\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.\newline
{\em Electronic Journal of Differential Equations},
Conference 12, 2005, pp. 87--101.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{7mm}}
\setcounter{page}{87}
\begin{document}
\title[\hfilneg EJDE/Conf/12 \hfil Using Lie symmetries in epidemiology]
{Using Lie symmetries in epidemiology}
\author[M. C. Nucci \hfil EJDE/Conf/12 \hfilneg]
{Maria Clara Nucci}
\address{Dipartimento di Matematica e Informatica,
Universit\`a di Perugia,
06123 Perugia, Italy}
\email{nucci@unipg.it}
\date{}
\thanks{Published April 20, 2005.}
\subjclass[2000]{34A05, 34C14, 92D30}
\keywords{Lie symmetries; mathematical epidemiology}
\begin{abstract}
Lie symmetry method has been and still is successfully applied in
different problems of physics for about a hundred years, but its
application in epidemiology has been rare perhaps because
the ordinary differential equations studied in this field are
generally of first-order in contrast with those in physics
which are usually of second-order.
Here we exemplify the use of Lie symmetry method in the study of
mathematical models in epidemiology, and show how it complements
the mathematical techniques (qualitative and numerical analysis)
traditionally used.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\section{Introduction}
In January 2001, the first Whiteman prize for notable exposition on
the history of mathematics was awarded to Thomas Hawkins by the
American Mathematical Society. In the citation, published in the
Notices of AMS {\bf 48} 416-417 (2001), one reads that Thomas
Hawkins ``\dots has written extensively on the history of Lie
groups. In particular he has traced their origins to [Lie's] work
in the 1870s on differential equations $\ldots$ the {\em id\'ee
fixe} guiding Lie's work was the development of a Galois theory of
differential equations \dots [Hawkins's book \cite{Hawkins2}]
highlights the fascinating interaction of geometry, analysis,
mathematical physics, algebra and topology \dots". Also Hawkins
had established ``the nature and extent of Jacobi's influence upon
Lie" \cite{Hawkins1}. This is particularly noteworthy since 2004
marks two hundred years since Jacobi's birth. ``Given the fact
that the Jacobi Identity is fundamental to the theory of Lie
groups, Jacobi's influence upon Lie will come as no surprise. But
the bald fact that he inherited the Identity from Jacobi fails to
convey fully or accurately the historical dimension of the impact
of Jacobi's work on partial differential equations"
\cite{Hawkins1}.
In the Introduction of his
book \cite{Olver} Olver wrote that ``it is impossible to
overestimate the importance of Lie's contribution to modern
science and mathematics. Nevertheless anyone who is already
familiar with [it] \dots is perhaps surprised to know that its
original inspirational source was the field of differential
equations".
Lie's monumental work on transformation groups,
\cite{Lie 1}, \cite{Lie 2} and \cite{Lie 3}, and in particular
contact transformations \cite{Lie 4}, led him to achieve his goal
\cite{Lie 5}.
Many books have been dedicated to this subject
and its generalizations \cite{Ame72,BluC74,Ovs,
Olver, BluK, RogA, Ste, Hil, crc1, crc2, ibra99, Hydon, BluA}.
Lie group analysis is indeed the most powerful tool to find the
general solution of ordinary differential equations. Any known
integration technique
(We mean those taught in most
undergraduate courses on ordinary differential equations.) can be
shown to be a particular case of a general integration method
based on the derivation of the continuous group of symmetries
admitted by the differential equation, i.e. the Lie symmetry
algebra, which can be easily derived by a straightforward although
lengthy procedure. As computer algebra software becomes widely
used, the integration of systems of ordinary differential
equations by means of Lie group analysis is becoming easier to
perform. A major drawback of Lie's method is that it is useless
when applied to systems of $n$ first-order equations\footnote{Any
undergraduate science/engineering student knows that a $n$-order
ordinary differential equation can be transformed into an
equivalent system of $n$ first-order equations. Less well-known to
students but common knowledge among experts in Lie group analysis
is the dramatic consequence that that transformation has on the
dimension of the admitted Lie symmetry algebra. In fact while the
maximum Lie symmetry algebra admitted by a single $n$-order
equation is finite \cite{gasgon} the dimension of the Lie symmetry
algebra admitted by a system of $n$ first-order equations is
infinite.}, because they admit an infinite number of symmetries,
and there is no systematic way to find even an one-dimensional Lie
symmetry algebra, apart from trivial groups like translations in
time admitted by autonomous systems. One may try to derive an
admitted $n$-dimensional solvable Lie symmetry algebra by making
an ansatz on the form of its generators but when successful
(rarely) it is just a lucky guess.
However, in \cite{kepler} we have remarked that any system of
$n$ first-order equations could be transformed into an equivalent
system where at least one of the equations is of second-order.
Then the admitted Lie symmetry algebra is no longer
infinite-dimensional, and nontrivial symmetries of the original
system could be retrieved \cite{kepler}. This idea has been
successfully applied in several instances. In \cite{kepler} it was
shown that Krause's symmetries \cite{Krause 94 a} for the Kepler
problem are actually Lie symmetries, and in \cite{harmony} how to
derive the harmonic oscillator from the Kepler problem by using
Lie symmetries. The Kepler problem and MICZ-Kepler problem were
also shown to be equivalent to an isotropic two-dimensional system
of linear harmonic oscillators in \cite{MICZ} thanks to Lie
symmetries. In \cite{marcelnuc} Lie group analysis -- when applied
to Euler-Poisson equations as obtained from the reduction method
\cite{kepler} -- unveiled the Kowalevski top \cite{Kowalevski} and
its peculiar integral without making use of
either Noether's theorem \cite{Noether} or the Painlev\'e method
\cite{Kowalevski}. In \cite{lorpoin} Lie group analysis related the
famous Lorenz system \cite{Lorenz} to the Euler equations of a
rigid body moving about a fixed point and subjected to a torsion
depending on time and angular velocity, namely Lie group analysis
transformed the ``butterfly" into a ``tornado". In
\cite{goldfish} a solvable many-body problem introduced by
Calogero \cite{calo01} was shown to be intrinsically linear by
means of Lie symmetries. In \cite{fc70} a three-body problem
derived and solved up to a quadrature by Jacobi \cite{Jacobic} was
shown to be reducible to the equation of motion of a single free
particle on the line.
