yet able to 
transmit it, infectious ($I$), and recovered ($R$).   
For simplicity, we assume permanent immunity after recovery.
As we look to examine the effect of vaccines being available to adults 
and children at different times, we separate each epidemiological 
compartment into two classes based on age ($A$ for adults, $K$ for 
children or ``kids''). Therefore, the model consists of the state variables 
$S_A, E_A, I_A,$ and  $R_A$ for susceptible, exposed, infectious, and 
recovered adults; $S_K, E_K, I_K,$ and $ R_K$ for susceptible, exposed,
infectious, and recovered kids; and is given by the equations
\begin{align} 
S_A' &=  -{S_A}\Big(p \beta_{AA}\frac{I_A}{N_A} 
  + p \beta_{KA}\frac{I_K}{N_k}\Big),\\
E_A' &= {S_A}\Big(p \beta_{AA}\frac{I_A}{N_A} 
  + p \beta_{KA}\frac{I_K}{N_k}\Big) - \gamma_1E_A, \\
I_A' &= \gamma_1E_A - \rho_1I_A - \delta_1I_A, \\
R_A' &= \rho_1I_A, \\
S_K' &=  -{S_K}\Big(p \beta_{KK}\frac{I_K}{N_k} 
 + p \beta_{AK}\frac{I_A}{N_A}\Big),\\
E_K' &= {S_K}\Big(p \beta_{KK}\frac{I_K}{N_k} 
 + p \beta_{AK}\frac{I_A}{N_A}\Big) - \gamma_2E_K, \\
I_K' &= \gamma_2E_K  - \rho_2I_K - \delta_2I_K, \\
R_K' &= \rho_2I_K,
\end{align}
where
 $N_A = S_A + E_A + I_A + R_A$ is the total population of adults and
 $N_K = S_K + E_K + I_K + R_K$ is the total population of kids.
Exposed adults and kids progress to become infectious at rates $\gamma_1$ 
and $\gamma_2$, respectively,
and recover from infection at rates $\rho_1$ and $\rho_2$, respectively.
Disease-induced mortality occurs at rates $\delta_1$ and $\delta_2$ 
for adults and children, respectively.
As we are modeling respiratory diseases with fast epidemiological dynamics 
(on the order of weeks or months),
we ignore demographics and assume a fixed population.
In other words, we do not consider birth/recruitment rates into the 
population or background mortality rates.

Transmission occurs both within and outside age groups at contact rates of
$\beta_{AA}$ between infected adults and susceptible adults,
$\beta_{KA}$ between infected kids and susceptible adults,
$\beta_{KK}$ between infected kids and susceptible kids, and
$\beta_{AK}$ between infected adults and susceptible kids.
For consistency with~\cite{Del Valle},
we consider these as contact rates sufficient to pass the disease and
multiply by the probability $p$ of infection actually being transmitted in 
such contacts.

\subsection{Basic reproductive number for epidemiological model}\label{S:5}

In disease modeling, the basic reproductive number is defined as the number
of new infections caused by a single infected person introduced into a 
completely susceptible population. Thus a reproductive number greater than 
one means that the disease will continue to grow and spread with time, 
while a reproductive number less than one means the disease will dwindle and 
eventually die out \cite{NIH2}. 

