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\title{\large {\bf{Using Adomian Decomposition Method for Solving Vector-Host Model}}
\author {\bf \small Ibrahim M. ELmojtaba\footnote{Email address:
elmojtaba@squ.edu.om}\\ \small{College of
Sciences, Sultan Qaboos University} \\ \small{P.O.Box 50, Muscat, Oman}\\ \\\small{Faculty of
Mathematical Sciences, University of Khartoum} \\ \small{P.O.Box 321, Khartoum, Sudan}\\ \\
}}
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\begin{center}
\section*{\small {\bf Abstract}}
\end{center}
\small{  ~  In this paper, we use Adomian decomposition method (ADM)
for solving vector-host model by using the alternate algorithm suggested by Biazar
\textit{et. al} \cite{Biazar2}. Some of the first terms were generated and plotted
against time and compared our results with the
regular Runge-Kutta numerical methods by using Matlab \verb"ode45" function.}\\
{\bf Keywords:} Adomian decomposition method, vector-host model, ODE
solvers, numerical simulation, stability analysis.
\section{Introduction}
  ~ The vector-host model is a mathematical model (framework) for the spread of
  a disease that transmits from human to another human throw another
  carrier (vector). To formulate this model we consider the dynamics
  of the disease into two different populations, human population
  and vector population. We assumed that the human population is
  divided into three different subgroups, susceptible $s_h(t)$,
  infected (and infectious) $i_h(t)$ and recovered $r_h(t)$, and
  the vector population into two subgroups susceptible $s_v(t)$
  and infected $i_v(t)$. It is assumed that susceptible individuals
  acquire infection following contacts with infected vectors at a per
  capita rate $a b\, i_v(t)$, where $a$ is the per capita
  biting rate of vectors on humans, and $b$ is the
  transmission probability per bite per human (as the case for
  malaria, \cite{Mac,Ross}). The per capita biting rate of vectors
  $a$ is equal to the number of bites received per human from
  vectors due to conservation of bites mechanism \cite{Gumel1,Zindoga}.
  Infected humans recover and acquire permanent immunity at an average
  rate $\beta$.
  ~ Susceptible vectors are acquire leishmaniasis infection following
  contacts with infected human at an average rate equal to $a c\, i_h(t)$,
  where $a$ is the per capita biting rate, and $c$ is the transmission
  probability for vector infection.
  ~ It is also assumed that there is no demographic effects on the
  model. Then our model is given by
  \begin{eqnarray}
  s_h^\prime & = & - a b\, s_h\,i_v\nonumber\\
  i_h^\prime & = & a b\, s_h\,i_v - \beta\,i_h \nonumber\\
  r_h^\prime & = & \beta\,i_h \label{Eq1}\\
  s_v^\prime & = & - a c\, s_v\,i_h\nonumber\\
  i_h^\prime & = & a c\, s_v\,i_h \nonumber
  \end{eqnarray}
  with initial conditions:
 \center{$s_h(0) = N_1,\quad i_h(0) = N_2,\quad r_h(0) = N_3,\quad s_v(0) = N_4,\quad
 i_v(0) = N_5$.}\flushleft
\section{Solving system (\ref{Eq1}) by Adomian decomposition method (ADM)}
