# Acoustic resonators: symmetry classification and multipolar content of the
eigenmodes.

Mariia Tsimokha  Vladimir Igoshin  Anastasiya Nikitina  Mihail Petrov  Ivan
Toftul<EMAIL_ADDRESS>Kristina Frizyuk<EMAIL_ADDRESS>The School
of Physics and Engineering, ITMO University

###### Abstract

Acoustics recently became a versatile platform for discovering novel physical
effects and concepts at a relatively simple technological level. On this way,
single resonators and the structure of their resonant modes play a central
role and define the properties of complex acoustic systems such as acoustic
metamaterials, phononic crystals, and topological structures. In this paper,
we present a powerful method allowing a qualitative analysis of eigenmodes of
resonators in the linear monochromatic acoustic domain based on multipole
classification of eigenmodes. Using the apparatus of group theory, we explain
and predict the structure of the scattered field knowing only the symmetry
group of the resonator by connecting the multipolar content of incident and
scattered fields. Such an approach can be utilized for developing resonators
with predesigned properties avoiding time-consuming simulations. We have
performed full multipole symmetry classification for a number of resonators
geometries, and tightened it with scattering spectra profiles.

Acoustic resonator eigenmodes, symmetry groups, irreducible representations,
spherical functions, Wigner’s theorem, multipole decomposition

##  I Introduction

Studying acoustic resonators is essential both for many technological
application and for fundamental research developing acoustic metamaterials
with established properties [1, 2, 3, 4], various opto-mechanical systems [5,
6],topological insulators [7, 8], and achieving bound states in the continuum
[9, 10]. One of the most important characteristics of any resonator is its
eigenmode spectrum and their filed structure. Commonly, this problems is
addressed with full-wave numerical simulations, while analytical solutions can
be defined solely for a limited number of resonator shapes. A spherical
scatterer is one of the examples and the plane wave scattering on an arbitrary
size sphere [11, 12] was considered for the first time more than 150 years ago
by Clebsch [13] and Lorentz [14] for elastic waves, which in electro-magnetic
theory is well-known as Mie-scattering [15]. However, the unified description
of eigenmodes in an acoustic resonator of arbitrary shape has not been made so
far. In solid-state physics, quantum chemistry, and optics a poweful method
based on group theory analysis has been widely utilized [16, 17, 18, 19, 20].
The mode structure is defined solely by the symmetry the system and can be
classified by the irreducible representations (irreps) of the system’s
symmetry group. The symmetry of the eigenmodes can also give an answer on
which modes are involved in physical processes such as linear and nonlinear
wave scattering, also referred to as selection rules [21, 22, 23, 24, 25, 26,
27].

Figure 1: The general approach and step-by-step algorithm to analysis of
acoustic resonators modes suggested in this work.

We build our approach on multipole decomposition of acoustic waves. Generally,
the multipole expansion is actively used in nanophotonics [28, 29, 30, 31] and
demonstrated that it can be effectively used for predicting the optical
properties of subwavelength resonators. In acoustics the multipoles approach
is not that widely spread however started to draw attention recently [32, 33,
34], enabling effective control over the wave propagation directions and
radiation reaction forces [35, 36, 37, 38, 39, 6, 40].

In our paper, we provide classification and multipole expansion of an acoustic
resonators’ eigenmodes of various symmetry, as well as its application to the
acoustic scattering. Throughout this work we will discuss analyze only
longitudinal acoustic pressure waves in monochromatic domain. Inspired by the
recent progress in nanophotonics, we will operate in terms of spherical
harmonics (multipoles) basis analyzing their symmetry [41, 42, 43, 44], which
could be even simpler and efficient due to scalar origin of fields. We show
how to connect the symmetry of the resonator with the particular multipolar
components of the eigenmode representation. Basing on this, one can
immediately interpret and predict the scattering spectra, directivity of the
scattering, and even acoustics forces acting on resonators due to interaction
with an arbitrary incident wave.

The manuscript is constructed as follows: in Section I we give introductory
part, which settles up the place of this manuscript in the current state of
linear acoustics and optics; in Section II we give some helpful basis of group
theory which is necessary for the understanding of the main results; in
Section III we explain results obtained for multipole expansion for resonators
of $D_{3h}$ symmetry group, and elaborate it on resonators of decreased
symmetry; in Section IV we discuss an influence of resonators’ symmetry on the
cross section of scattered wave. Finally, in Section V we compare it with the
case of optical resonators, whilst demonstrating some similarities and
dissimilarities, eventually revealing ways of better understanding both types
of scattering and other processes.

##  II The basics of group symmetry and acoustic modes analysis

For better understanding of the theory the brief summary of several topics is
presented below. Here we will introduce the concepts of irreducible
representation, functions transformed under irreducible representation (basis
of the representation), spherical harmonics, multipole expansion and Wigner
theorem.

Figure 2: The real spherical functions $Y_{p\ell m}(\theta,\varphi)$ up to
$\ell=6$. At the top of the line — the color shows the value of the function
depending on the angles $\theta,\varphi$, the graph is shown on the sphere. At
the bottom — the radius of the sphere is deformed in proportion to the modulus
of the function value. Figure 3: Spherical functions with $\ell=1$
transforming through each other under the same irreducible representation. For
complex form of spherical harmonics $D(g)$ matrix can be obtained using Wigner
D-matrixes [45].

###  II.1 Group and representation theory

Group is a set equipped with a binary operation that holds three axioms:
associativity, identity, and invertibility. The following study focuses mainly
on the systems’ symmetry groups that consist of the elements that transform
the system to itself, i.e. symmetry operations [46, 47]. A representation of a
group $\mathbf{G}$ on a vector space $\mathbb{V}$ is a homomorphism
$\mathrm{T}$ of $\mathbf{G}$ to the group of automorphisms of $\mathbb{V}$:
$\mathrm{GL}(\mathbb{V})$ [48].

$\mathrm{T}:\mathbf{G}\rightarrow\mathrm{GL}(\mathbb{V})$

In simpler words, group representation is a matrix group with box matrixes,
where we assign a matrix $D(g)$ to each element $g\in G$ such that
$D(g_{1}g_{2})=D(g_{1})D(g_{2})$, i.e. the matrixes satisfy the group’s
multiplication table [18]. Representation is considered irreducible if there
is no any nontrivial invariant subspace in space $\mathbb{V}$ [17, 18]. In
other words, all matrixes $D(g)$ of any representation can be simultaneously
reduced (by a linear basis transformation) to a block-diagonal form, which
consists of irreducible blocks, that turn out to be in fact irreducible
representations of that group [18].

The term set of functions $\psi_{i}(\mathbf{r})$ transforming through each
other under the irreducible representation (or the basis of representation)
describes that after the group element action (rotations or reflections in out
case) functions are transformed to the particular linear combinations of
themselves.