First, we calculated a reproductive number for each transmission category. 
These calculations are as follows: 
\begin{gather*} 
\mathcal{R}_0^{AA} = \frac{B_{AA}}{\delta_1 + \rho_1}, \quad
   \mathcal{R}_0^{AK} = \frac{B_{AK}}{\delta_1 + \rho_1}, \\
 \mathcal{R}_0^{KA} = \frac{B_{KA}}{\delta_2 + \rho_2}, \quad
 \mathcal{R}_0^{KK} = \frac{B_{KK}}{\delta_2 + \rho_2}.
\end{gather*}
We then used these individual reproductive numbers to find an overall 
reproductive number for the entire population. 
This was found by finding the next-generation matrix and evaluating its 
eigenvalues at the disease-free equilibrium. 
The overall reproductive number is 
\[ 
\mathcal{R}_0 = \frac{\Big(\frac{B_{AA}}{\delta_1 + \rho_1} 
+ \frac{B_{KK}}{\delta_2 + \rho_2} + \sqrt{(\frac{B_{AA}}
{\delta_1 + \rho_1}+\frac{B_{KK}}{\delta_2 + \rho_2})^2
 - 4(\frac{B_{AA}}{\delta_1 + \rho_1}\frac{B_{KK}}{\delta_2 + \rho_2}
 -\frac{B_{AK}}{\delta_1 + \rho_1}\frac{B_{KA}}{\delta_2
  + \rho_2})}\Big)}{2}.
\]

\section{Clinical trial modeling}\label{S:3}

With the epidemiological model complete, we proceed to establish the system 
to describe clinical trials for a vaccine for a new respiratory infection. 
At this point, it is important to express the principal aspects of the 
clinical trial process that our model will consider and those it does not. 
 
Clinical trials for new vaccines typically involve three separate phases.
Phase 1 involves a small group of healthy volunteers to study dosage, 
safety, and immune response.  Phase 2 trials expand the study population 
and further evaluate safety and immunogenicity while also exploring factors 
like age and underlying health conditions. 
Phase 3 trials, the largest phase, involve thousands of participants and 
assess vaccine efficacy by comparing vaccinated individuals to a 
control group receiving a placebo or alternative. 
These trials closely monitor participants for the given respiratory 
infection, aiming to determine if the vaccine prevents or reduces the 
severity of the disease. Importantly, given the logistical challenge and 
size of Phase 3 trials, trial participants are not monitored for infection 
continuously but rather a fixed periods (e.g.\ weekly, biweekly, etc.).

The end of a Phase 3 trial can be determined in a few different ways.
First, statistical significance can be used to indicate the successful 
completion of a trial. This occurs when the vaccine's performance is 
statistically significant enough to support the vaccine's approval. 
The analysis must determine whether the observed results are due to the 
effectiveness of the vaccine and not random chance~\cite{JTO}. 
In another method, clinical trials follow a predetermined protocol that 
outlines specific endpoints and goals, and the trials need to collect 
enough data to reach these goals. 
The extensive and complicated details of clinical trial design, execution, 
and regulation are beyond the scope of this modeling work~\cite{NIH}, so 
we focus on the central characteristics related to epidemiological dynamics.

One last feature of clinical trials that is central to our modeling work is 
that clinical trials of new vaccines are typically conducted for adults 
before subsequent clinical trials for the vaccine in kids. 
This occurs for multiple reasons~\cite{Paturel}.
First, it is obvious that adults are in the workforce, and interact with 
others much more than children. Thus, curbing the spread starts with them 
since they come in contact with both adults and children alike. 
Children are often only in contact with other children, and their parents, 
whom they rely on for basic needs. Furthermore, it is important to test 
adults first to verify safety and will not have adverse effects that could 
affect their development. Overall, parents are also very weary of giving 
something to their kids that has not been proven safe~\cite{Paturel}.
 
For our mathematical model, we focus only on Phase~3, the large-scale trial 
that happens at the end of the testing process. We use a discrete-time 
model (i.e.\ difference equation) to track the number of infections in the 
control and treatment arms
due to the fact that trial participants are monitored for new infection 
a fixed time periods (e.g.\ weekly, biweekly, etc.).
The clinical trial model tracks the number of cumulative infections in 
the control arms, $C$, and treatment or vaccinated arms, $V$, of the 
clinical trial and $e$ is the vaccine efficacy. Subscripts $u$ and $i$ 
refer to the numbers of uninfected and infected individuals in each 
clinical trial arm, respectively.