  ~ Adomian decomposition method (ADM) (see \cite{ADOM1,ADOM2}),
  considers $s_h, i_h, r_h, s_v$ and $i_v$ as the sums of the
  following series:
  \begin{displaymath}
  s_h = \sum_{i=0}^{\infty} s_h^{i}, \quad i_h = \sum_{i=0}^{\infty}
  i_h^{i}, \quad r_h = \sum_{i=0}^{\infty} r_h^{i}, \quad s_v = \sum_{i=0}^{\infty}
  s_v^{i}, \quad i_v = \sum_{i=0}^{\infty} i_v^{i}
  \end{displaymath}
  By applying inverse of the operator $\frac{d(.)}{dt}$, which is
  the integration operator $\int_{0}^{t}(.) dt$ to each equation in
  the system (\ref{Eq1}) we have
  \begin{eqnarray}
  s_h(t) & = & s_h(t=0) - a\,b \int_{0}^{t}s_h(t)\,i_v(t)
  \, dt \nonumber\\
  i_h(t) & = & i_h(t=0) + a\,b \int_{0}^{t}\big(s_h(t)\,i_v(t) -
  \beta\, i_h(t) \big)\,dt \nonumber\\
  r_h(t) & = & r_h(t=0) + \beta \int_{0}^{t} i_h(t)\,dt \label{Eq2}\\
  s_v(t) & = & s_v(t=0) - a\,c \int_{0}^{t}s_v(t)\,i_h(t)
  \,dt \nonumber\\
  i_v(t) & = & i_v(t=0) + a\,c \int_{0}^{t}s_v(t)\,i_h(t)\,dt \nonumber
  \end{eqnarray}
  Using the alternate method for computing Adomian polynomials
  suggested by Biazar {\it et. al} \cite{Biazar2}, and
  substituting the initial conditions, we would have the following
  scheme
  \begin{eqnarray}
  s_h(t) & = & N_1 - a\,b \int_{0}^{t} \sum_{i=0}^{n}
  s_h^{(i)}(t)\,i_v^{(n-i)}(t)\, dt \nonumber\\
  i_h(t) & = & N_2 + a\,b \int_{0}^{t}\sum_{i=0}^{n}\big(s_h^{(i)}(t)\,i_v^{(n-i)}(t) -
  \beta\, i_h^{(n)}(t) \big)\,dt \nonumber\\
  r_h(t) & = & N_3 + \beta \int_{0}^{t} i_h^{(n)}(t)\,dt \label{Eq2}\\
  s_v(t) & = & N_4 - a\,c \int_{0}^{t} \sum_{i=0}^{n} s_v^{(i)}(t)\,i_h^{(n-i)}(t)
  \,dt \nonumber\\
  i_v(t) & = & N_5 + a\,c \int_{0}^{t} \sum_{i=0}^{n} s_v^{(i)}(t)\,i_h^{(n-i)}(t)
  \,dt\nonumber
  \end{eqnarray}
  From the above method we can calculate some first few terms
  \begin{eqnarray}
  s_h^{(1)} & = & -a\, b\,N_1\,N_5\, t\nonumber\\
  i_h^{(1)} & = & (a\, b\,N_1\,N_5 - \beta\, N_2)\, t\nonumber\\
  r_h^{(1)} & = & \beta\, N_2\, t\nonumber\\
  s_v^{(1)} & = & -a\, c\,N_4\,N_2\, t\nonumber\\
  i_h^{(1)} & = & a\, c\,N_4\,N_2\, t\nonumber\\
  s_h^{(2)} & = & -\frac{1}{2}\,a\,b \Big[N_1 (a\,c\, N_2\, N_4) - a\,b\,N_1\, N_5^{2}
  \Big]\, t^{2}\nonumber\\
  i_h^{(2)} & = & \frac{1}{2}\,a\,b \Big[N_1 (a\,c\, N_2\, N_4) - a\,b\,N_1\,
  N_5^{2} - \beta(a\,b\,N_1\,N_5 - \beta\,N_2) \Big]\, t^{2}\nonumber\\
  r_h^{(2)} & = & \frac{1}{2}\,\beta\Big[a\,b\,N_1\,N_5 - \beta\,N_2\Big]\,
  t^{2}\nonumber\\
  s_v^{(2)} & = & -\frac{1}{2}\,a\,c \Big[N_4 (a\,b\,N_1\,N_5 - \beta\,N_2) -
  a\,c\,N_2^{2}\,N_4\Big]\,t^{2}\nonumber\\
  i_v^{(2)} & = & \frac{1}{2}\,a\,c \Big[N_4 (a\,b\,N_1\,N_5 - \beta\,N_2) -
  a\,c\,N_2^{2}\,N_4\Big]\,t^{2}\nonumber\\
  \vdots\nonumber
  \end{eqnarray}
\section{Convergence of the method}
  ~ Since after the first step, applying the inverse operator $\int_{0}^{t}
  (.)dt$, we drive a system of Volterra integral equations of second
  kind, and the convergence of these systems is discussed in
  \cite{Biazar1}.