$\psi_{i}\left(g^{-1}\mathbf{r}\right)=\sum_{j}D_{ji}(g)\psi_{j}(\mathbf{r})$
(1)

To illustrate it, let us introduce spherical harmonic functions also referred
as multipoles. A real form in terms of complex spherical harmonics (Fig. 2) is
set as [49, 37]:

$Y_{p\ell
m}=\begin{cases}\displaystyle\frac{i}{\sqrt{2}}\left(Y^{m}_{\ell}-(-1)^{m}Y^{-m}_{\ell}\right)&\text{
}\ p=o,\\\ Y_{l}^{0}&\text{ }\ m=0,\\\
\displaystyle\frac{i}{\sqrt{2}}\left(Y^{-m}_{\ell}+(-1)^{m}Y^{m}_{\ell}\right)&\text{
}\ p=e.\end{cases}$ (2)

Spherical functions with a particular $\ell$ are basis functions of
$2\ell+1$-dimensional irreducible representation of the rotation group of
sphere $SO(3)$, therefore under an arbitrary angle rotation they transform
into the linear combination of functions with the same $\ell$ ( Fig. 3) [45].

The results of our work are based on a Wigner’s theorem [50]. It is formulated
as follows:

$\mathcal{H}(\mathbf{r})\psi(\mathbf{r})=\epsilon\psi(\mathbf{r})$ (3)

Suppose that an eigenvalue equation, describing a system is invariant under
the transformations of a symmetry group, then the eigenfunctions are
transformed under irreducible representations of the group.

Applied to acoustic waves among medium characterized by compressibility
$\beta(r)$ and density $\rho(r)$ Helmholtz equation is set as:

$-c(r)^{2}\nabla^{2}p=\omega^{2}p,\;c(r)^{2}=\displaystyle\frac{1}{\beta(r)\rho(r)}$
(4)

here $p$ is the pressure function, $c(r)$ is the coordinate dependent speed of
sound, and the operator $\mathcal{H}(\mathbf{r})\equiv-c(r)^{2}\nabla^{2}$. At
first, We restrict ourselves to considering the hard boundary condition,
$\partial_{n}p|_{\partial\Omega}=0$, at the resonator surface. Alternatively,
the radiative, Sommerfeld-type, boundary condition at $r\to\infty$ can be
considered, however from the symmetry point of view they will provide the same
result.

From the Wigner’s theorem it immediately follows that the degree of degeneracy
of an energy level equals to the dimension of the corresponding irreducible
representation. One of the simplest examples of this approach, is that the
each mode of a spherical resonator can be portrayed as a particular spherical
function (Fig.2)[18], while modes with identical $\ell$ are
$(2\ell+1)$-degenerate.

###  II.2 Multipole expansion of resonators’ eigenmodes

While the eigenmodes of a spherical resonator are defined by only one
spherical harmonic, the situation becomes much more complex for the resonators
of an arbitrary shape. Now, their modes cannot be defined by a specific
spherical harmonic but rather can be decomposed over a multipoles set. At this
stage, defining the multipole content becomes a complex numerical problem,
however, one immediately identify it using the Wigner’s theorem. Indeed, the
modes’ behavior under all symmetry transformations defines under which
irreducible representation it transforms. Now, the multipoles contained in the
mode should only belong to the same irreducible representation. Thus, one
needs to know, under which irrep of the resonator’s symmetry group each
spherical harmonics transforms.

##  III Theory and results of multipole expansion

###  III.1 Multipole analysis using group theory

If there is an acoustic resonator with a defined symmetry, then the equation
describing the system is invariant under the symmetry transformations of the
group. For the compact resonator which can be considered to be a perturbation
of a spherical resonator [51, 52] its eigenmodes are similar to the sphere’s
eigenmodes.

Let us consider a particular eigenmode of an acoustic resonator which is
described by a complex amplitude of pressure $p(\mathbf{r})$. We note that
real observed field are defined as
$p(\mathbf{r},t)=\real[p(\mathbf{r})e^{-i\omega t}]$. We can write a multipole
decomposition as a sum of scalar spherical functions [53]:

$p(\mathbf{r})=\sum_{p,\ell,m}c_{p\ell m}(r)Y_{p\ell m}(\vartheta,\varphi),$
(5)

where the summation is taken over all the indexes as follows
$\sum_{p,\ell,m}\equiv\sum_{p=e,o}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}$,
$r=|\mathbf{r}|$ is the magnitude of the radius vector, $\vartheta$ is the
polar angle and $\varphi$ is the azimuthal angle in the spherical coordinate
system (see Fig 2 inset). Indeed, the real spherical functions form a basis in
space with scalar product given as integral of the solid angle

$\langle Y_{p\ell
m},Y_{p^{\prime}\ell^{\prime}m^{\prime}}\rangle=\int_{4\pi}Y_{p\ell
m}Y_{p^{\prime}\ell^{\prime}m^{\prime}}d\Omega=\delta_{p\ell
m}^{p^{\prime}\ell^{\prime}m^{\prime}}$ (6)

where $\delta_{p\ell m}^{p^{\prime}\ell^{\prime}m^{\prime}}$ is a Kronecker
delta,
$\int_{4\pi}d\Omega\equiv\int_{\varphi=0}^{2\pi}\int_{\theta=0}^{\pi}\sin{\theta}d\theta
d\varphi$. The multipole content of the mode $c_{plm}$ can fully describe the
mode properties, however we leave the discussion of multipole series
convergence and accuracy out of the scope of this paper [54] saying that the
precision is high enough for our purposes.

According to the Wigner theorem a specific eigenmode $p(\mathbf{r})$ is
transformed under particular irreducible representation of the resonator’s
symmetry group. Thus, in the expansion (5), only spherical functions which are
transformed under this irreducible representation are presented. There cannot
be any spherical functions transformed under different representation in the
expansion. Accordingly, the problem of multipole expansion of eigenmodes of
the resonator is reduced to determining from symmetry considerations under
which irreducible representation a particular mode is transformed and finding
a set of spherical harmonics $Y_{p\ell m}$, transformed through each other
under the same one. To find such a set, the projection operator on the
irreducible representation is introduced $\hat{P}_{\alpha}$ [55, 56]:

$\hat{P}_{\alpha}=\frac{n_{\alpha}}{|G|}\sum_{g}\chi^{*}_{\alpha}(g)\hat{D}(g)$
(7)

where $|G|$ is the order of the group, $g$ is the group element, $\alpha$ is
the number of irrep, $n_{\alpha}$ is the dimension of irrep,
$\chi^{*}_{\alpha}(g)$ is the character of $g$, and $\hat{D}(g)$ is the
transformation operator.