Lie group analysis is successfully applied in
different problems of physics (and has been for about a hundred
years), but rarely in biology (or epidemiology) maybe because the
ordinary differential equations studied in these fields are
generally of first-order in contrast with those in physics which
are usually of second-order. Yet when Lie group analysis is
successfully applied to epidemiological models then several
instances of integrability even linearity are found which lead to
the general solution of the model. Thus the dynamics of epidemics
can be exactly described. The purpose of this paper is to promote
the use of Lie symmetry method among bio-mathematical
practitioners. We present three examples \cite{valenu},
\cite{core}, \cite{flu} where Lie symmetries
have been found, and the general solution of the
epidemiological model consequently derived whenever appropriate
conditions among the involved parameters are satisfied. In
\cite{SIS} and \cite{jlm05} one can find an instance where Lie
group analysis leads to the general solution of a SIS model
formulated in \cite{Brauer 02} without any condition on the
involved parameters. Moreover each example epitomizes a different
situation, i.e. hidden linearity of the model, an anomalous
behavior in the dynamics of the infectives, and a general periodic
solution in apparent contrast with prediction by qualitative
analysis, respectively.
In section 2 we show
that for a certain relationship among the involved parameters, Lie
group analysis unveils the hidden linearity \cite{valenu} of a
seminal model given by Anderson, which describes HIV
transmission in male homosexual/bisexual cohorts \cite{ruoland}.
In section 3 we show that for an appropriate relationship among
the involved parameters, Lie group analysis leads to the general
solution of a core group model for sexually transmitted disease
formulated by Hadeler and Castillo-Chavez \cite{HadC}, and gives a
deeper insight on the strange behavior of the number of infectives
\cite{core}. In section 4 we show that for a certain relationship
among the involved parameters Lie group analysis, when applied to
a SIRI disease transmission model formulated by Derrick and van
den Driessche \cite{derrick}, leads to a periodic general solution
\cite{flu} in apparent contrast to the qualitative analysis
performed in \cite{derrick}. In section 5 we conclude with some
final remarks.
\section{An HIV-transmission model}
In \cite{valenu}, Lie group analysis was applied to a seminal
model formulated by Anderson, which describes HIV transmission in
male homosexual/bisexual cohorts \cite{ruoland}. This
compartmental model divides the population at time $t$ into
susceptibles (HIV negatives), infecteds (HIV positives), and AIDS
patients, represented by $u_{1}(t)$, $u_{2}(t)$, and $u_{3}(t)$,
respectively. HIV infecteds are individuals who test positive for
specific antibodies to the virus \cite{Andbk}. AIDS patients are
persons exhibiting characteristic clinical manifestations of
full-blown AIDS, the end-stage of the disease \cite{Murray}. In
this model, the population is not subject to recruitment and
individuals are removed only by death. An individual may belong to
only one compartment at any specified time. However, individuals
move from one compartment to the next according to the following
flow diagram:
\begin{center}
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\linethickness{0.4pt}
\begin{picture}(70.00,110.00)(45,5)
\put(20.00,90.00){\framebox(30.00,20.00)[cc]{$u_{1}(t)$}}
\put(20.00,50.00){\framebox(30.00,20.00)[cc]{$u_{2}(t)$}}
\put(20.00,10.00){\framebox(30.00,20.00)[cc]{$u_{3}(t)$}}
\put(35.00,90.00){\vector(0,-1){20.00}}
\put(35.00,50.00){\vector(0,-1){20.00}}
\put(50.00,100.00){\vector(1,0){20.00}}
\put(50.00,60.00){\vector(1,0){20.00}}
\put(50.00,20.00){\vector(1,0){20.00}}
\put(70.00,100.00){\makebox(0,0)[lc]{Natural death}}
\put(70.00,60.00){\makebox(0,0)[lc]{Natural death}}
\put(70.00,20.00){\makebox(0,0)[lc]{AIDS-related death}}
\put(60.00,105.00){\makebox(0,0)[cc]{$\mu$}}
\put(60.00,65.00){\makebox(0,0)[cb]{$\mu $}}
\put(60.00,25.00){\makebox(0,0)[cc]{$\alpha $}}
\put(40.00,40.00){\makebox(0,0)[lc]{$\nu $}}
\put(40.00,80.00){\makebox(0,0)[lc]{$\lambda $}}
\end{picture}
\end{center}
The parameter $\mu$ is the per capita natural death rate (non-AIDS
related) of both susceptibles and infecteds, and $\alpha $ is the
AIDS-related death rate. The term $\lambda $ is the per capita
force of infection and is defined as:
$$
\lambda =\frac{ \beta c u_{2}(t)}{u_{1}(t)+u_{2}(t)+u_{3}(t)},
$$
where $\beta $ is the average probability that an infected
individual will infect a susceptible partner over the duration of
their relationship \cite{ruoland}, \cite{Grant}, and $c$ is the
effective rate of partner change within the specified risk
category \cite{ruoland}.
In the model, all infecteds are supposed to develop AIDS with an
average
incubation period $1/\nu $ \cite{ruoland}, \cite{Murray}.
The system of nonlinear ordinary differential equations derived
from this model is
\begin{gather}
\frac{{\rm d}u_{1}}{{\rm d}t}= \frac{-\beta c u_{1} u_{2}}{u_{1}+u_{2}+u_{3}} -\mu u_{1}\label{1} \\
\frac{{\rm d}u_{2}}{{\rm d}t}=
\frac{\beta c u_{1} u_{2}}{u_{1}+u_{2}+u_{3}} -(\nu
+\mu)u_{2}\label{2}\\
\frac{{\rm d}u_{3}}{{\rm d}t}= \nu u_{2} - \alpha u_{3}.