The equations for the clinical trial model are as follows:
\begin{align} 
V_i[j+1] &= V_i[j] + V_u[j](1-e)(\text{\small probability of infection 
for time step $j+1$}),\label{CT1} \\ 
 V_u[j+1] &= V_u[j] - V_u[j](1-e)(\text{\small probability of infection 
 for time step $j+1$}),\\
 C_i[j+1] &= C_i[j] + C_u[j](\text{\small probability of infection 
 for time step  $j+1$}),\\
 C_u[j+1] &= C_u[j] - C_u[j](\text{\small probability of infection 
 for time step  $j+1$}). \label{CT4}
\end{align} 
Importantly,  the probability of infection in a given time step 
(e.g.\ 2 weeks) is determined by the rate of new infections over that 
period in the general population 
which, in our modeling framework, is governed by the solution to the 
epidemiological component of the model. 


\subsection{Coupling the epidemiological and clinical trial models}
\label{S:3d}
As our goal is to model the interplay between epidemiological dynamics and 
clinical trials for new vaccines, it is essential to formulate the 
relationship between the two systems. Fortunately, the relatively small 
size of a clinical trial population compared to the much larger size of the 
general population simplifies the process.
While the dynamics of the ongoing epidemic certainly affect infection rates 
experienced by the clinical trial population, we can ignore any impact that 
the clinical trial population will have on the trajectory of the epidemic 
in the overall population.
Clinical trials can be large and include thousands of individuals.  
However, the protective effect experienced by those individuals will not 
change the experience of the millions of individuals in the general 
population.

To couple the two models, we must express the probability of infection for 
a given time step in the discrete model in terms of the disease incidence 
resulting from the continuous epidemiological model. We do so using the 
formula
\begin{align*}
&\text{probability of infection for time step} \\
&= \frac{\text{new infections for time step}}
 {\text{number of uninfectious individuals at the beginning of the time step}} \\
&= \frac{\int_{t_0}^{t_0+\Delta t} \gamma E(t)\,dt}{S(t_0)+E(t_0)+R(t_0)},
\end{align*}
where the time step $j+1$ in the clinical trial model corresponds to the 
interval $(t_0,t_0+\Delta t)$ in the epidemiological model.
In all of our simulations, we use a clinical trial timestep of 
$\Delta t = 14$ days.

With the linked epidemiological and clinical trial models, we can state the 
plan for our general simulation as illustrated in Figure~\ref{plan}.

\begin{figure}[htb]
 \centering
\includegraphics[width=0.9\textwidth]{fig1} % simulationgraphic.png
 \caption{General simulation linking epidemiological dynamics to those 
 of vaccine clinical trials in adults and children.}\label{plan}
 \end{figure}
 
For simplicity, we assume that the respiratory infection of interest is
transmitted in waves (as was true with COVID and for seasonal respiratory 
infections).
For a clear comparison, we will also assume that these waves occur a year 
apart and are identical except for the aspects that are the focal point 
of this study.

We assume that a Phase 3 vaccine clinical trial begins on the 
100\textsuperscript{th} day of the first wave of the epidemic. 
This first clinical trial is conducted to measure vaccine efficacy in adults 
and infection rates of the clinical trial population (both experimental and 
control groups) are governed by the state of the ongoing epidemic in the 
general population at large.
These infection rates in the adult clinical trial are tracked by the model 
in Equations~\ref{CT1}-\ref{CT4}. We assume that the vaccine is approved 
for use in adults and that a certain level of vaccine coverage has been 
achieved by the beginning of the second wave of the epidemic.
 For simplicity, this second wave begins exactly one year after the first
wave.  Also for simplicity, the clinical trial for vaccination in children 
begins 100 days into this second wave of the epidemic. The only difference 
between the first and second waves is the effect of adults being vaccinated 
in the population at large during the second wave.


