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\title{The Mathematical Model of the Optimized Use of Ebola Virus Drug}
\team{38403}
\question{A}
%\date{February 8, 2015}
\begin{document}
\begin{summary}
A new medication may prevent the spread of Ebola virus and cure
those patients in the early affected period. How to make full
use of the new medication to fight Ebola is of great concern
and top priority. We are going to establish two mathematical
models: one to predict the infection trend of Ebola virus, the
right quarantine time and appropriate dosage; the other to
explore the optimal distribution in transportation systems and
production rate.
\par Model One will center on Ebola virus infection trends,
quarantine time and the amount of medication. According to the
characteristics of Ebola virus transmission, An Ordinary
Differential Equation (ODE) model of virus transmission is
built, along with the relevant graphics. On the basis of phase
trajectories, discussions are made about the nature of ODE
model of the spread of disease, the entire process of virus
transmission and the best time for initial virus isolation is
calculated. A nonlinear regression analysis is performed to
test the reliability of the model.
\par Model Two will explore the optimal medication distribution
within transport systems and the optimal drug production rate.
Some specific mathematical models are created by linear
programming in order to save the transportation time.
\end{summary}
\maketitle
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\section{Introduction}
\subsection{The basic situation of Ebola virus}
Ebola is a cause in humans and other primate and mammals, with
a high mortality rate of 50\% to 90\% \cite{bib1}. It is of
great importance to develop an effective medicine. World
Medication Association declared that the newly develop drugs
can cure the less seriously affected patients. Therefore the
key of our model is to help with the effective use of the
life-saving drugs.
\subsection{The Direction of Research}
The main direction of our research is to use a variety of
mathematical methods and means to achieve the optimal use
of the drug and the full value of drug, thereby reducing the
Ebola virus infection rate and mortality.
\section{Problem Analysis}
\subsection{The Restatement of the Problem}
Ebola virus appeared resulted in the death of many patients,
the Medical Association developed a drug to cure patients yet
to come late, so that more patients get second chance to see
the hope of life.
\par The model we build contributes to the crucial role the
medicines play at the moment.
\par Factors that play a crucial role in the influence of drugs,
including the speed of the disease, isolation time, the amount
of drug required drug, delivery systems and drug production
speed. Thus our tasks are gradually clear.
\par The task is as follows:
\begin{itemize}
\item The establishment of the Model One is to be more
intuitive to see the trends in the spread of Ebola virus.The
model not only provides important information for Ebola
virus' prevention and treatment but also a similar model for
the spread of other virus.
\item Using the conclusion of Model One, we can calculate the
isolation time roughly and then calculated the drug dosage
further.
\item Searching the database of the infected cases and using
the nonlinear regression ``nlinfit'' analysis to test the
reliability of Model One.
\item The establishment of the Model Two is to explore the
best transportation system for drug distribution, so that we
can save drug delivery time and strive to save more patients.
\item Using the Model Two elaborating the drug delivery rate
and drug production speed is to calculate vaccines and drugs
production speed.
\item Using the speed of medicine transport and
production in Model Two to further calculate the rate of
vaccine and medicine production.
\end{itemize}
\subsection{The Layout of Problem Solving}
\begin{itemize}
\item First of all, we need to establish the ordinary
differential equation models of the spread of Ebola
virus(Model One), draw images, describe
propagation of Ebola virus and discuss the characteristics
concerned the solution. At the same time, using of the model
we can calculate the best time to isolate rough.
\item Secondly, use the results of Model One to identify
positive relationship between dosage and the number of
infections including time parameter variables, and then we
can calculate the total dose.
\item Thirdly, according to the database provided by World
Health Organization(WHO), drawing scatter plots and using the
nonlinear regression ``nlinfit'' analysis to test the
reliability \cite{bib6}.
\item Fourthly, using linear programming model in the use of
transport,we establish transport system of Ebola drug.
\item Fifthly, using the results of optimal transport from
Model Two to determine the nonlinear relationship between
production rate and optimal distribution and calculate the
vaccine and medicine production rate.
\item Finally, according to the results of our research,
write a non-professional letter for the World Medical
Association's announcement.
\end{itemize}
\section{Model Design}
\subsection{The Spread of the Ebola Virus}
\label{sec:m1}
\input{m1spread}
\subsection{Required Drug}
\input{m1demand}
\subsection{Transportation System}
\input{m2trans}
\subsection{Produnction Speed}
\input{m4output}
\section{Testing and Analysis}
\input{mtest}
\section{Model Evaluation}
\subsection{Advantage}
\begin{itemize}
\item This model gives a result of a general, not only for
Ebola virus infection case analysis, but also for other
infections virus analysis.
\item Model use a method of combination of numbers and shapes
to present Ebola virus propagation visually objectively.
\item Model not only takes into account more than two origin,
more requirements to the case but also combines drug
production speed with drug delivery speed to get the optimal
allocation scheme.
\item The model takes into account the recovered cases and
death (no more infection) and greatly improve the
predictability and applicability of the model.
\item The principle of the model is straightforward. It
simplify algorithm. Model has good practicability.