\label{3}
\end{gather}
We can easily transform this system into a system of one
equation of second order in $u_1$, and one of first order in
$u_2$. Indeed, if we derive $u_3$ from (\ref{1}), i.e.:
\begin{equation}
u_3 = \frac{- {\frac{{\rm d}u_1}{{\rm d}t}} (u_2-u_1)
+ \beta c u_1 u_2 + \mu u_1^2 + \mu u_1 u_2}{
{\frac{{\rm d}u_1}{{\rm d}t}} + \mu u_1}
\end{equation}
then we obtain the following system in $u_1$ and $u_2$:
\begin{equation}\begin{split}
\frac{{\rm d}^2 u_{1}}{{\rm d} t^2}
=&\Big[\alpha \beta c \mu u_1^2 u_2 +
\alpha \beta c u_1 \frac{{\rm d}u_1}{{\rm d}t} u_2 + \alpha \mu^2 u_1^3
+ \alpha\mu^2 u_1^2 u_2\\
&+ 2 \alpha \mu u_1^2 \frac{{\rm
d}u_1}{{\rm d}t} + 2 \alpha \mu u_1 \frac{{\rm d}u_1}{{\rm d}t}
u_2 + \alpha u_1 \big(\frac{{\rm d}u_1}{{\rm d}t}\big)^ 2 +
\alpha \big(\frac{{\rm d}u_1}{{\rm d}t}\big)^2 u_2 \\
& - \beta c \mu^2 u_1^3 - \beta c \mu^2 u_1^2 u_2 - \beta c \mu \nu
u_1^2 u_2
- 2 \beta c \mu u_1^2 \frac{{\rm d}u_1}{{\rm d}t}\\
& - \beta c \mu u_1 \frac{{\rm d}u_1}{{\rm d}t} u_2 - \beta
c \nu\, u_1
\frac{{\rm d}u_1}{{\rm d}t} u_2
- \beta c u_1 \big(\frac{{\rm d}u_1}{{\rm d}t}\big)^2
+ \beta c \big(\frac{{\rm d}u_1}{{\rm d}t}\big)^2 u_2 \\
& - \mu^3 u_1^3 -
\mu^3 u_1^2 u_2 - 2 \mu^2 u_1^2 \frac{{\rm d}u_1}{{\rm d}t}
- 2 \mu^2 u_1
\frac{{\rm d}u_1}{{\rm d}t} u_2\\
& - \mu u_1 \big(\frac{{\rm d}u_1}{{\rm d}t}\big)^2 - \mu \big(\frac{{\rm
d}u_1}{{\rm d}t}\big)^2 u_2\Big]/(\beta c u_1 u_2)
\label{u1r}
\end{split}\end{equation}
\begin{equation}
\frac{{\rm d}u_{2}}{{\rm d} t}= -\left(\mu u_1+\mu u_2 +\nu u_2
+\frac{{\rm d}u_1}{{\rm d}t}\right) \label{u2r}
\end{equation}
When Lie group analysis of this system is performed using
\cite{man2}, a linear partial
differential equation of parabolic structure is obtained. Its
characteristic curve is given by $u_1+u_2$. Consequently, we
introduce the new dependent variable:
\begin{equation}
v_2 = u_1+u_2
\end{equation}
to obtain a new system, which in the case $\alpha=\mu +\beta c$
admits an eight-dimensional Lie symmetry algebra. Actually, it
becomes separable, i.e.:
\begin{equation}\begin{split}
\frac{{\rm d}^2u_1}{{\rm d}t^2}
= &\big[\beta c \mu u_1^2 + \beta
c u_1 \frac{{\rm d}u_1}{{\rm d}t} + \mu^2 u_1^2 - \mu\, \nu u_1^2
+ 2 \mu u_1 \frac{{\rm d}u_1}{{\rm d}t} -\\
&- \nu u_1
\frac{{\rm d}u_1}{{\rm d} t} + 2 (\frac{{\rm d}u_1}{{\rm d}
t})^2 \big]/u_1\label{u1rr}\end{split}
\end{equation}
\begin{equation}\frac{{\rm d}v_2}{{\rm
d}t}=- (\mu + \nu) v_2 + \nu u_1
\end{equation}
Therefore, equation (\ref{u1rr}) is linearizable by means of a
point transformation \cite{Lie 5} because it admits an
eight-dimensional Lie symmetry algebra generated by the following
eight operators:
\begin{gather*}
X_1=e^{-(\beta c+\mu-\nu)t} \big(\frac{1}{u_1}\,\partial_t-\mu
\,\partial_{u_1}\big), \quad X_2 = e^{-\mu
t}\big(\frac{1}{u_1}\,\partial_t-(\beta
c+\mu-\nu)\,\partial_{u_1}\big),\\
X_3=e^{(\beta c+\mu-\nu)t}u_1^2\,\partial_{u_1}, \quad
X_4=e^{\mu t}u_1^2\,\partial_{u_1},\quad X_5= u_1\,\partial_{u_1},
\quad X_6= \partial_t,\\
X_7= e^{(\beta c-\nu)t}\left(-\partial_t+(\beta
c+\mu-\nu)u_1\,\partial_{u_1}\right),\quad
X_8=-e^{(\beta c-\nu)t}\left(-\partial_t+\mu
u_1\,\partial_{u_1}\right),
\end{gather*}
if $\beta c \neq \nu$, or
\begin{gather*}
\hat X_1=e^{-\mu t} \big(\frac{1}{u_1}\,\partial_t-\mu
\,\partial_{u_1}\big), \quad \hat X_2 = e^{-\mu
t}\big(\frac{t}{u_1}\,\partial_t-(\mu t +1)\,\partial_{u_1}\big), \\
\hat X_3=e^{\mu t} u_1^2\,\partial_{u_1}, \quad \hat X_4=e^{\mu t}
t u_1^2\,\partial_{u_1},\quad \hat X_5= u_1\,\partial_{u_1}, \quad
\hat X_6= \partial_t, \\
\hat X_7= t\left(t\,\partial_t-(\mu
t+1)u_1\,\partial_{u_1}\right),\quad \hat
X_8=t\left(\partial_t-\mu u_1\,\partial_{u_1}\right),
\end{gather*}
if $\beta c=\nu$.
To find the linearizing
transformation we have to look for a two-dimensional abelian
intransitive subalgebra, and, following Lie's classification of
two-dimensional algebras in the real plane \cite{Lie 5}, we have
to transform it into the canonical form
\begin{equation}
\partial_{\tilde u},\quad \tilde x\partial_{\tilde u}\label{canvar}
\end{equation}
with $\bar u$ and $\bar t$ the new dependent
and independent variables, respectively. We find that one such subalgebra
is that generated by $X_3$
and $X_4$, if $\beta c \neq \nu$, or $\hat X_3$ and $\hat X_4$, if
$\beta c=\nu$. Then it is easy to derive that the transformation
which changes (\ref{u1rr}) into a linear ordinary differential
equations is either:
\begin{equation}
\bar t=e^{(\nu-\beta c)t}, \quad
\bar u=-\frac{e^{(\nu-\beta c-\mu)t}}{u_1}
\end{equation}
if $\beta c \neq \nu$, or
\begin{equation}
\bar t=t, \quad \bar u=-\frac{e^{-\mu t}}{u_1}
\end{equation}
if $\beta c=\nu$.