\section{Simulation results}\label{4}
In this work, our goal is to examine the interplay between background 
epidemiological dynamics and the outcomes of clinical trials for a new vaccine.
More specifically, we are interested in the situation where a vaccine is 
tested and approved for adults before a clinical trial for the vaccine in 
children begins. To do so, we proceed to numerical simulations. 
First, we will examine this interplay when no heterogeneity between adults 
and children is assumed in population size, contact rates, and disease 
outcomes. We then conduct the same analyses when heterogeneity is 
incorporated.

Importantly, the epidemiological model stated in Section~\ref{epimodel} 
does not include vaccination. For the second wave of the epidemic,
we assume that a vaccine coverage of $c$ in the adult population because 
the vaccine was tested and approved
for adults during the first wave.  To account for this most clearly, 
we adapt the first two equations for susceptible and exposed adults to 
become
\begin{align} 
S_A' &=  \Big(\frac{cS_A(1 - e) + (1-c)S_A}{N_A}\Big)p(\beta_{AA}I_A 
 + \beta_{KA}I_K),\\
E_A' &= \Big(\frac{cS_A(1 - e) + (1-c)S_A}{N_A}\Big)p(\beta_{AA}I_A 
 + \beta_{KA}I_K) - \gamma_1E_A,
\end{align}  
where $c$ is the vaccine coverage in the adult population during the 
second wave of the epidemic and 
$e$ is the protective efficacy of the vaccine.

\subsection{Results assuming homogeneous populations}
As mentioned above, our model represents a general respiratory infection 
for which a new vaccine is developed. We assume parameter values relevant 
to such infections but not for a specific disease. We begin the simulation 
of each wave of the epidemic using the initial values in Table~\ref{homogICs}.
Note that we assume there are an equal number of adults and children in the 
population and seed the epidemic with equal numbers of 
exposed and infectious individuals.

 \begin{table}[htb]
 \caption{Initial values of state variables used at the beginning of both 
the first and second wave of the epidemic
  when assuming homogeneous populations.}\label{homogICs}
  \centering
  \begin{tabular}{|l|l|l|} \hline
 \textbf{Variable} & \textbf{Description} & \textbf{Initial Value} \\
 \hline
   $S_A$ & Susceptible adults & 50,000 \\
   $S_K$ & Susceptible children & 50,000 \\
   $E_A$ & Exposed adults & 50 \\ $E_K$ & Exposed children & 50 \\
 $I_A$ & Infected adults & 2 \\
 $I_K$ & Infected children  & 2\\
 $R_A$ & Recovered adults  & 0 \\
 $R_K$ & Recovered children & 0  \\
 \hline
  \end{tabular}

\end{table}

For the simulation carried out under the homogeneity assumption, we use 
the model parameters summarized in Table~\ref{homogpars}.
Importantly, we note that identical contact rates are used between the 
age groups in the population. Moreover, disease progression, mortality and 
recovery rates for adults and children are identical.

 \begin{table}[htb]
 \caption{Parameter values for simulations when assuming homogeneous 
  populations.
  $^{\dagger,\ddagger}$ Values used for first and second waves of epidemic 
  respectively.
  $^*$ Value adapted from~\cite{Del Valle} by averaging age-specific contact 
  rates.  } \label{homogpars} 
 \centering
  \begin{tabular}{|l|l|l|}\hline
 \textbf{Parameter} & \textbf{Description} & \textbf{Value} \\ \hline
 $c$ &  Vaccine coverage in adults & 0{$^\dagger$}, 0.80{$^\ddagger$}\\
 $e$ &  Vaccine efficacy & 0$^\dagger$, 0.80$^\ddagger$\\
 $p$ & Probability of transmission  & 0.10 \\
 $\beta_{AA}, \beta_{KA}, \beta_{KK}, \beta_{AK}$ & Contact coefficient & 0.974$^*$ \\
 $\gamma_1$ & Adult progression rate  & 1/21 \\
 $\gamma_2$ & Child progression rate  & 1/21 \\
 $\rho_1$ & Adult recovery rate   & 1/21 \\
 $\rho_2$ & Child recovery rate   & 1/21 \\
 $\delta_1$ & Adult disease death rate   & 0.001\\
 $\delta_2$ & Child disease death rate   & 0.001\\
 $\Delta t $ & Days between disease  & 14 days \\ & monitoring in CT &  \\
 \hline
  \end{tabular} 
  