The operator $\hat{P}_{\alpha}$ projects arbitrary basis functions onto a
linear combination of functions which are transformed under particular
irreducible representation $\alpha$ [57]. Commonly, this procedure allows for
identification of the basis functions. However, for the case of spherical
functions set we can simply use the already obtained results and address to
the readymade character tables [44, 58]. With this, knowing that a particular
eigenmode transforms under irrep $\alpha$, one can immediately determine the
multipole composition of this mode by simply finding the spherical functions,
which are also transformed under the same irrep $\alpha$. This set will
determine the non-zero components in the expansion Eq. (5). For the resonators
of $D_{3h}$ symmetry, the multipolar content is shown in Fig. 4. The first
column shows the results of numerical simulation obtained using an eigenmode
solver in COMSOL Multiphysics™, where the color denotes the acoustical
pressure at the surface of the resonator.

Figure 4: a) Character table for symmetry group $D_{3h}$ group. b) Table of
multipole composition of eigenmodes for a closed acoustic resonator of
symmetry group $D_{3h}$: examples of modes transformed under particular
irreducible representation (first column) and their multipolar content (third
column).
---
Figure 5: Illustration of the analysis of the multipole composition of a
particular eigenmode of a closed resonator. The integration of the pressure
function of the eigenmode multiplied by the spherical function was performed
over an auxiliary sphere inside the structure. The nonzero value of the
integral (filled colored cells in the table) indicates that such spherical
function is included in the eigenmode expansion with a certain coefficient.

###  III.2 Numerical simulations

The group theory approach provides the non-zero multipole coefficients in the
expansion Eq. (5). The quantitative analysis of these coefficients is usually
carried out by direct multipole decomposition of the modes of open
resonators[42]. Here, for a closed resonator under consideration, we are
rather interested in testing the predicted non-zero multipole components
basing on the numerical simulations. For that, firstly we numerically compute
the pressure distributions of the eigenmode $\psi(\mathbf{r})$ inside the
resonator with help of commercially available COMSOL Multiphysics™ software.
Then, the eigenmode function is multiplied by spherical harmonics $Y_{p\ell
m}$ and integrated over a sphere surface embedded inside the resonator and
with center matching the resonator center of symmetry as shown in Fig. 5:

$\langle\psi,Y_{p\ell
m}\rangle=\sum_{p^{\prime},\ell^{\prime},m^{\prime}}c_{p^{\prime}\ell^{\prime}m^{\prime}}\int_{4\pi}Y_{p^{\prime}\ell^{\prime}m^{\prime}}Y_{p\ell
m}d\Omega=\\\
=\sum_{p^{\prime},\ell^{\prime},m^{\prime}}c_{p^{\prime}\ell^{\prime}m^{\prime}}\delta_{p\ell
m}^{p^{\prime}\ell^{\prime}m^{\prime}}=c_{p\ell m}$ (8)

A non-zero result of the integration represents the fact that the spherical
function is included to the eigenmode expansion (see Fig. 5 for a particular
eigenmode of a prism resonator). The result will depend on the size of the
sphere however we are interested in zero values of the coefficients due to
symmetry restriction which will stay zero for any sphere size.

Figure 6: A part of the infinite table of coefficients at spherical functions
for the symmetry group $D_{3h}$. On the vertical axis, the product of the
wavevector $k$ of eigenmodes by the height $h$ of the prism is given. The
color saturation corresponds to the value of the coefficient $c_{p\ell m}$ in
the expansion. The checkerboard arrangements correspond to degenerate modes
transformed under two-dimensional representations of $E^{\prime}$,
$E^{\prime\prime}$. The length of the horizontal base of the prism (object of
symmetry group $D_{3h}$) is 44.68 mm, and its height is 50 mm. However
geometrical properties are unimportant to this end. Figure 7: Degenerated
modes transformed under the irreducible representation $E^{\prime\prime}$.

Fig. 6 summarizes the results of the symmetry analysis for different eigen
modes of $D_{3h}$ group symmetry. It shows that for each eigenfrequency
coefficients deviate from zero in one representation only. Moreover, there are
degenerated modes in two-dimentional representations: two lines correspond to
a single frequency, and coefficients are in blocks or chequerwise. While
finding the degenerate modes, the numerical solver oftenly selects them in
arbitrary manner, however by slightly violating the symmetry, we can force it
to select a particular linear combination. The case of the checkboard
positioning corresponds to the case, where in each line every spherical
function is either odd or even (Fig. 7), therefore one of the two degenerate
modes is even when reflected in $y=0$ plane, and the second one is odd. The
chequerwise distribution can be achieved artificially by stretching the shape
out along the $x$-axis. In this case modes would not be truly degenerated due
to the lower symmetry of the system, but their frequencies would be close
enough.

Figure 8: Two symmetry groups $C_{4v}$, $C_{2v}$, one of which is a subgroup
of the other. Each irreducible representation corresponds to explicit examples
of eigenmodes of the resonator transformed under given irreducible
representation and expressed by a linear combination of spherical functions
transformed under the same one. The blue arrows illustrate the reduction of
the number of irreducible representations with decreasing symmetry, and the
red arrows show the cancellation of degeneracy with breaking the rotational
symmetry around the $z$-axis by 90 degrees. Geometry of studies closed
resonators: $a=50$ mm, $b=20$ mm, overall height is 50 mm. For each resonator:
density $\rho=1190\;\text{kg}/\text{m}^{3}$, speed of sound
$c=2500\;\text{m}/\text{c}$ However geometrical properties are unimportant to
this end.

###  III.3 Eigenmodes decomposition of closed resonators of different symmetry
groups

As a next step, we have performed the classification of resonators of other
symmetry groups of $D_{2h}$, $D_{3h}$, $D_{3d}$, $D_{4h}$, $D_{6h}$,
$D_{\infty h}$, $C_{2v}$, $C_{3v}$, $C_{4v}$, $C_{6v}$, $C_{\infty v}$ with
customized geometrical parameters (see Appendix A). The tables for all
symmetries are given in Appendix B in the form similar to Fig. 4.

###  III.4 Multipole content in the resonators of decreased symmetry

In this part, we would like to focus attention on an interesting behavior when
lowering the symmetry of a resonator and correspondent multipoles content
evolution. In Fig. 8 the multipoles set of pyramid resonator ($C_{4v}$-group
symmetry) is shown. Under particular transformation one can decrease its
symmetry to $C_{2v}$, which is a subgroup of $C_{4v}$. When the symmetry is
being lowered the number of irreducible representations is reduced: two one-
dimensional representations are merged, forming a different one-dimensional
representation [59]. Moreover, degeneracies are cancelled when the rotational
symmetry around the $z$-axis is broken: two-dimensional representation turns
to two one-dimensional, in other words two degenerated modes turn into two
modes with different energies. In the following section, we will demonstrate
the effect it has on the spectral features of acoustic wave scattering.