Thus, equation (\ref{u1rr}) becomes:
\begin{equation}
\frac{{\rm d}^2\bar u}{{\rm d}\bar t^2}=0 \label{ubareq}
\end{equation}
and its general solution is trivially\footnote{Here $a_1, a_2$ are arbitrary
constants.}
\begin{equation}
\bar u=a_1 \bar t+a_2,
\end{equation}
which yields the following general solution of system
(\ref{1})-(\ref{3}):
\begin{equation}
u_{1}=\frac{e^{\nu t} c_{2}}{e^{\mu t} \left[e^{\nu t}
\left(\beta\,c-\nu \right) c_{1}+e^{\beta\,c\: t}
\beta\,c\:\right]}, \label{u1} \end{equation}\begin{equation}
u_{2}=\frac{\left(\beta\,c-\nu\right) {
\int}{ \frac{e^{\beta\,c\: t+2 \nu t}}{
\left(e^{\beta\,c\: t } \beta\,c\: +e^{\nu t} \beta\,c\:
c_{1}-e^{\nu t} c_{1} \nu\right)^{2}}} \,dt\ \beta\,c \:
c_{2}+c_{3}}{e^{\mu t+\nu t}}, \label{u2}
\end{equation}
\begin{equation} \label{u3}
\begin{aligned}
u_{3}&=\frac{\left[e^{\nu t}
\left(\beta\,c-\nu\right) c_{1} +e^{\beta\,c\: t} \nu\right] c_3}{
e^{ \beta\,c\: t+\mu t+\nu t} \left(\beta\,c-\nu\right)} +
\frac{-e^{\nu t} \: c_{2}}{e^{ \mu t} \left [e^{\nu t}
\left(\beta\,c-\nu\right) c_{1}+e^{\beta\,c\: t} \beta\,c\:\right
]}\\
&\quad +\frac{ \beta\,c\: c_{2} \left[e^{\nu t} \left(\beta\,c-\nu\right)
c_{1}+e^{\beta\,c\: t} \nu \right] {
\int}{ \frac{e^{\beta\,c\: t+2 \nu t}}{
\left(e^{\beta\,c\: t } \beta\,c\: +e^{\nu t} \beta\,c\:
c_{1}-e^{\nu t} c_{1} \nu\right)^{2}}} \,dt}{e^{ \beta\,c\:
t+\mu t+\nu t} }.
\end{aligned}
\end{equation}
where $c_1,c_2,c_{3}$ are arbitrary constants. If $\beta c=2\nu $,
the general solution assumes a simpler form:
\begin{equation}
u_{1}=\frac{c_{1}}{e^{\mu t} \left(2 e^{\nu t}+c_{1}
c_{2}\right)}, \label{u1bis} \end{equation}\begin{equation}
\begin{split}u_{2}=&\left[2 e^{\nu t} \log \left(2 e^{\nu t}+c_{1} c_{2}\right)
c_{1}-2 e^{\nu t} c_{1}+4 e^{\nu t} c_{3}\right.+\\&+\left.\log
\left(2 e^{\nu t}+c_{1} c_{2}\right) c_{1}^{2} c_{2}+2 c_{1} c_{2}
c_{3}\right]/\left[ 2 e^{\mu t+\nu t} \left(2 e^{\nu t}+c_{1}
c_{2}\right)\right], \label{u2bis}
\end{split}\end{equation}
\begin{equation} \begin{split}
u_{3}=&\left[e^{\nu t} \log \left(2
e^{\nu t}+c_{1} c_{2}\right) c_{1}-2 e^{\nu t} c_{1}+2 e^{\nu t}
c_{3}\right.+\\&+\left.\log \left(2 e^{\nu t}+c_{1} c_{2}\right)
c_{1}^{2} c_{2}+2 c_{1} c_{2} c_{3}\right]/\left[ 2 e^{\mu t+2 \nu
t}\right]. \label{u3bis}\end{split}
\end{equation}
In \cite{valenu} the
solution was tested on data from three U.S. epidemiologic studies,
and found to closely match observed epidemic data.
\section{A core group model}
In \cite{HadC} Hadeler and Castillo-Chavez presented a model for
sexually transmitted diseases which takes into consideration an
active and relatively small core group of constant size. The
core group recruits individuals from the non-core group, and the
rate of recruitment may depend on the state of the core group. The
non-core group is completely inactive. The total population has
size $P(t)$, and the non-core group has size $A$. The population
of the core group $C$ is further divided into susceptibles $S$,
educated (or vaccinated) $V$, and infecteds $I$. The birth rate is
$b>0$, the birth rate of infecteds is $\tilde b\leq b, \tilde
b\geq 0$, the death rate is $\mu>0$, the death rate of infecteds
is $\tilde{\mu}\geq\mu$, the recovery rate is $\alpha\geq 0$, the
education (vaccination) rate is $\psi\geq 0$, the transmission
rate from infecteds to susceptibles is $\beta\geq 0$, the
transmission rate from infecteds to educated (vaccinated) is
$\tilde{\beta}, 0\leq\tilde{\beta}\leq \beta$. At recovery
individuals may either pass into the educated class at the rate
$\alpha\gamma, 0\leq\gamma\leq 1$, or return to the susceptible
class at the rate $\alpha(1-\gamma)$. Recruitment into the core
group is described by a function $r(I,C)$. Hadeler and
Castillo-Chavez focused on the situation where the disease has no
demographic effects and population size is constant, i.e.
$P=$const, $b=\tilde b=\mu=\tilde \mu$. Thus their model assumes
the form
\begin{gather}
\dot A= \mu P -A r(I,C)-\mu A,\label{allA}\\
\dot S=A r(I,C)-\beta \frac{S I}{C}-\psi S+\alpha (1-\gamma) I-\mu S,\label{allS}\\
\dot V=\psi S-\tilde{\beta} \frac{V I}{C}+\alpha \gamma I-\mu V,\label{allV}\\
\dot I=\frac{\beta S I+\tilde\beta V I}{C}-\alpha I-\mu
I.\label{allI}
\end{gather}
where the overdot denotes differentiation with respect to $t$.