\end{table}

For illustration, we assume each vaccine clinical trial consists of 100 
participants; 50 randomly assigned to the treatment (i.e.\ vaccination) 
arm and 50 randomly assigned to the control group. The number of 
infections in the two groups is tracked using the clinical trial model of
 Equations~\ref{CT1}-\ref{CT4}. Through the course of the clinical trial 
simulation, we track the $p$-value of the different infection rates using 
the Fisher exact test;  a test for determining whether the proportions of 
data described by two or more categorical variables are random. While our 
modeling work is deterministic with no random effects, the $p$-value 
quantifies the probability of our observed infection rates if the vaccine 
had no efficacy at all. In a true clinical trial, many details and 
considerations can go into deciding when a clinical trial will stop. 
Even in the most basic plans, researchers would want statistical evidence 
that the vaccine efficacy was greater than a certain value 
(e.g.\ 50\%, 70\%, etc.). Our simplified estimates looks simply for evidence 
that the vaccine is showing any positive efficacy at all, regardless of how 
small. While clinical trials would continue longer to achieve more rigorous 
results, our simplification illustrates the effects that would most likely 
be amplified with more-detailed stopping conditions.

The results of our numerical experiment are illustrated in 
Figure~\ref{homogsim}.
In Figure~\ref{homogsim}a, we see the first wave of the epidemic through 
an entirely unvaccinated population. Notably, the figure illustrates curve 
for both adults and children, but the dashed curves for children are entirely 
obscured by the curves for adults. This is to be expected as we are assuming 
complete homogeneity between adults and kids. Therefore, with identical 
initial conditions and parameters, the epidemic plays out the same for both 
sub populations. In Figure~\ref{homogsim}b, we see the results of the vaccine 
clinical trial in adults which starts 100 days into the first wave.
The black line tracks the cumulative number of infections in the control 
group and the red line tracks those in the vaccine group. In blue, we 
calculate the $p$-value of the protective efficacy of the vaccine on a 
logarithmic scale. Using our simplified approach, we see that a $p$-value 
of $10^{-4}$ is achieved after 6 steps of the clinical trials 
(i.e.\ 12 weeks, 84 days).

\begin{figure}[htb]
 \centering

 \includegraphics[width=.46\textwidth]{fig2a} % homogeneous/1a- General Model All Classes.pdf}}
\includegraphics[width=.46\textwidth]{fig2b} \\ % homogeneous/2a- Clinical Trial Plot.pdf}}
\parbox{55mm}{(a) First wave of epidemic; no one vaccinated} \hfil
\parbox{55mm}{(b)  Clinical trial for adults that begins on Day 100 of the first wave}
\\
\includegraphics[width=.46\textwidth]{fig2c} % homogeneous/3a- General Model All Classes.pdf}}
\includegraphics[width=.46\textwidth]{fig2d} % homogeneous/4a- Clinical Trial Plot.pdf}}
\\
 \parbox{55mm}{(c) Second wave of epidemic; 80\% of adults are vaccinated} \hfil
 \parbox{55mm}{(d) Clinical trial for kids that begins on Day 100 of the second wave}

  \caption{Simulation results when assuming population homogeneity between 
 adults and kids 
 (i.e.\ equal size populations and random mixing between populations) and 
 a vaccine efficacy of $e=0.80$.}\label{homogsim}
\end{figure}