##  IV Acoustic wave scattering

Figure 9:  Scattering cross section expansion of two objects with $C_{2v}$ (a)
and $C_{4v}$ (b) symmetries. Plane wave propagates along $z$-axis and
therefore contains spherical harmonics with $m=0$ only in its expansion. (b)
The figure illustrates how slight change of a symmetry alters the scattering
spectrum multipolar content. Additional contributions of $Y_{e\ell 2}$ are
shown. Geometry parameters are $a=1.5\;$m, $b=0.6\;$m, overall height
$h=1.5\;$m; parameters of the resonators: speed of sound $c=2\;$m/s, density
$\rho=1/\sqrt{6}\;$kg/m3; parameters of the host medium: speed of sound
$c_{0}=1\;$m/s, density $\rho_{0}=1\;$kg/m3

The scattering of an acoustic wave on the resonator having particular set of
eigenmodes results in appearance of resonant features in the scattering
spectrum. At this stage, the information on the particular multipole content
of resonators can be extremely helpful to analyze and predict the spectral
response of the resonant object.

###  IV.1 Multipoles expansion

Radiation pressure of a plane acoustic wave, propagating along $z$-axis is in
the form of [53, 6, 11, 12, 60]

$p^{i}=p_{0}e^{ikr\cos\theta}=\sum_{\ell=0}^{\infty}p_{\ell}j_{\ell}(kr)Y_{e\ell
0}(\cos\theta)$ (9)

where $p_{\ell}=p_{0}i^{\ell}\sqrt{4\pi(2\ell+1)}$, $p_{0}$ is the incident
wave amplitude, $j_{\ell}(kr)$ is the spherical Bessel function. By virtue of
symmetry, there are solely spherical harmonics $m=0$ presented in the
expansion, however for the scattered wave there might be all components in the
expansion:

$p^{s}(\mathbf{r},\omega)=\sum_{p,\ell,m}a_{p\ell
m}(\omega)h_{\ell}^{(1)}(kr)Y_{p\ell m}(\cos\theta),$ (10)

where $h_{\ell}^{(1)}$ is the Spherical Hankel function of the first kind
[61]. The scattering cross section can be expressed through the scattering
coefficient as:

$\sigma_{sc}=\frac{1}{k^{2}}\sum_{p,\ell,m}|a_{p\ell
m}(\omega)|^{2}=\sum_{p,\ell,m}\sigma_{p\ell m}(\omega)$ (11)

where $\sigma_{p\ell m}(\omega)=\frac{1}{k^{2}}|a_{p\ell m}(\omega)|^{2}$. The
scattering cross section also can be obtained by direct integration of the
scattered wave energy flux over any surface surrounding the resonant object,
for instance a sphere of radius $R$,

$\sigma_{sc}=\frac{R^{2}}{I_{0}}\int_{4\pi}\frac{1}{2}\real({p^{s}}^{*}v_{r}^{s})d\Omega$
(12)

where $I_{0}$ is the incident wave intensity, $v_{r}^{s}$ is the component of
the local velocity of the scattered wave [62]. The answer does not depend on
the integration sphere radius since the energy flux of the scatterer power
$\frac{1}{2}\real p^{s*}\mathbf{v}^{s}\propto 1/R^{2}$. The multipole
expansion coefficients can be found by projecting the pressure of the
scattered wave onto the corresponding spherical harmonic:

$\langle p^{s}(\mathbf{r},\omega),Y_{p\ell
m}\rangle=\sum_{p^{\prime},\ell^{\prime},m^{\prime}}a_{p^{\prime}\ell^{\prime}m^{\prime}}(\omega)h_{\ell^{\prime}}^{(1)}(kr)\int_{4\pi}Y_{p^{\prime}\ell^{\prime}m^{\prime}}Y_{p\ell
m}d\Omega=\sum_{p^{\prime},\ell^{\prime},m^{\prime}}a_{p^{\prime}\ell^{\prime}m^{\prime}}(\omega)h_{\ell^{\prime}}^{(1)}(kr)\delta_{p\ell
m}^{p^{\prime}\ell^{\prime}m^{\prime}}=a_{p\ell m}(\omega)h_{\ell}^{(1)}(kr)$
(13)

If the spherical function $Y_{p\ell m}$ is included both in the incident wave
expansion and in an eigenmode’s expansion, than this eigenmode will be excited
along with all others, which are transformed by the same irrep as $Y_{p\ell
m}$. Therefore in the scattered wave the whole basis of spherical harmonics
transformed under that irreducible representation is constituted. The incident
plane wave propagating along the $z$-axis contains only spherical functions
with $m=0$ and, thus, it excites all modes which contain at least one
spherical function with $m=0$.

###  IV.2 Acoustic scattering results

Let us now consider a particular case of wave scattering over an open acoustic
resonator of $C_{2v}$ and $C_{4v}$ symmetry groups. In accordance with the
multipole analysis results mentioned earlier (Fig. 8), the incident wave
excites only one irreducible representation, $A_{1}$. However, irreducible
representation $A_{1}$ for $C_{2v}$ symmetry is a result of merging
irreducible representations $A_{1}$ and $B_{1}$ of $C_{4v}$ symmetry group,
therefore irreducible representation $A_{1}$ for $C_{2v}$ symmetry group
corresponds to $Y_{e\ell(2m)}$ spherical functions, while in $C_{4v}$ group
there are only $Y_{e\ell(4m)}$ functions present. As a result, the multipole
expansion of the wave scattered by the object with $C_{2v}$ symmetry shows
contributions of modes missing in the scattering on an object with $C_{4v}$
symmetry (Fig. 9). The used parameters of models and environment are presented
in Appendix B.

###  IV.3 Analogy to optics

Recently, the group symmetry analysis of eigen modes and their multipolar
content of subwavelength optical resonator has been carried out in optics[42],
where multipolar theory is a powerful tool for describing optical response of
resonant nanostructures, predict, and design their optical properties[63, 28,
64, 34]. The main difference between the optics and acoustics is that the
electromagnetic fields are described by vector functions which adds a
polarization degree of freedom. When considering the farfield multipolar
content of the modes, one should consider two types of vector spherical
harmonics — electric and magnetic ones. However, despite of complexity of
electromagnetic modes, one can note that content of the acoustic and optical
modes is rather similar [42] as the electric multipoles behave almost exactly
as scalar spherical harmonics. The magnetic harmonics have opposite behavior
under inversion and reflection, and can excite, for example, modes which
transform under irreducible representations $A_{2g}$ and $A_{2u}$ in cylinder,
which is impossible in acoustics. Another important difference between
acoustic and optical scattering is that the plane wave, incident along
$z$-axis, has only $m=1$ due to vector nature of the electromagnetic fields
[65] and, thus, transforms under a different irreducible representation.
Cylindricaly symmetric optical fields with $m=0$ can be achieved in vector
beams. We summarize the correspondence between acoustics and optics in Table
1.

Acoustics | Optics |  |  |
---|---|---|---|---
Scalar spherical functions $Y$ | Electric multipoles $\mathbf{N}$ |  |  |
- | Magnetic multipoles $\mathbf{M}$ |  |  |
m=0 for a plane wave k——z | m=1 for a plane wave k——z |  |  |
Table 1: Relation between the multipole analysis in acoustics and optics.