They point out that this system is closely related to a model for
an isolated population of constant size $C=1$, i.e.
\begin{gather}
\dot S=\mu-\beta S I-\psi S+\alpha (1-\gamma) I-\mu S,\label{coreS}\\
\dot V=\psi S-\tilde{\beta} V I+\alpha \gamma I-\mu V,\label{coreV}\\
\dot I=\beta S I+\tilde\beta V I-\alpha I-\mu I.\label{coreI}
\end{gather}
In \cite{HadC} the stationary solutions of system
(\ref{coreS})-(\ref{coreI}) are found and their qualitative
features discussed. Then the stationary solutions of system
(\ref{allA})-(\ref{allI}) are also discussed. Qualitative
conclusions are finally drawn. \\
In \cite{core} Lie group analysis is applied to system
(\ref{coreS})-(\ref{coreI}) in order to determine under which
physical conditions on the parameters Lie point symmetries exist
and, when
possible, deduce the general solution in closed form. Also in
\cite{core} a
discussion of
the solutions that have been found is presented to show
how Lie group analysis complements Hadeler and Castillo-Chavez's
qualitative analysis.
System (\ref{coreS})-(\ref{coreI}) is composed of three first
order ordinary differential equations which can be easily reduced
to two equations by using the following condition
\begin{equation}
C\equiv 1=S+V+I. \label{pop1}
\end{equation}
Then we can easily transform the system of two equations so
obtained into one equation of second order. We derive $I$ from
(\ref{pop1}), i.e.
\begin{equation}I=1-S-V,\label{Iw3}\end{equation}
and then deduce $V$ from equation
(\ref{coreS}), i.e.
\begin{equation}
V=\frac{\dot S - \mu S+\psi S -
\mu }{\alpha \gamma - \alpha + \beta S}-S+1.\label{Iw2}
\end{equation}
Consequently a second order equation for $S$ is
obtained. When we apply Lie group analysis to this
equation\footnote{We look for Lie operators of the
form $\Gamma=v(t,S)\partial_t
+G(t,S)\partial_{S}$.}
then we obtain a first-order linear partial differential
equation for $v(t,S)$; its characteristic curve suggests to make
the following simplifying transformation
\begin{equation}
S=\frac{ - \alpha \gamma + \alpha + u}{\beta},\label{w1r1}
\end{equation}
where $u(t)$ is the new
dependent variable. Then (\ref{Iw2}) transforms into
\begin{equation}
V=1+\frac{\dot u +\alpha(1-
\gamma)( \mu +\psi) + (\alpha \gamma
-\alpha+\mu+\psi-u)u - \beta \mu}{\beta u},
\end{equation}
and we have to study the following equation in $u=u(t)$:
\begin{equation}
\begin{aligned}
\ddot u =& (\alpha^2 \beta \gamma^2 \mu u + \alpha^2 \beta
\gamma^2 \psi u - \alpha^2 \beta \gamma \mu u
- \alpha^2 \beta \gamma \psi u + \alpha^2 \tilde \beta
\gamma^2 \mu^2 + 2 \alpha^2 \tilde \beta \gamma
^2 \mu \psi \\
&- \alpha^2 \tilde \beta \gamma^2 \mu u +
\alpha^2 \tilde \beta \gamma^2 \psi^2 - \alpha^2 \tilde \beta
\gamma^2 \psi u - 2 \alpha^2 \tilde \beta \gamma \mu^2 - 4
\alpha^2 \tilde \beta \gamma
\mu \psi + 2 \alpha^2 \tilde \beta \gamma \mu u \\
&- 2 \alpha^2 \tilde \beta \gamma \psi^2 + 2 \alpha
^2 \tilde \beta \gamma \psi u + \alpha^2 \tilde \beta \mu^2 + 2
\alpha^2 \tilde \beta \mu \psi - \alpha ^2 \tilde \beta \mu u +
\alpha^2 \tilde \beta \psi^2 \\
&- \alpha^2 \tilde \beta
\psi u + \alpha \beta^2 \gamma \mu u + 2 \alpha \beta \tilde \beta
\gamma \mu^2 + 2 \alpha \beta \tilde \beta \gamma \mu \psi - 2
\alpha
\beta \tilde \beta \gamma \mu u - \alpha \beta \tilde \beta \gamma \psi u\\
& - 2 \alpha \beta \tilde \beta \mu^2 - 2
\alpha \beta \tilde \beta \mu \psi + 2 \alpha \beta \tilde \beta
\mu u + \alpha \beta \tilde \beta \psi u + \alpha \beta
\gamma \mu^2 u \nonumber \\
&+ \alpha \beta \gamma \mu \psi u - 2 \alpha \beta \gamma \mu u^2 - \alpha \beta
\gamma \mu \dot u - 2 \alpha \beta \gamma \psi u^2 - \alpha \beta
\gamma \psi \dot u - \alpha \beta \gamma u
\dot u - \alpha \beta \mu^2 u \\
&- \alpha \beta \mu \psi u + \alpha \beta \mu u^2 + \alpha \beta
\mu \dot u + \alpha \beta \psi u^2 + \alpha \beta \psi \dot u - 2
\alpha \tilde \beta \gamma \mu^2 u -
4 \alpha \tilde \beta \gamma \mu \psi u \\
& + 2 \alpha \tilde \beta \gamma \mu u^2 - 2 \alpha \tilde \beta \gamma \mu
\dot u - 2 \alpha \tilde \beta \gamma \psi^2 u + 2 \alpha \tilde
\beta \gamma \psi u^2 - 2 \alpha \tilde \beta \gamma \psi \dot u +
\alpha \tilde \beta \gamma u \dot u \\
&+ 2 \alpha
\tilde \beta \mu^2 u + 4 \alpha \tilde \beta \mu \psi u - 2 \alpha
\tilde \beta \mu u^2 + 2 \alpha \tilde \beta \mu \dot u + 2 \alpha
\tilde \beta \psi^2 u \\
&- 2 \alpha \tilde \beta \psi u^2 + 2 \alpha \tilde \beta \psi \dot u
- \alpha \tilde \beta u \dot u + \beta^2
\tilde \beta \mu^2 \\
&- \beta^2 \tilde \beta \mu u +
\beta^2 \mu^2 u - \beta^2 \mu u^2 - \beta^2 \mu \dot u - 2 \beta
\tilde \beta \mu^2 u - 2 \beta \tilde \beta \mu \psi u \\
&+ 2 \beta \tilde \beta
\mu u^2 - 2 \beta \tilde \beta \mu \dot u + \beta \tilde \beta \psi u^2
+ \beta \tilde \beta u \dot u -
\beta \mu^2 u^2 - \beta \mu \psi u^2 + \beta \mu u^3 - \beta \mu u
\dot u \\
&+ \beta \psi u^3 + \beta u^2 \dot u + \beta \dot u^2
+ \tilde \beta \mu^2 u^2 + 2 \tilde \beta \mu
\psi u^2 - \tilde \beta \mu u^3 + 2 \tilde \beta \mu u \dot u \\
& + \tilde \beta \psi^2 u^2 -
\tilde \beta \psi u^3 + 2 \tilde \beta \psi u \dot u - \tilde
\beta u^2 \dot u + \tilde \beta \dot u^2)/(\beta u).