\end{itemize}
\subsection{Weaknesses}
\begin{itemize}
\item The study population N is the total number of dynamic
change, there is birth, death, flow and so on. This makes the
model and the actual situation have a tolerance.
\item Ebola virus in the human body have latency and Ebola
virus carriers is not contagious, and this is where the model
is not perfect.
\item Drug has half-life in human body. Patients must adhere
to medication for some time to recover, so the dose
calculations and the actual have a tolerance.
\item Several factors are not taken into account, such as
technology level of every medicine manufacturer, the
different production rate and difference in transport
vehicle, road condition, which might result in the gap
between optimal distribution plan and the reality.
\item In actual circumstances, defective medicines in
production, in transport and in medicine-taking will affect
the reliability of the model.
\end{itemize}
\subsection{Improvement and Value of the Model}
In the process of model predictions, program one in Model One
ignores many factors, does not comply with the later development
of Ebola virus infection. Based on program one, we consider more
factors and then establish program two. Development of Ebola
virus infection and the development of other infections were
similar. However we know that both the use of drug and isolation
measures for Ebola virus infection trend have important
effect. Therefore, our model can give a more comprehensive
account of multiple factors and address the real problems. In
other words, the model prediction in the number of infections is
accurate, but due to virus variation and other uncontrollable
factors, it might not be predicable in the long period. To
develop in line with the actual situation of the Ebola
virus-term forecast, we must improve the above model further.
\par Model Two are simplified in many factors, so our model
should consider all factors affecting the transport logistics
and distribution, but the reality is very complex,multifaceted
consideration is needed to improve our model of development
direction.
\par Generally, infection intensity can be perceived through
the new added cases between intervals yet the infection trend
can hardly be determined. The model we build can solve the
problem and present a clearer picture of the transmission and
seize the best time to quarantine. In terms of distribution and
transport, our model can calculate the shortest time of
transport and production, hence saving more time and life by
providing medicine to disease-stricken areas.
Our model can be applied not only to the
establishment of the Ebola virus, but also be applied to a virus
similar to Ebola.
\section{A Letter for the World Medical Association}
\centerline{A letter for the World Medical Association}
Dear the World Medical Association:
\par Thanks to the world medical association has manufactured a
new medication for the treatment of Ebola virus to control the
spread of Ebola, more and more patients can be treated. In
order to make better use of the medication, we also need to
optimize the amount of drugs, drug distribution transportation
systems, drug production speed.
\par According to our research, we got two findings from our
research and modeling. In terms of the trends of Ebola virus
infection, we find that Ebola cases will continue to increase
dramatically if no control is made at early stage, but
infection can be control if prevention is made such as
therapeutic agents. When drug application is optimized,
infection will cease, even decline. Also, the infection will
spread only when the number of infected cases is over a certain
threshold. That's why Ebola transmit more rapidly in those
densely-populated areas, where prevention is poorly managed and
excluding rate is high. Reversely, Ebola can be better
controlled in less populated areas with a lower infection rate.
In terms of drug dosage, we must make sure sufficient medicine
supply for early infected patients but avoid possible wastes.
The dosage of most severe cases is used as the base and
multiplied by the number of patients. The calculated result is
the best dose.
\par We also studied the best drug distribution transport
systems. The purpose is to guarantee the amount of medicine for
each infected areas and fast acquisition of the medicine. Given
that the needed amount and production volume is constant, we
calculate the minimum time for medicine production and the
amount of time for distribution. This is the best distribution
plan that can save more time and lives. Regarding the
production rate, the amount of needed medicine divided the time
spent in production is the best production rate.
Mentioned above is the result of our study. We hope that the
World Medical Association will seriously consider and adopt the results of our study, and
show an announcement on better optimize the use of drugs in
order to do a better job in the fight against Ebola!
\par Our team wishes Ebola patients can recover rapidly and
people all around the world are happy and healthy!\\\nline%
\rightline{Team \#38403}
\rightline{February 9, 2015}
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{bib1}Ebola virus, Baidu Encyclopedia,
\url{http://baike.baidu.com/subview/188032/5072671.htm}
\bibitem{bib2}Wei Li, On Mathematical Model of the Spread of
SARS Virus, Journal of Bijie Teachers College (volume 22,
issue 2, June 2004)
\bibitem{bib3}Jiang Hua,Pan Haixia,\ldots, Simulating the Ebola
virus disease transmission and outbreak in China by using computational
epidemiological model, Chinese Journal of emergency
medicine(volume 23, issue 9, September 2014)
\bibitem{bib4}Xueqin Wu,Linear Programming in logistics and
transportand application of mathematical models, Journal of
Jiangxi Vocational and Technical College of Electricity(volume
20, issue 1, March 2007)
\bibitem{bib5}The latest data of deaths from Ebola,
\url{http://snap.windin.com/ns/findsnap.php?id=232970537}
\bibitem{bib6}Shaohui Zhang, The mathematical model, Science
Press(August 2010)
\end{thebibliography}
\section*{Appendix}
\input{appendix}
\end{document}