##  V CONCLUSION

Finally, in conclusion, we applied the machinery of group theory in order to
classify and analyze the modes of subwavelength acoustic resonators in a way
similar to nanophotonics. We considered the resonators eigenmodes of several
symmetry groups ($D_{2h}$, $D_{3h}$, $D_{3d}$, $D_{4h}$, $D_{6h}$, $D_{\infty
h}$, $C_{2v}$, $C_{3v}$, $C_{4v}$, $C_{6v}$, $C_{\infty v}$) and presented the
classification tables of their eigenmodes and the multipolar content for each
class. The proposed approach of multipole classification approach can be
extended to any resonator shape and material. We have connected the multipolar
classification with the acoustic scattering problem, since knowing only the
multipolar structure of the incident wave and the symmetry group of the
resonator one can predict the exact multipolar composition of the scattered
field. We studied in detail the lift of degeneracy in a resonator of
$C_{4v}\to C_{2v}$ symmetry and traced the evolution of multipolar content as
well as reconfiguration the scattering spectrum. In particular, we showed that
symmetry decreasing leads to expanding the range of multipoles in the
scattered field. We believe, that our result will find an application in
rapidly developing acoustic of metamaterials and metaatoms.

###### Acknowledgements.

The authors would like to thank Yuri Kivshar for the fruitful discussions.
This work was supported by Russian Science Foundation (project 20-72-10141).
KF acknowledges support from the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS” (Russia)

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##  Appendix A COMSOL Multiphysics™ model

We used Eigenvalue Solver to observe electromagnetic fields of closed
resonators. We established spherical domain inside the resonator (Fig. 5) to
estimate $c_{p\ell m}$ coefficient in the multipole expansion (8). We
integrate pressure and spherical harmonics (6) over the surface of the sphere:

$c_{p\ell m}=\int_{4\pi}p(\mathbf{r})Y_{p\ell m}d\Omega$ (14)

We used a built-in complex form of spherical harmonics to get a real solution
(2).

For open resonator we used Frequency Domain Solver. The model geometry is a
sphere with a Perfect Matched Layer (PML) domain and resonator at the center.
To get $a_{p\ell m}(\omega)$ (13) we integrated scalar product of far-field
scattering pressure $p^{s}$ and spherical harmonics over concentric parametric
surface with a radius $R$ outside of the resonator, but inside the PML sphere:

$a_{p\ell
m}(\omega)=\frac{1}{h_{\ell}^{(1)}(kR)}\int_{4\pi}p^{s}(\mathbf{r},\omega)Y_{p\ell
m}d\Omega$ (15)

We also note that in Eq. (15) one should use $h_{\ell}^{(2)}$ in COMSOL
Multiphysics™ instead of $h_{\ell}^{(1)}$. This comes from the fact the
outgoing wave should have the asymptotics as $e^{-ikr}$ once the $e^{+i\omega
t}$ convention is used, which is the case for the COMSOL Multiphysics™. Sum of
squared absolute values of coefficients equals $\sigma_{sc}$ (11).

We used the same concentric parametric surfaces to get $\sigma_{sc}$ by (12).
The following expression was used as an integrand in COMSOL Multiphysics™
model (built-in acoustics interfaces notation)

    0.5*realdot(acpr.p_s,(
    Ψ(-(d(acpr.p_s,x))*x)+
    Ψ(-(d(acpr.p_s,y))*y)+
    Ψ(-(d(acpr.p_s,z))*z)
    )/(acpr.rho_c*acpr.iomega))

##  Appendix B Models and geometry of studies closed resonators

---
Parameter | $D_{2h}$ | $D_{3h}$ | $D_{4h}$ | $D_{6h}$ | $D_{\infty h}$ | $D_{3d}$
---|---|---|---|---|---|---
$h$, mm | 60 | 50 | 40 | 40 | 30 | 40
$a$, mm | 40 | 44.68 | 25 | 25 | 12.5 | 25
$b$, mm | 20 |  |  |  |  | 30
Parameter | $C_{2v}$ | $C_{3v}$ | $C_{4v}$ | $C_{6v}$ | $C_{\infty v}$ |
$h$, mm | 50 | 50 | 50 | 37.5 | 21.65 |
$a$, mm | 50 | 50 | 50 | 50 | 12.5 |
$b$, mm | 20 |  |  |  |  |

Figure 1: Geometry of studies closed resonators. $h$ – overall height. For each resonator: density $\rho=1190\;\text{kg}/\text{m}^{3}$, speed of sound $c=2500\;\text{m}/\text{c}$ Parameter | $C_{4v}$ | $C_{2v}$ | Host
---|---|---|---
$h$, m | 1.5 | 1.5 |
$a$, m | 1.5 | 1.5 |
$b$, m |  | 0.6 |
$c$, m/s | 2 | 2 | 1
$\rho$, kg/m3 | $1/\sqrt{6}$ | $1/\sqrt{6}$ | 1
Figure 2: Parameters of the studied resonators and media in the Mie scattering
problem

##  Appendix C Eigenmodes’ multipole decomposition results

Figure 3: Tables of irreducible representations and examples of eigenmodes transformed under them, and tables of characters of irreducible representations for symmetry groups $D_{4h}$, $D_{2h}$, $D_{3d}$, $D_{3h}$ |
---|---
|
Figure 4: Tables of irreducible representations and examples of eigenmodes transformed under them, and tables of characters of irreducible representations for symmetry groups $C_{\infty v}$, $C_{6v}$, $D_{\infty h}$, $D_{6h}$ |
---|---
|
Figure 5: Tables of irreducible representations and examples of eigenmodes
transformed under them, and tables of characters of irreducible
representations for symmetry groups $C_{2v}$, $C_{3v}$, $C_{4v}$
---


11institutetext:  School of Computing, Information and Mathematical
Statistics,
The University of the South Pacific, Laucala Campus, Suva, Fiji.
22institutetext: Geography, Earth Science and Environment,
The University of the South Pacific, Laucala Campus, Suva, Fiji.
33institutetext: Artificial Intelligence and Cybernetics Research Group,
Software Foundation, Nausori, Fiji
<EMAIL_ADDRESS>
%\institute{School␣of␣Geography,␣Earth␣Science␣and␣Environment,\\\The␣University␣of␣the␣South␣Pacific,\\\%Laucala␣Campus,␣Suva,␣Fiji\\\http://www.usp.ac.fj

Authors’ Instructions

# Mobile Application for Dengue Fever Monitoring and Tracking via GPS: Case
Study for Fiji

Emmenual Reddy 1133 Sarnil Kumar 1133 Nicholas Rollings 2233 Rohitash Chandra
1133

###### Abstract

The 2013 outbreak of Dengue in Fiji resulted in an alarming number of deaths
and has been is a matter of serious concern. Dengue fever is a disease caused
by the four types of the Dengue virus serotypes and transmitted mostly from
mosquito bites. In Fiji, dengue diagnosis is only done in hospitals which are
slow and time consuming. It is also important to monitor the spread of Dengue.
Fiji needs an convenient method of monitoring the spread of Dengue. With
increase in affordable smartphones and better Internet coverage, there is
scope for a mobile application for Dengue fever monitoring and tracking. This
paper proposes a mobile application for Dengue monitoring based on global
positioning system (GPS) enabled mobile phone technology. It also provides an
information network that shows the spread of dengue which will allow health
authorities to quickly identify dengue infected areas in Fiji. A mobile
application prototype is developed and tested and the scope for further
testing and implementation is also given.