\end{aligned}\label{Iueq}
\end{equation}
In \cite{core} Lie group analysis
was applied to (\ref{Iueq}) and non-trivial Lie point symmetries
were obtained in five cases: in the first case
an eight-dimensional Lie symmetry algebra was
obtained, which means that equation (\ref{Iueq}) is
linearizable, while in the other four cases a two-dimensional Lie
symmetry algebra was found.
Here we present Case (5) of \cite{core} in order to show how to
get the general solution if a two-dimensional Lie
symmetry algebra is found and which new insights -- not detected
by qualitative analysis - on the dynamics of the epidemics can be
obtained from it. Case (5) corresponds to the following
relationship among the involved parameters:
$$
\tilde\beta=0,\;\;
\psi=\alpha \gamma,\;\;
\beta=\frac{\alpha(1-\gamma)(\alpha\gamma+\mu)}{\mu}
$$
Then (\ref{Iueq}) admits a two-dimensional Lie algebra
generated by the following operators:
\begin{equation}\Gamma_1=e^{(\mu + \alpha\gamma)
t}\Big(\partial_t-(\mu+\alpha\gamma)u\partial_u\Big),
\quad\Gamma_2=\partial_t \label{I5gen}
\end{equation}
A basis of the differential invariants of order $\leq 1$ for
operator $\Gamma_1$ in (\ref{I5gen}) is
\begin{equation}
\tilde t=u\, e^{(\mu+\alpha\gamma)t},\quad \tilde u=(\dot
u+u\mu+u\alpha\gamma)\,e^{2(\mu+\alpha\gamma)t},\label{I5tra}
\end{equation}
and therefore (\ref{Iueq}) becomes the first
order equation:
\begin{equation}
\frac{{\rm d}\tilde u}{{\rm d}\tilde t}=\frac{\tilde u + \tilde
t^2}{\tilde t},\label{I5uteq}
\end{equation}
which can be easily integrated, i.e.:
\begin{equation}
\tilde u=a_1\tilde t + \tilde t^{\,2}. \label{I5uts}
\end{equation}
The integration of (\ref{I5uteq}) is not haphazard but derives
from Lie symmetry method itself \cite{Lie 5}. In fact replacing
(\ref{I5tra}) into (\ref{I5uts}) yields the following first order
equation:
\begin{equation}
\dot u= \frac{u\left( - e^{(\alpha\gamma+
\mu)t}(\alpha\gamma+\mu-u) +
a_1\right)}{e^{(\alpha\gamma+ \mu)t}} \label{I5udot}
\end{equation}
which is easy to integrate because it admits the Lie symmetry
generated by $\Gamma_1$ in (\ref{I5gen}). Lie proved that if one
knows a symmetry $\tau(t,u)\partial_t+\xi(t,u)\partial_u$ of a
first-order ordinary differential equation, say $\dot u=f(t,u)$,
then an integrating factor for the corresponding linear
differential form, say ${\rm d}u-f(t,u){\rm d}t=0$, is
$1/(\xi-f(t,u)\tau)$ \cite{Lie 5}. Thus the
general solution of ({\ref{Iueq}) is
\begin{equation}
u= \frac{a_1} {e^{(\alpha\gamma + \mu)t}
(e^{a_1/ (e^{(\alpha\gamma +\mu)t} (\alpha\gamma +\mu))}
a_1a_2 - 1)}.
\end{equation}
and consequently the general solution of
(\ref{coreS})-(\ref{coreI}) is:
\begin{gather}
S= \frac{\left(e^{(\alpha \gamma + \mu) t}
\left(e^{a_1/\left(e^{(\alpha \gamma + \mu) t} (\alpha \gamma +
\mu)\right)} a_1 a_2 - 1\right) (\gamma - 1) \alpha - a_1\right)
\mu}{e^{(\alpha \gamma + \mu) t} \left(e^{a_1/\left(e^{(\alpha
\gamma + \mu) t} (\alpha \gamma + \mu)\right)} a_1 a_2 - 1\right)
(\alpha \gamma + \mu) (\gamma - 1) \alpha}, \label{I5S}\\
V = \frac{e^{(\alpha \gamma + \mu) t} \alpha^2 \gamma(\gamma -1)
-a_1\mu}{e^{(\alpha \gamma + \mu) t} (\alpha \gamma + \mu) (\gamma - 1) \alpha},\label{I5V}\\
I = \frac{e^{a_1/\left(e^{(\alpha \gamma + \mu) t} (\alpha
\gamma + \mu)\right)} a_1^2 a_2 \mu}{e^{(\alpha \gamma + \mu) t}
\left(e^{a_1/\left(e^{(\alpha \gamma+ \mu) t} (\alpha \gamma +
\mu)\right)} a_1 a_2 - 1\right) (\alpha \gamma + \mu) (\gamma - 1)
\alpha}. \label{I5I}
\end{gather}
In \cite{core} the effectiveness of a disease management program
in the core group was simulated by plotting the solutions with the
help of the graphing capability of MAPLE 7. In some instances a
temporary increase of the number of infecteds followed by a
decrease occurs despite the presence of the vaccination/education
program. This outcome is in agreement with the qualitative
description
by Hadeler and Castillo-Chavez. The quantitative description in \cite{core}
provides a further insight
on the strange behavior of the number of infecteds in the core group as can be
seen in several instances. In fact as it was shown in \cite{core}
if the prevalence of infecteds is initially small in the core
group then an increase occurs before the number of infecteds
actually decreases.
