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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION
\else STUDENT VERSION \fi\\Modeling Social Campaigns}



\markboth{Modeling Social Campaigns}{Modeling Social Campaigns}

\author{Hyunsun Lee\\
Hawaii Pacific University\\
Honolulu HI 96813 USA\\
hlee@hpu.edu
}

\makeStitlePDFLaTex


\begin{abstract} 
Mathematical epidemic models are crucial tools to understand, analyze, predict, and control infectious diseases.
The Susceptible-Infected-Recovered (SIR) model is a basic compartment model, describing how an infectious disease propagates through a population. The problem is formulated as a system of three nonlinear, first order differential equations in which  three compartments (S, I, and R) of the population are linked. The application of this model is not limited to epidemic modeling. It has been applied to describe malware propagation, viral marketing, social networking, etc. In this project, we explore an application of the SIR model in a social campaign scenario. 

\end{abstract}


\section*{SCENARIO DESCRIPTION}

The Ice bucket challenge was one of the most successful social media campaigns. What do you think made the ice bucket challenge so extremely successful? More generally, why do some campaigns become successful while some fail terribly? How can we characterize a successful campaign versus an unsuccessful campaign? How can we make an unsuccessful campaign into a successful one? What is your strategy?  


To tackle these problems, we could  elaborate the similarities between a social campaign and an infectious disease transmission by exploring a compartment model, Susceptible-Infected-Recovered (SIR) model, that is often used to describe how infectious disease propagates through a population. In an SIR model, the problem is formulated with a system of three nonlinear, first order differential equations where three compartments (S, I, and R) of the population are linked. The application of this model is not limited to epidemic modeling. It has been applied to describe malware propagation, viral marketing, and social networking.  


The state variables are defined as:
$S(t)$ is the number of susceptible individuals at time $t$, $I(t)$ is the number of infected individuals at time $t$, and
$R(t)$ is the number of post-infective individuals removed from the population at time $t$.
The basic governing equations with a constant population $P=S(t)+I(t)+R(t)$ can be written as: 
\begin{eqnarray}
S'(t)&=&-\beta I(t) S(t)/P, \label{eq:S} \\
I'(t)&=&\beta I(t) S(t)/P - \gamma I(t),\\
R'(t)&=&\gamma I(t), \label{eq:R}
\end{eqnarray}
where $\beta$ is the contact rate between the infected and the susceptible compartments, and $\gamma$ is the recovery rate which is the reciprocal of the mean recovery time, $m$. In this system, the rate of change in the susceptible class is non-positive and proportional to the product of the infected and the susceptible. The rate of change in the recovered class is non-negative and proportional to the infected class. Assuming that there is no change in the entire population $P$, i.e. $P'=(S(t)+I(t)+R(t))'=0$. In the case that people in the recovered group are temporarily immune and return to the susceptible class, then the extra term $\nu R$ should be considered in the right side of (\ref{eq:S}) and (\ref{eq:R}), where $1/\nu$ is an average length of time for an individual in the recovered class to rejoin to the susceptible group. See the Project 1 in Chapter 9 in \cite{BoyMea17}.\\


In this study, we apply the SIR model to describe and predict a social campaign. The scenario of our interest is that a person nominates $k$ people within his/her social media group and each nominee is expected to nominate $k$ people within $m$ days, as seen in the ice bucket challenge. 


Consider the following questions: 

\begin{itemize}

\item[] Question 1. We first consider how to modify  (\ref{eq:S})-(\ref{eq:R}) to describe the given scenario of social campaign. What are the compartments  in the social campaign scenario? How  do you define or interpret the constants $\beta$ and $\gamma$?

\item[] Question 2. Since the status of a social campaign is evaluated per day or per week, convert the system of differential equations  (\ref{eq:S})-(\ref{eq:R})  into a system of appropriate difference equations. What should be considered to discretize the differential equations? See \cite{LinAll1994} for details. 

\item[] Question 3. How can we model a joining (quitting) process and integrate the process into the system of difference equations in Question (2)? Does the process follow any particular probability density distribution?  

\item[] Question 4. What do you think is the major difference in the joining (quitting) process for  distinguishing a successful campaign and an unsuccessful campaign? What is your strategy to lead a social campaign to a success? 

\end{itemize}

\begin{thebibliography}{98}

\bibitem{BoyMea17} Boyce, W.~E and DePrima, R.~C and Meade, D.~B. 2017. {\it Elementary Differential Equations and Boundary Value Problems}. Wiley Global Education, 2017.