In Figure~\ref{homogsim}(c), the simulation moves to the second wave of the 
epidemic where we assume that 80\% of the adult population is vaccinated with 
the 80\% effective vaccine.
We see this protective effect as the second wave moves more slowly through 
the adult population than in children.
In Figure~\ref{homogsim}(d), we see the results of the vaccine clinical trial 
for kids.
Infections in the clinical trial for kids accumulate more slowly than in the 
adult clinical trial. 
As these simulations assume complete homogeneity between adults and kids, 
we know that this difference is caused entirely by the vaccine protection 
in adults during the second wave
(and consequently fewer infections in the population overall).
While fewer infections during the second wave is clearly a desirable outcome,
we do see that the clinical trial for children must continue for a longer 
time period to achieve the same $p$-value as the adult clinical trial.
Specifically, to get to the same $10^{-4}$ level of significance, 
the clinical trial in kids would need to run 10 steps 
(i.e.\ 20 weeks, 140 days).

\subsection{Results assuming differences between adults and children}
In the previous section, we saw that a vaccine clinical trial in children 
can be affected by vaccination among adults in the general population. 
More specifically, the clinical trial would need to run longer to establish 
statistically significant protection.
However, this behavior was exhibited under an assumption of identical
populations of adults and children.
For some respiratory infections, such an assumption does not hold.

In this section, we investigate the same ideas when assuming heterogeneity 
(both in population size and epidemiological characteristics) among adults 
in children.
In Table~\ref{heteroICs}, we see the initial values for our simulations.
Notably, we assume that there are significantly more adults than children 
(70\% vs.~30\%).

\begin{table}[htb]
\caption{Initial values of state variables used at the beginning of both 
the first and second wave of the epidemic
  when assuming heterogeneous populations.}\label{heteroICs}
  \centering
  \begin{tabular}{|l|l|l|} \hline
 \textbf{Variable} & \textbf{Description} & \textbf{Initial Value} \\
 \hline
   $S_A$ & Susceptible adults & 70,000 \\
   $S_K$ & Susceptible children & 30,000 \\
   $E_A$ & Exposed adults & 50 \\ $E_K$ & Exposed children & 50 \\
 $I_A$ & Infected adults & 2 \\
 $I_K$ & Infected children  & 2\\
 $R_A$ & Recovered adults  & 0 \\
 $R_K$ & Recovered children & 0 \\  
 \hline
  \end{tabular}

\end{table}

In Table~\ref{heteropars}, we now use age-specific contact rates to estimate 
mixing patterns between children and adults.
To establish these parameters,  we use the results of 
Del Valle, et al.\ \cite{Del Valle} who built a detailed agent-based model 
from data on daily interactions between different age groups.
More specifically, we used the averages of the age-structured contact rates 
in \cite[Table 2]{Del Valle} for adults and children to establish our values 
for $\beta_{AA}, \beta_{KA}, \beta_{KK}$, and $\beta_{AK}$ in 
Table~\ref{heteropars}.

 \begin{table}[htb]
 \caption{Parameter values for simulations when assuming heterogeneous 
 populations.   $^{\dagger,\ddagger}$ Values used for the first and 
 second waves of epidemic respectively.
  $^*$ Value adapted from~\cite{Del Valle} by averaging age-specific 
  contact rates.  }\label{heteropars} 
 \centering
  \begin{tabular}{|l|l|l|} \hline
 \textbf{Parameter} & \textbf{Description} & \textbf{Value} \\ \hline
 $c$ &  Vaccine coverage in adults & 0{$^\dagger$}, 0.80{$^\ddagger$}\\
 $e$ &  Vaccine efficacy & 0$^\dagger$, 0.80$^\ddagger$\\
 $p$ & Probability of transmission  & 0.10 \\
 $\beta_{AA}$ & Adult-to-adult contact coefficient  & 1.15 \\
 $\beta_{KA}$ & Child-to-adult contact coefficient  & 0.438 \\
 $\beta_{KK}$ & Child-to-child contact coefficient  & 1.61 \\
 $\beta_{AK}$ & Adult-to-child contact coefficient  & 0.697\\
 $\gamma_1$ & Adult progression rate  & 1/21 \\
 $\gamma_2$ & Child progression rate  & 1/14 \\
   $\rho_1$ & Adult recovery rate   & 1/21 \\
 $\rho_2$ & Child recovery rate   & 1/14 \\
 $\delta_1$ & Adult disease death rate   & 0.01\\
 $\delta_2$ & Child disease death rate   & 0.001\\
 $\Delta t $ & Days between disease  & 14 days \\ & monitoring in CT &   \\
 \hline
 \end{tabular} 
 