The same numerical value of $\alpha$, $\mu$
and $\gamma$ as given in \cite{HadC} are used, i.e.:
\begin{equation}
\alpha=4,\quad \mu=0.2, \quad \gamma=0.025. \label{values}
\end{equation}
The numerical values of the other parameters are derived from the
relationships that Lie group analysis has discerned.\\
The dynamics of the core group is simulated by taking into
consideration
two different initial conditions at time $t=0$:
(A) no vaccinated/educated are present and the prevalence of infecteds
is relatively small, i.e. $S(0)=0.9, V(0)=0, I(0)=0.1$;
(B) no vaccinated/educated are present and the prevalence of
infecteds is high being nearly half the size of the core group,
i.e. $S(0)=0.6, V(0)=0, I(0)=0.4$
In the figures, the solid line represents the plot of $S$, the lighter
dashed line represents the plot of $V$, and the darker dashed line
represents the plot of $I$.
In this case the numerical values of the remaining parameters are
as follows:
$$
\tilde\beta=0,\quad
\psi=\alpha \gamma= 0.1, \quad
\beta=\frac{\alpha(1-\gamma)(\alpha\gamma+\mu)}{\mu}= 5.85
$$
Note that if the initial prevalence of infecteds is small
($I(0)=0.1$), then there is a delay in the effectiveness of the
vaccination/education program as can be seen in Figure \ref{fig1}, even in
this case when there is no exchange between vaccinated/educated
and infecteds (i.e., $\tilde\beta=0$). Instead if the initial
prevalence of infecteds is high ($I(0)= 0.4$), then the
vaccination/education program immediately takes effect (Figure \ref{fig2})
as expected. Further discussion can be found in \cite{core}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1}
\end{center}
\caption{ \label{fig1} $\tilde\beta=0$, $\psi=0.1$,
$\beta=5.85$, $S(0)=0.9$, $V(0)=0$, $I(0)=0.1$}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig2}
\end{center}
\caption{\label{fig2}
$\tilde\beta=0$, $\psi=0.1$,
$\beta=5.85$, $S(0)=0.6$, $V(0)=0$, $I(0)=0.4$}
\end{figure}
\section{A SIRI model}
In \cite{derrick} Derrick and van den Driessche formulated a model
of disease transmission in a nonconstant population of size $N$
divided into three classes: susceptibles (S), infectives (I) and
recovereds (R).
Individuals move from one compartment to the next according to the
following flow diagram:
\begin{flushleft}
\unitlength=0.5mm \special{em:linewidth 0.4pt}
\linethickness{0.4pt}
\begin{picture}(200.00,100.00)
\put(20.00,40.00){\framebox(20.00,20.00)[cc]{$S$}}
\put(80.00,40.00){\framebox(20.00,20.00)[cc]{$I$}}
\put(140.00,40.00){\framebox(20.00,20.00)[cc]{$R$}}
\put(200.00,40.00){\framebox(20.00,20.00)[cc]{$S$}}
\put(30.00,90.00){\vector(0,-1){30.00}}
\put(30.00,40.00){\vector(0,-1){30.00}}
\put(90.00,40.00){\vector(0,-1){30.00}}
\put(150.00,40.00){\vector(0,-1){30.00}}
\put(40.00,50.00){\vector(1,0){40.00}}
\put(100.00,54.00){\vector(1,0){40.00}}
\put(140.00,46.00){\vector(-1,0){40.00}}
\put(160.00,50.00){\vector(1,0){40.00}}
\put(23.00,25.00){\makebox(0,0)[cc]{$dS$}}
\put(23.00,73.00){\makebox(0,0)[cb]{$bN $}}
\put(61.00,55.00){\makebox(0,0)[cc]{$I\Phi(S,I,N)$}}
\put(120.00,59.00){\makebox(0,0)[cc]{$\gamma I $}}
\put(121.00,39.00){\makebox(0,0)[cc]{$I\Psi(R,I,N) $}}
\put(180.00,55.00){\makebox(0,0)[cc]{$\rho R $}}
\put(75.00,25.00){\makebox(0,0)[cc]{$(d+\epsilon)I$}}
\put(167.00,25.00){\makebox(0,0)[cc]{$(d+\delta)R$}}
\end{picture}
\end{flushleft}
The parameter $b$ is per capita birth-rate, $d$ per capita disease
free death rate, $\epsilon$ excess per capita death rate of
infectives, $\delta$ excess per capita death rate of recovereds,
$\gamma$ per capita recovery rate of infectives, and $\rho$ per
capita loss of immunity rate of recovereds. The incidence of
disease in the susceptible class is given by the function
$I\Phi(S,I,N)$, while $I\Psi(R,I,N)$ is the transfer rate of the
recovered class into the infective class. The above hypotheses
lead to the following differential equations, where $'$ denotes
differentiation with respect to $t$,
\begin{gather}
S' = {bN-dS+{\varrho}R-I{\Phi}(S,I,N)} \label{s1a}\\
I' = {I[{\Phi}(S,I,N)+{\Psi}(R,I,N)-(d+{\epsilon}+{\gamma})]}
\label{s1b}\\
R' = {{\gamma}I-(d+{\delta}+{\varrho})R-I{\Psi}(R,I,N)}
\label{s1c}
\end{gather}
The analysis in
\cite{derrick} was mainly dedicated to show existence (or
nonexistence) of periodic solutions for the SIRS model
(\ref{s1a})-(\ref{s1c}) when proportions of individuals in the
three epidemiological classes are considered, i.e.
\begin{equation}
s=S/N,\quad i=I/N,\quad r=R/N\,.
\end{equation}
With these variables system (\ref{s1a})-(\ref{s1c}) becomes
\begin{gather}
s' = b(1-s)+{\varrho}r+{\epsilon}si+{\delta}sr-i{\Phi}(s,i)\label{s2a}\\
i' = -(b+{\epsilon}+{\gamma})i+{\epsilon}i^2+{\delta}ir+
i{\Phi}(s,i)+i{\Psi}(r,i)\label{s2b}\\
r' = {\gamma}i-(b+{\varrho}+{\delta})r+{\epsilon}ri+{\delta}r^2
-i{\Psi}(r,i) \label{s2c}
\end{gather}
where
${\Phi}(s,i)={\Phi}(s,i,1)={\Phi}(S/N,I/N,N/N)={\Phi}(S,I,N)$ and
${\Psi}(r,i)={\Psi}(r,i,1)$ $={\Psi}(R/N,I/N,N/N)={\Psi}(R,I,N)$.