\bibitem{LinAll1994} Allen, L.~J.~S. {\it Some Discrete-Time SI, SIR, and SIS Epidemic Models\/}. Elsevier Science Inc., Mathematical Biosciences, 1994.

\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

For success in this Modeling Scenario we presume the students are familiar with probability density and distribution  functions from the study of probability.

This project is designed to encourage students to develop skills to associate a well-known differential equation model to another situation by exploring and processing the similarities and interpreting the problem in context. The solution for this project may not be unique; rather it depends on how students process the scenario. 


In the given social campaign scenario, we consider three different classes: Susceptible (S), Nominated (N), Action (A) classes similarly to the SIR model. In the given social campaign scenario of this problem, the susceptible (S) class is a group of people who are connected to their friends or family or any acquaintances through any type of social media so that they could be exposed to the campaign. The nominated (N) class is the group of people who were in the S class but moved to the N class as they were nominated by other people in their social network. It is interesting to discuss with students how people nominated multiple times can be differently weighed in the equation. The action (A) class is the group of people who were previously in the N class but moved to the A class, having decided to participate in the campaign, i.e.~ice bucketing and nominating other people in the ice bucket campaign. We assume that any member in the A class will not go back to the S class. 

This will lead us to follow the same system of equations in  (\ref{eq:S})-(\ref{eq:R}) by replacing the classes in the system, based on the different situational explanation above:
\begin{eqnarray}
S'(t)&=&-\beta N(t) S(t)/P, \label{eq:S1} \\
N'(t)&=&\beta N(t) S(t)/P - \gamma N(t),\\
A'(t)&=&\gamma N(t), \label{eq:A}
\end{eqnarray}
where the growth rate $\beta$ depends on the given parameter $k$ that is the expected number of people nominated by a participant. The recovery rate $\gamma$ depends on the given parameter $m$, the average length of the time period from the date of becoming a nominee to the date of becoming a participant in the campaign, i.e.~ $m$  is  the average delay time for a random person in the N class to decide to participate in the campaign by committing the action and nominating other people. In the sense, the last equation for the A class can be re-written as:
\begin{equation}
A'(t)=\int_0^t N'(x)f(t-x)dx, \label{eq:Join}
\end{equation} 
where $f(t)$ is a probability density function describing the distribution of time spent for a random person to make the decision to participate (delay time), i.e.~the time period from the date of becoming a nominee to the date of becoming a participant in the campaign. The mean of this probability distribution is expected to be $m$ days since it was set as a rule for the campaign in the given problem description. The right side of (\ref{eq:Join}), the change of the A class at time $t$, represents the cumulative number of previous nominees who become participants at time $t$. 


The probability distribution of the delay time, $f(t)$, can be expressed in the form of the exponential distribution function that describes the time between Poisson or independent events: 
\begin{equation}
f(t)=\frac{1}{m}\exp \left( {-\frac{t}{m}} \right)=\gamma \exp \left ({-\gamma t} \right),
\end{equation}  
by the definition of $\gamma=1/m$ for the given $m$ in the problem. The probability density function describes the joining process of the campaign. However, one may consider a quitting class instead of the A class proposed in  (\ref{eq:A}), where a quitting process should be explored accordingly, rather than a joining process in the equation. 

If students do not have any statistics background, the probability density function $f(t)$ of delay time can be derived as done below. 

\begin{theorem}
Let $f(t)$ be the probability density function of delay time $t$ in the social campaign model, then $f(t)=\gamma \exp \left ({-\gamma t} \right).$
\end{theorem}

\begin{proof}
Let $F(t)$ be the cumulative probability density function such that 
\begin{equation}
F(t)=P(T\le t)=1-P(T>t). \nonumber
\end{equation}
By the definition of $f(t)$, 
\begin{equation}
f(t)=\frac{dF(t)}{dt}=\frac{dP(T\le t)}{dt}=\frac{d\{1-P(T>t)\}}{dt}=-\frac{dP(T>t)}{dt}. \label{eq:pdfDlay}
\end{equation}
Since $1/\gamma$ is the average recovery time, then $\gamma$ can be interpreted as the probability of recovery over one unit time period. 
Therefore, the last term in  (\ref{eq:pdfDlay}) can be rewritten as 
\begin{equation}
f(t)=\gamma P(T>t). \label{eq:pdfDlay1}
\end{equation}
Consequently, by (\ref{eq:pdfDlay}) and (\ref{eq:pdfDlay1}), we have
\begin{equation}
\frac{df(t)}{dt}=\gamma \frac{dP(T>t)}{dt}=-\gamma f(t). \label{eq:eqn}
\end{equation}
We solve for $f(t)$ with $f(0)=\gamma$ in (\ref{eq:eqn}):
\begin{equation}
f(t)=\gamma \exp \left ({-\gamma t} \right). \nonumber
\vspace*{-.5 true cm}
\end{equation}