###### Keywords:

Dengue, Dengue virus, Dengue monitoring, Dengue tracking, GPS, Mobile
application

##  1 Introduction

Health information technology, which includes computers, mobile devices and
other devices used in the management of medical information, has great
potential to promote health and support health care around the world [5],
[32], [1] [14], [35]. Mobile phones are more preferred in the delivery of
health information technology because of its widespread adaption and its
increasing processing power.People tend to carry mobile phones with them
everywhere and unlike desktop computer, mobile phone are nearly always with
the person [16] [28].

In developing countries there are many challenges faced when designing and
implementing ways to improve the health information technology. The problems
faced are limited health care resources, access to technology infrastructure
and technical expertise to support technology [18] [3]. Other difficulties
faced by people in developing countries when using mobile phone are Internet
connectivity, mobile network signal and access to electricity and charges
[34].

There are challenges also faced when deploying the health information
technology in the healthcare workforce. One major challenge is the lack of
knowledge in information technology. Illiteracy itself may be one of the
central barriers that could prevent the effective use of these phones in this
situation; Illiteracy is often associated with poverty, which coupled with the
inability to read may limit an individual’s access and ability to benefit from
such technologies [8]. Healthcare workforce is often scarce and overburdened
already with responsibilities. Technology interventions in the health care
should not require high learning curves, it should be flexible and easily
understood so it will not consume their time [5].

According to [7], traditional medicine offers an alternative solution and
could be explored as a safer treatment option. One such treatment includes
using the Papaya leaf remedy to cure dengue which has been proven to be
successful in a research carried out in Pakistan [2].

Dengue fever is a major health concern in Fiji with a number of dengue
outbreaks occurring in the last three decades [27]. The most recent outbreak
was declared in December 2013 [36]. In the latest outbreak there were over 446
suspected cases of Dengue notified to the Ministry of Health and there have
been 12 confirmed deaths [23]. Dengue fever is a disease caused by any of the
four types of the dengue virus serotypes (DEN 1, DEN 2, DEN 3, DEN 4) [31] and
transmitted mostly from mosquito bites [22].

Traditionally, the diagnosis of dengue infection is a clinical determination
by a medical professional. However, during the early infection period, when
symptoms may not be as severe as later in the course, the patient with dengue
fever can appear to have a less serious illness and therefore may not be given
appropriate treatment [19]. The early recognition of dengue symptom is very
important, however, dengue detection kits are only accessible in hospitals
because of the high costs [21]. Therefore, a more convenient method of
diagnosing dengue in patients is needed.

The Global Positioning System (GPS) is a satellite-based navigational system
that was developed by the United States Department of Defence in the early
1980s [30]. The GPS technology allows the accurate positioning of an object
using satellite signals [20]. GPS technology has been used in many health
applications [37] [38] [39] [40] .

In this paper, a mobile based dengue fever monitoring and tracking application
is proposed that monitors the spread of dengue via global positioning system
(GPS). The application allows likely dengue patients to enter their symptoms
and receive immediate vital feedback. The application provides users with
advice on how to prevent dengue fever. The application is informational and
not a substitute for visiting a doctor. In the case of detected or diagnosed
fever or any related symptoms, the users are advised to immediately see a
physician. The users’ current location which is tracked through the GPS
tracker and other important user information is stored in a database that can
be further be integrated with existing electronic medical record systems.
Using the data collected, a map showing the locations of likely dengue
infected people is then produced. A mobile application prototype is developed
and tested and the scope for further testing and implementation is also given.
Relevant authorities such as Ministry of Health or the town councils can use
this information to identify dengue outbreak areas in Fiji and precautionary
measures can be taken.

The rest of the paper is organised as follows. Section 2 gives a background of
the area. Section 3 presents the proposed design and Section 4 shows
Implementation and results. Section 5 concludes the paper with discussion of
future work.

##  2 Background

According to the World Health Organisation, primary prevention is the most
effective measure in dengue prevention and control since no vaccine is
currently available [31].

There is a dire need for a dengue vaccine to further prevent the spread and
contain the growing pandemic. Several vaccine companies and research groups
are actively pursuing the development of a vaccine to prevent dengue [4].
While a licensed dengue vaccine is not yet available, several vaccine
candidates are currently being evaluated in clinical trials [26].

Since no effective vaccine is available for treating dengue, the eradication
or control of the main mosquito vector is regarded as essential. Research has
also been done in devising control methods for the Aedes Aegypti mosquitoes
which is the main type that spreads dengue virus [41].

In developing countries, viral outbreak of Dengue causes a heavy financial
constrain in terms of manpower (medical staff) and resources such as hospitals
and equipment [15]. In addition, there is little awareness about the symptoms
of Dengue fever.

A number of studies have been conducted to determine the usefulness of herbal
medicine in curing dengue. Researches have indicated that the juice of the
leaves of the Carica papaya plant from the family Caricaceae could help to
increase the platelet levels in these patients. [42] [43] [44]

The World Health Organization has estimated that out of the 2.5 billion world
population who are currently at risk of suffering dengue in over 100
countries, two-thirds reside in the Asia Pacific Region [31]. Fiji, a multi-
cultural south pacific island nation with a population of around 800,000, had
its share of dengue fever outbreaks in the past. Major dengue outbreaks
occurred in 1998 and 2003 that resulted in loss of lives, substantial health
care stress and considerable economic costs [27]. The latest dengue outbreak
was declared in December 2013 [36].

Electronic Medical Records (EMR) are used in medical sector in various ways
which includes easy access to medical record, reduction of potential medical
error, appointments and billing management [25] However, there are obstacles
to a successful adoption of electronic records and to the transition from a
paper-based paradigm. These obstacles are not only technological, but often
also related to the deployment procedures followed [45] [46]. The purpose of
EMR system is to have a centralized location of all information regarding a
patient which can be shared by doctors, hospitals and patients. [13].