In \cite{derrick} a theorem was presented and proved in order to
establish under which conditions system (\ref{s2a})-(\ref{s2c})
does not possess periodic solutions in the feasibility region
\begin{equation}
{\mathcal D } = \{ s \geq 0, i \geq 0, r \geq 0 : s+i+r=1 \}
\label{region}
\end{equation}
An example of the nonexistence of periodic solutions was then
introduced, namely a special SIRI case of the general model
(\ref{s2a})-(\ref{s2c})
with ${\varrho}={\delta}=0$,
${\Phi}(s,i)={\phi}s$, and ${\Psi}(r,i)={\psi}r$.
Since $s+i+r=1$ it is possible to eliminate $r$ and finally
obtain the following system:
\begin{gather}
s' = b(1-s)-({\phi}-{\epsilon})si \label{s3a}\\
i' = i[({\phi}-{\psi})s+({\epsilon}-{\psi})i-({\epsilon}+b+{\gamma}-{\psi})]
\label{s3b}
\end{gather}
In \cite{flu} Lie group analysis was applied to system
(\ref{s3a})-(\ref{s3b}), namely to either the second-order
equation in the unknown $s$ that one obtains by deriving $i$ from
(\ref{s3a}) or the second-order equation in the unknown $i$ that
one obtains by deriving $s$ from (\ref{s3b}). Several cases were
found, even instances of hidden linearity. Here we show that when
$$
b=0, \quad \phi=2\epsilon-\psi, \quad \gamma=\psi-\epsilon
$$
then a two-dimensional Lie symmetry algebra
is admitted by equation
\begin{equation} \label{ca2}
\begin{aligned}
i''=& - ((\psi i^2 - i')i' + \gamma\phi i^3 + b^2
i^2 + (i - 1)\epsilon^2 i^3 + ((i - 1)\psi i + i')\phi i^2 \\
&- (\gamma i + 2i' + (\phi + \psi)(i - 1)i)\epsilon i^2
+ (\psi i^2 + i' + \gamma i \\
&+ (i - 1)\phi i - (2i - 1)\epsilon i) b i)/i
\end{aligned}
\end{equation}
which is obtained from system (\ref{s3a})-(\ref{s3b}) by deriving
$s$ from equation (\ref{s3b}), i.e.
\begin{equation}
s=\frac{[b +\gamma - (\epsilon - \psi)(i - 1)]i + i'}{(\phi - \psi)i}
\label{U1}
\end{equation}
and substituting it into equation (\ref{s3a}). The
Lie symmetry algebra is generated by the operators
\begin{equation}
\Gamma_1=t\partial_t-i\partial_i,\quad \Gamma_2=\partial_t\,.
\end{equation}
This means that equation (\ref{ca2}) can be easily integrated by
quadrature as was shown in the previous section. Its general
solution is
\begin{equation}
i= { \frac {\mathit{a_1}}{\sin\big(
{ \frac {\mathit{a_1}\,\mathit{a_2} -
\mathit{a_1}\,t}{ \mathit{\epsilon} - \psi }}
\big)\,(\mathit{\epsilon}^{2} - 2\,\mathit{\epsilon}\, \psi +
\psi ^{2})}}\label{w2c4aa-w2}
\end{equation}
and from (\ref{U1}) one obtains:
\begin{equation}
s={ \frac {1}{2}} \,{ \frac
{\mathit{a_1} \,\left(\cos\big({ \frac
{\mathit{a_1}\,\mathit{a_2} - \mathit{a_1}\,t}{\mathit{\epsilon} -
\psi }} \big) - 1\right)}{\sin\big( {
\frac {\mathit{a_1}\,\mathit{a_2} - \mathit{a_1}\,t}{
\mathit{\epsilon} - \psi }} \big)\,(\mathit{\epsilon}^{2} -
2\,\mathit{\epsilon}\, \psi + \psi ^{2})}} \label{w2c4aa-w1}
\end{equation}
This general solution of system (\ref{s3a})-(\ref{s3b}) is clearly
periodic in apparent contrast with the findings in \cite{derrick}.
Note that the functions (\ref{w2c4aa-w2})-(\ref{w2c4aa-w1}) are
neither bounded nor positive nor continuous, and do not belong to
the feasibility region (\ref{region}). In fact $b$ must be
positive for nonexistence of periodic solutions. However in
\cite{derrick} the condition $b=0$ was allowed in order to show
that system (\ref{s2a})-(\ref{s2c}) has periodic solutions if
${\Phi}(s,i)={\phi}s i$, and ${\Psi}(r,i)=0$.
\section{Final remarks}
In the Introduction to his Principia, Newton stated
\cite{principia, cajori}:
\begin{quote}
I wish we could derive the rest of the phenomena of
nature by the same kind of reasoning from mechanical principles.
\end{quote}
However, Pulte \cite{pulte}
has reminded us that in his lectures on analytical mechanics
Jacobi wrote \cite{jac48}:
\begin{quote}
Wherever Mathematics is mixed up with anything, which is
outside its field, you will find attempts to demonstrate these
merely propositions {\em a priori}, and it will be your
task to find out the false deduction in each case \dots
Mathematics cannot invent how the relations of system of
points depend on each other.
\end{quote}
In 1964 Arscott in the Preface to his book on periodic
differential equations wrote \cite{ars64}:
\begin{quote} Only
rarely does one find mention, at post-graduate level, of any
problem in connection with the process of actually solving such
equations. The electronic computer may perhaps be partly to blame
for this, since the impression prevails in many quarters that
almost any differential equation problem can be merely ``put on
the machine", so that finding an analytical solution is largely a
waste of time. This, however, is only a small part of the truth,
for at higher levels there are generally so many parameters or
boundary conditions involved that numerical solutions, even if
practicable, give no real idea of the properties of the equation.
Moreover, any analyst of sensibility will feel that to fall back
on numerical techniques savours somewhat of breaking a door with a
hammer when one could, with a little trouble, find the
key.
\end{quote}
In conclusion, Lie group analysis should be
considered an essential tool for anyone who wants to ``comprehend"
differential equations of relevance in physics and other
scientific fields. As brilliantly stated by Ibragimov
\cite{ibrap} \begin{center} cherchez le groupe! \end{center}
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\end{thebibliography}
\end{document}