\end{table}
In line with several respiratory infections, we include additional 
heterogeneity by assuming faster disease dynamics in children 
(i.e.\ $\rho_2 = \gamma_2 =\frac{1}{14}$) 
and higher disease-induced mortality in children~(i.e.\ $\mu_1 = 0.01$).


The results of this simulation are illustrated in Figure~\ref{heterosim}.
In Figure~\ref{heterosim}a, we see disease dynamics in the first wave that 
differ for adults and children.
In addition to different population sizes, we see the epidemic peaks 
slightly earlier in children.
In Figure~\ref{heterosim}(b), the clinical trial in adults follows an almost 
indistinguishable pattern from that of Figure~\ref{homogsim}(b),
with the clinical trial reaching a $p$-value of $10^{-4}$ in 6 steps 
(i.e.\ 12 weeks, 84 days).
Figure~\ref{heterosim}(c) shows the second wave of the epidemic with 80\% of 
adults vaccinated and receiving 80\% efficacy from the vaccine.
The most notable results are illustrated in the clinical trial in children
shown in Figure~\ref{heterosim}(d).
With heterogeneity included, we see that the clinical trial in children 
would reach a $p$-value of $10^{-4}$ in roughly the same number of steps 
as the adult clinical trial;
specifically 7 steps (i.e.\ 14 weeks, 98 days).
Therefore, we see that the result of vaccine clinical trials taking longer 
in children than in the adult population can be largely erased due to 
epidemiological differences between adults and children.

\begin{figure}[htb]
 \centering
\includegraphics[width=.46\textwidth]{fig3a} % heterogeneous/1a- General Model All Classes.pdf}}
\includegraphics[width=.46\textwidth]{fig3b} \\ % heterogeneous/2a- Clinical Trial Plot.pdf}}
\parbox{55mm}{(a) First wave of epidemic; no one vaccinated} \hfil
\parbox{55mm}{(b) Clinical trial for adults that begins on Day 100 of the first wave}

\includegraphics[width=.46\textwidth]{fig3c} % heterogeneous/3a- General Model All Classes.pdf}}
\includegraphics[width=.46\textwidth]{fig3d} \\ %heterogeneous/4a- Clinical Trial Plot.pdf}}
\parbox{55mm}{(c) Second wave of epidemic; 80\% of adults are vaccinated }\hfil
\parbox{55mm}{(d) Clinical trial for kids that begins on Day 100 of the second wave}

 \caption{Simulation results when assuming population heterogeneity between 
 adults and kids 
 (i.e.\ population is 70\% adult and 30\% kids, and age-based mixing
  patterns adapted from \cite{Del Valle})
 and a vaccine efficacy of $e=0.80$.}\label{heterosim}
\end{figure}

\section{Conclusions and Further Research}\label{S:7}

Our objective was to model the interaction between epidemiological dynamics 
and clinical trials for a new vaccine targeting a general respiratory 
infection. The clinical trial for children was initiated only after the 
completion of the adult trials, utilizing the data from the adult trials 
to inform the children's trials. 
A key finding was that the timeline for children's clinical trials is 
affected by adult vaccination rates and the resultant herd immunity.

Although the timing of another pandemic event similar to COVID-19 is 
unpredictable, preparedness is crucial. Our model simulates a general 
scenario and could be valuable for future pandemic preparedness. 
Further refinement of our results using more precise parameters is an 
area for future research.

\subsection*{Acknowledgements}
This research wass partially supported by 
Xavier University's Summer Undergraduate Research funding during 
Summer 2023.
 


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