Fiji has used PATIS (Patient Information System) which was developed with
AusAid funding for the Pacific nations for managing patient records.
Initially, PATIS was still paper based and not web based. PATIS is a low-cost
application software system that provides functionality generally found in
commercial patient administration systems modules of a Health Information
System. There have been significant enhancements to Fiji’s PATIS, which now
has functionality usually only provided by specialised clinical system modules
[47]. A review team in 2010 was informed that the PATIS system is being
transferred to one that is web based and this will help encourage wider use of
the very valuable data being generated by the system [48]. The latest version
of PATIS with improved accessibility and speed is referred to as PATISPlus
[49].

OpenEMR is an open source software used in health information systems [11]
[12]. Open source software solutions are becoming popular in developing
countries as they can be deployed taking into consideration the cost of
purchasing licensed software [50] [51].

##  3 Proposed System

The motivation to develop a mobile based dengue monitoring system came from
the alarming rate of dengue deaths in the recent dengue outbreak in Fiji [52].
There is a need for a faster and convenient method of checking for possibility
of dengue infection.

We propose a mobile application that can be used by mobile devices such as
smartphones and Tablets as these devices are becoming more affordable
smartphones and there is better network coverage in Fiji when compared to past
years.

During an outbreak of Dengue in Fiji, the proposed mobile application can be
used by the Ministry of Health to monitor the spread of the disease and take
precautionary measures by identifying affected areas and provide support with
help of the media. The proposed mobile application enables a symptoms check of
Dengue and in order to take the precautionary steps that is advised by the
mobile application and can reduce a significant amount of workload of the
medical practitioners.

There is no vaccine cure for dengue, treatment consist of early identification
of the disease with intensive surveillance and fluid support [19]. The
symptoms of the dengue which will be assessed in this mobile application are
fever, severe headache, severe eye pain, joint and muscle pain(myalgia), skin
rash, mild bleeding manifestation, low white blood cell count, nausea/vomiting
and joint swell [15] [31].

There are some limitations in using mobile application since not everyone has
a smartphone. Some people are not smartphone literate, meaning they do not
know how to install and use mobile applications. In addition, using the
application requires the user to have internet credit in his smart phone and
also to have the mobile data enabled. Despite this limitations, the proposed
system has many advantages as it can be used to identify and control the
spread of dengue. The proposed system’s features are discussed in the
following subsections.

Figure 1: Overall architecture of the system showing the path of data
transmission from the mobile application through the mobile carrier and the
Internet Service Provider (ISP) to the application server and finally to one
of the computing devices.

###  3.1 Overview

The objectives of the dengue fever monitoring and tracking application are to
provide the general public with a dengue symptoms checker, receive feedback of
the papaya leaf remedy [2], self-report a Dengue case and to store the
location of the user via GPS. The data (GPS coordinates) which is stored in
the server will be featured in the website showing the location of users
infected by Dengue.

Figure 2: The proposed mobile application system showing the different
features of the system including the features to register and login, checking
symptoms for dengue, reporting a confirmed case of dengue and giving papaya
leaves feedback.

###  3.2 Authentication

The access to the dengue symptoms monitoring and tracking application is given
to registered users. The mobile application has a feature where the general
public can register and login to the system. In the registration process, the
user is asked the following details; name, email address as username,
password, date of birth, gender, height and weight. To login to the
application, the user needs to enter their email address and password.

###  3.3 Dengue Symptoms Monitoring

Once the user has successfully logged in the application, they can use the
Dengue Systems Check feature to verify their symptoms for possibility of
dengue infection. The mobile application determines the likelihood of the
users being infected with dengue based on their symptoms. If the users have
selected fever and any two other symptoms from the eight other symptoms, they
are informed that they are highly likely to be infected with Dengue fever
[31]. They are also advised to visit a physician immediately.

###  3.4 Dengue Case Reporting and Tracking

As discussed earlier, there is no vaccine cure for dengue but there are ways
in which we can control dengue outbreak. One way to control dengue is through
early detection and surveillance [22] [9]. The dengue surveillance allows
public health workers in resources-limited areas to identity areas with high
Dengue infections and potential larval potential larval development areas such
as garbage piles and large pools of standing water. [6].

In this mobile application, there is a feature where the patient can report a
case of Dengue. Once the user reports a case, the location of the user is
updated on the database. The location of the user is based in terms of GPS
coordinates; latitude and longitude referenced to WGS84.

The locations of the likely dengue infected people are used to generate a
Dengue map that is given on the Dengue tracker website. Relevant authorities
such as the Ministry of Health in Fiji and town councils can use the
information on the website to track the possible dengue outbreak and mosquito
breeding areas and take appropriate action. This deployment in mobile
application can be a very efficient and cost effective way of controlling the
spread of Dengue [6].

###  3.5 Feedback on Papaya Leaves

Dengue fever alternative medical treatment can be done with papaya leaf
extracts [44]. Studies have shown that platelet count increases by taking
papaya leaves extracts [42] [43]. In a case study in Pakistan, a 45 year old
man infected with dengue virus who was given Carica papaya leaves extracts
every morning and afternoon noted an increase in platelet counts in the blood
cells [2].

In Fiji, dengue infected patients have also tried papaya leaf remedy, however,
it is not known how effective it has been in terms of platelet counts. The
proposed mobile application ensures that such alternative medicine treatment
results are deposited in the system for further research. The feature allow
users who have tried papaya leaf to provide feedback by checking their
platelet counts through blood test by doctor before and after the papaya leaf
treatment and feeding it into the mobile application. For example, if a person
visits a doctor and after examination it is found out that the patient has
dengue fever with low platelet counts. The same patient takes the advice of
taking papaya leaf remedy. On the patient’s next medical examination if the
platelets count increases, the users can give feedback via mobile application.

This would assist relevant authorities such as researchers and the Ministry of
Health officials to determine whether papaya leaf remedy can really help
Dengue infected patients in Fiji. The feedback can be also used for further
research in drug discovery. If it works, it would be a breakthrough in Dengue
research as currently there is no effective vaccine to cure Dengue fever. It
would also greatly assist in tackling Dengue cases and saving lives with a
very cost effective way.

###  3.6 Website

The website features data which is uploaded on the database from the users of
the mobile application. On the website, a map can be generated using the GPS
coordinates from the database. This will show markers of the location of users
who have reported through the mobile application who have been infected with
Dengue. When a researcher zooms in to a particular marker on the map, they
will be able to see the exact location of the infected users. Moreover, a
particular area concentrated with many markers would suggest that it could be
possible breeding grounds for mosquitoes. Health authorities can use this
vital information to quickly identify areas in Fiji which needs to be
fumigated and for health workers to visit people living in this areas to
control further spread of Dengue.

Figure 3: Map showing markers which indicates the location of people infected
with Dengue Fever in the island of Viti Levu, Fiji. Figure 4: Map when zoomed
in showing markers which indicates the location of people infected with Dengue
Fever in the area of Ba, Fiji.