\end{proof}


If the campaign is evaluated on a daily basis, then the time $t$ can be discretized into days and the system becomes:
\begin{eqnarray}
\triangle S (n+1)&=&-\beta N(n) S(n)/P, \label{eq:S2} \\
\triangle N(n+1)&=&\beta N(n) S(n)/P - \gamma N(n),\\
\triangle A(n+1)&=&\sum_{t=0}^{t=n} \ \triangle N(t) \ f(n+1-t), \label{eq:Join1}
\end{eqnarray}
where the right side of  (\ref{eq:Join1}) can be interpreted, in a discrete sense, that the change of the A class on day $(n+1)$ is the average total number of people as a product of the number of new nominees on Day $t$ and the probability that the nominees takes $(n+1-t)$ days to make the decision to participate at Day (n+1).


It is a good practice to simulate how each class grows with a given initial condition for each class as well as a growth rate $\beta$ and a recovery rate $\gamma$. Using various constant rates, students can observe the corresponding change in resulting class sizes. The graph in Figure~\ref{fig:6-010-SIRex} is an example with $\beta=0.2$, $\gamma=0.3$, $S(0)/P=0.85$, $N(0)/P=0.15$ and $A(0)/P=0$. If there are any available data sets for social campaigns, one successful campaign and one unsuccessful campaign, it would be interesting to compare and discuss the difference. Non-diffusive terms may be considered in the equations to learn how external impacts can be formulated.\cite{ZhaYan2017}

\begin{figure} [h!]
\vspace{- 5 true cm}
\centering
\includegraphics[width=12 cm]{6-010-SIRex}
\vspace{- 5 true cm}
\caption{\label{fig:6-010-SIRex} An example of social campaign model from Day 1 to Day 50.} 
\end{figure}


The equilibria of the system, found by setting the right side of (\ref{eq:S2})-(\ref{eq:Join1}) to zero, are categorized into three cases: $(P, 0, 0)$, $(0,0,P)$ and $(P_1, 0, P_2)$ where $0<P_1, P_2<P$ and $P_1+P_2=P$. For $(P,0,0)$, there is no campaign, i.e.~there is a  campaign-free state. The equilibrium point at $(0,0,P)$ represents the post-campaign state where everybody in the population participated in the campaign. The equilibrium at $(P_1,0,P_2)$ also represents the post-campaign state where not everybody in the population participated in the campaign. 

Finally, we point to \cite{TweedleSmith2012}, a reference suggested by a referee, in which the paper discusses the spread of Justin Bieber fever using social media, offering a simpler model than presented here.

\begin{thebibliography}{99}

\bibitem{LinAll1994} Allen, L.~J.~S. 1994.  Some Discrete-Time SI, SIR, and SIS Epidemic Models. {\it Mathematical Biosciences\/}. 124: 83-105.

\bibitem{BoyMea17} Boyce, W.~E and R.~C~DePrima, and D.~B.~Meade.  2017. {\it Elementary Differential Equations and Boundary Value Problems}. New York: Wiley Global Education.

\bibitem{TweedleSmith2012} Tweedle, V. and R~J.~Smith. 2012. A mathematical model of Bieber Fever. In, {\it Understanding the Dynamics of Emerging and Re-Emerging Infectious Diseases Using Mathematical
Models\/}. Editors: S.~Mushayabasa and C.~P.~Bhunu. 157-177. 


\bibitem{ZhaYan2017} Zhang, T. and P.~Cui, C.~Faloutsos, C. , W.~Zhu, and S~Yang. 2017.  Come-and-Go Patterns of Group Evolution: A Dynamic Model. {\it Proceedings of the 22nd ACM International Conference on Knowledge Discovery and Data Mining\/}.

\end{thebibliography}
}
\end{document}