\section{Numerical simulation and discussion}
  ~ In this section we give numerical simulation for our model using (ADM) and the
  regular Runge-Kutta numerical method by applying
  Matlab$^{\circledR}$ \verb"ode45" function, and then we compare between the results.\\
  ~  The parameters values used are in Table \ref{table1}.
    \begin{table}[!h]
 \center\begin{tabular}[!h]{|l|l|c|}
 \hline
 parameter & parameter description & value\\
 \hline \hline
 $N_1$ & Initial value of population $s_h(t)$, susceptible individuals & 100\\
 $N_2$ & Initial value of population $i_h(t)$, infected individuals & 6\\
 $N_3$ & Initial value of population $r_h(t)$, recovered individuals & 1\\
 $N_4$ & Initial value of population $s_v(t)$, susceptible vectors & 80\\
 $N_5$ & Initial value of population $i_v(t)$, infected vectors & 12\\
 a & Biting rate of vectors & 0.01\\
 b & Progression rate of the disease in the vector & 0.2\\
 c & Progression rate of the disease in human & 0.2\\
 $\beta$ & Human recovery rate & 0.3\\
 \hline
 \end{tabular}
\caption{\small{Parameter values for the model
simulation}}\label{table1}
\end{table}\\
  ~ We calculate three and four terms approximations for the variables
  are calculated and presented below.\\
  \flushleft Three terms approximation:
  \begin{eqnarray}
  s_h^{(3)} & = & 100 - 2.4\,t - 0.0672\,t^{2} - 0.0007\,t^{3} \nonumber\\
  i_h^{(3)} & = & 6 - 0.6\,t - 0.0228\,t^2 + 0.003\,t^{3} \nonumber\\
  r_h^{(3)} & = & 1 + 1.8\,t + 0.09\,t^{2} - 0.0023\,t^{3}\nonumber\\
  s_v^{(3)} & = & 80 - 0.96\,t - 0.0422\,t^{2} + 0.0018\,t^{3} \nonumber\\
  i_h^{(3)} & = & 12 + 0.96\,t + 0.0422\,t^{2} - 0.0018\,t^{3} \nonumber
  \end{eqnarray}
  \newpage
  Four terms approximation:
  \begin{eqnarray}
  s_h^{(4)} & = & 100 - 2.4\,t - 0.0672\,t^{2} - 0.0007\,t^{3} + 0.0002\,t^{4}\nonumber\\
  i_h^{(4)} & = & 6 - 0.6\,t - 0.0228\,t^2 + 0.003\,t^{3} - 0.0004\,t^{4} \nonumber\\
  r_h^{(4)} & = & 1 + 1.8\,t + 0.09\,t^{2} - 0.0023\,t^{3} + 0.0002\,t^{4}\nonumber\\
  s_v^{(4)} & = & 80 - 0.96\,t - 0.0422\,t^{2} + 0.0018\,t^{3} - 0.0001\,t^{4}\nonumber\\
  i_h^{(4)} & = & 12 + 0.96\,t + 0.0422\,t^{2} - 0.0018\,t^{3} + 0.0001\,t^{4}\nonumber
  \end{eqnarray}
\center\begin{figure}[h!]
 \center \includegraphics[width=4.2in,height=4.2in]{Adomian3terms.eps}
\caption[Simulation results using three terms
approximation]{Simulation results using three terms
approximation}\label{Fig1}
\end{figure}

\center\begin{figure}[p] \vspace{-2cm} \center
 \includegraphics[width=4.2in,height=4.2in]{Adomian4terms.eps}
\caption[Simulation results using four terms
approximation]{Simulation results using four terms
approximation}\label{Fig2}
\end{figure}

\center\begin{figure}[p]
 \center \includegraphics[width=4.2in,height=4.2in]{ode.eps}
\caption[Simulation results using ode45]{Simulation results using
ode45}\label{Fig3}
\end{figure}
\flushleft
  ~  We noticed that the three terms approximation of Adomian
  decomposition method is very similar to the simulation results
  generated using Matlab$^{\circledR}$ \verb"Ode45" function, which
  is reasonable compared to reality because it is clear that the number of susceptible (humans and vectors) decrease as the number of infected (humans and
  vectors) increase, and the number of recovered humans increases, as seen
  from Figures \ref{Fig1},\ref{Fig3}. However, as seen from Figure \ref{Fig2},
  using four terms approximation we found that the number of susceptible humans
  decrease first and then increase again, which coincide with reality, and this case needs further investigation, and it may happens due to the use of alternate method for computing Adomian polynomials.
\newpage
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