##  4 Implementation and Results

In this section, we provide details of implementing the proposed application
built on an Android platform. Firstly, the implementation details are
discussed which includes the selected programming language and other
supporting software required to develop the mobile application. Global
Positioning System (GPS) is also discussed as it is a key feature used in the
mobile application to determine the location of the user. Finally, details of
testing the mobile application are discussed.

###  4.1 Implementation Details

The Android application is developed using Java programming language. Java is
a class-based object orientated language and is one of the most popular
programming languages in client-server web application [53]. Once Java program
is complied, it can run on any platform regardless of the computer
architecture [54]. The Android SDK (Software Development Kit) provides the API
(Application Program Interface) libraries and developer tools necessary to
build, test, and debug applications for Android [55].

The data sent from the application is stored on a LAMP (Linux Apache MySQL
PHP) server [29]. The data is then sent in JSON111JavaScript Object Notation
format [10] from the android application and updated on the database using
PHP222Hypertext Preprocessor - PHP is a server-side scripting language
designed for web development but also used as general purpose programming
language and MySQL333My Structured Query Language - MySQL is an open source
relational database management system (RDBMS) based on Structured Query
Language [24].

The website444http://www.denguetracker.com is designed in WordPress [33] which
is a content management system built on PHP and MySQL. Since WordPress also
uses PHP and MySQL, it is easy to get data from the database and display it on
the website.

###  4.2 Global Positioning Systems (GPS)

GPS allows devices to track the position of individuals in latitude and
longitude along with timestamps. The limitation of the GPS is that signals are
usually blocked in indoor or underground places, GPS devices get interference
from tall buildings and continuously collecting GPS data normally consume
devices energy quickly [17].

The in-built GPS receiver in the smartphone is used to get the GPS coordinates
of the users who reports to be likely infected with dengue fever. The GPS
coordinates are stored in the server and then used to generate the map showing
the areas where people infected with dengue are located.

---

(a) Functions of the mobile application.  (b) Feature that allow users to
select symptoms and check whether they are likely infected with Dengue Fever.

Figure 5: The mobile application provides a menu of functions once the user
has successfully logged in. A user may select the first option to carry out
dengue symptoms check.

---

(a) Feature that allow users to report a confirmed case of dengue.  (b)
Feature that allow users to give feedback on the use of Papaya Leaves Remedy.

Figure 6: The mobile application also provides features to report a confirmed
case of dengue and to give feedback on the use of the Papaya Leaves Remedy.

###  4.3 Testing the Mobile Application

Testing is vital as it determines the features of the application that can be
reliable and useful under heavy load and features that can be accessible with
limited or no Internet [56]. The mobile application was made available to 10
android smartphone users for testing. Users were asked to perform the role of
someone who is suffering from Dengue. Following the trial, the users were
interviewed regarding their experience. Users in general, gave positive
feedback for user performance and satisfaction in areas with good Internet
connectivity. However, they indicated that they were not able to register and
log in to the mobile application when the internet connectivity was poor.

The application needs further improvement and wider testing to evaluate system
performance which can be done in future work. There are many challenges in
mobile application testing including variety of mobile devices, diversity in
mobile platforms, mobile network operators and scripting. Server load testing
can be performed in future work once the application is implemented for use by
more than 100 users at one particular time.

##  5 Discussion

The mobile based dengue fever monitoring and tracking application is developed
that allow likely dengue patients to enter their symptoms and receive
immediate vital feedback. The application provides users with advice on how to
prevent dengue fever and basic treatment. The application for is health
information only and not a substitute for visiting a doctor. In the case of
fever or any two other related symptoms, the users are advised to immediately
see a physician. The users current location which is tracked through the GPS
tracker and other important user information is stored in a database. Using
the data collected, a map is generated showing dengue infected areas. Relevant
authorities such as the Ministry of Health and town councils can use this
information to identify dengue outbreak areas in Fiji and notify their workers
to visit these areas. One potential problem with this tracking system is the
patient may realise the symptoms of dengue when they are some distance from
the actual infection point. This will give a false impression of where the
patient was actually infected. In future editions it is suggested that during
peak Dengue season users movements are recorded hourly for the previous 20
days. After 20 days the older positions are overwritten allowing fro a balance
between the amount of data collected and ensuring the incubation period for
Dengue is covered. If multiple infected persons have visited the same
geographic location it raises the possibility this is a potential site of
infection.

There are several important implications on how this would affect the
monitoring and control of Dengue. Due to the portable nature of the
smartphones and Tablets, Dengue symptoms check can be done at the patient’s
home. The mobile based symptoms check aids in the identification of patients
infected with dengue in rural areas. Subsequently, patients will be provided
with advice on how to prevent dengue infection. The mobile symptoms checkup
will have the largest impact on children who are the most susceptible to the
serious side effects of dengue.

One of the drawbacks of the dengue application prototype is the application
platform. The dengue application is created using Android platform, therefore,
it will only work in an Android based devices. Hence, it will not work in an
iOS555iPhone OS is a mobile operating system developed by Apple Inc. or
windows platform based smartphones and Tablets. However, the Android
application can be developed in other platforms using the proposed system
design so that it could work in all types of phones. Secondly, the application
will only work if the user has mobile data enabled. Without enough mobile
credit, a user will not be able to activate mobile data and therefore will not
be able to use the application.

The system that is developed is not going to replace physicians, however, it
will greatly assist them in making their work easier in controlling disease
outbreaks.

For mobile phones that do not have internet access, an alternative SMS based
system can be used whereby users send their options and receive feedback
through text messages. This alternative system can also be used on smartphones
in case they do not have internet or when internet connectivity is poor. Other
solutions include having wireless enabled in hospitals and health centres
where patients can easily download and install the mobile application free of
charge. In this case, their GPS location can be updated later and an option to
do so can be given.

In future, if Ministry of Health in Fiji adopts OpenEMR for storing all
patient data in electronic form, this Android application then can easily play
an important role in updating issues related to dengue. The migration of the
database will also be easy.

##  6 Conclusion and future work

In this paper, we present a design of a Dengue monitoring and tracking system
based on GPS. The proposed application provides an alternative and more
convenient method for checking symptoms in likely dengue patients. GPS
technology is also used to track the location of the dengue infected patients.
Using the data that is collected, a dengue map is generated which helps health
authorities to identify dengue infected areas.

The application can be improved to ensure real-time tracking of dengue
outbreaks and compatibility of the application with iOS and windows mobile
platform. To further enhance the capabilities of the application, features can
be added that will be enable users to upload multimedia such as videos and
photos of Dengue breeding sites. The data can be then retrieved from the
database and displayed on the Dengue tracker website and be used by the
relevant authorities to clear and fumigate affected areas.

Future work can involve implementation of the application for large scale use
of thousands of users and further testing to ensure that the features work
when hundreds of users access the application simultaneously. Feedback from
the relevant authorities through surveys can be used to validate the system
and improve it further. Data gathered can be published for further analysis by
medical researchers and authorities that can build preventative systems.

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