# DeCom: Deep Coupled-Factorization Machine for Post COVID-19 Respiratory
Syncytial Virus Prediction with Nonpharmaceutical Interventions Awareness

Xinyan Li  Analytics Center of Excellence, IQVIA Inc., United States  Cheng
Qian  Analytics Center of Excellence, IQVIA Inc., United States  Lucas Glass
Analytics Center of Excellence, IQVIA Inc., United States

###### Abstract

Respiratory syncytial virus (RSV) is one of the most dangerous respiratory
diseases for infants and young children. Due to the nonpharmaceutical
intervention (NPI) imposed in the COVID-19 outbreak, the seasonal transmission
pattern of RSV has been discontinued in 2020 and then shifted months ahead in
2021 in the northern hemisphere. It is critical to understand how COVID-19
impacts RSV and build predictive algorithms to forecast the timing and
intensity of RSV reemergence in post-COVID-19 seasons. In this paper, we
propose a deep coupled tensor factorization machine, dubbed as DeCom, for post
COVID-19 RSV prediction. DeCom leverages tensor factorization and residual
modeling. It enables us to learn the disrupted RSV transmission reliably under
COVID-19 by taking both the regular seasonal RSV transmission pattern and the
NPI into consideration. Experimental results on a real RSV dataset show that
DeCom is more accurate than the state-of-the-art RSV prediction algorithms and
achieves up to 46% lower root mean square error and 49% lower mean absolute
error for country-level prediction compared to the baselines.

††footnotetext: Corresponding Author: Xinyan Li (xinyan.li3@iqvia.com).

##  1 Introduction

Respiratory syncytial virus (RSV) is the most common cause of bronchiolitis
and pneumonia in infants, young children, and seniors (Hall et al., 2009).
According to the Centers for Disease Control and Prevention (CDC), about
58,000 children under the age of five are hospitalized in the United States
every year as a result of RSV infection (CDC, 2022). Worldwide, this number
can reach more than 3 million every year (Shi et al., 2017), and the true
burden is likely to be much more, with approximately half of RSV-associated
deaths estimated to occur outside of hospitals (Shi et al., 2017).

Prior to the 2020–21 season, RSV has had consistent and predictable seasonal
epidemics in both temperate and tropical regions. Such epidemics often begin
in the tropics of each hemisphere in the late summer and reach most temperate
locations in the winter and early spring. For instance, in the United States,
RSV activity typically begins in Florida in the fall and gradually spreads
from the southeast to the northwest. However, nonpharmaceutical interventions
(NPI), such as physical separation, stay-at-home directives, school closings,
travel bans, and border closures have been put in place as a result of the
current COVID-19 outbreak.While NPI minimizes COVID-19 transmission, it has
also had substantial effects on the circulation of RSV.

Since the implementation of NPIs, which are utilized to contain the COVID-19
pandemic, many regions have experienced uncertainty over the RSV outbreaks.
For instance, significant declines in RSV infection have been observed
globally in 2020. Even during the RSV seasons, when the number of infectious
cases was often consistently high, this number was close to zero in many
countries. However, nothing has yet returned to normal in 2021, when most
countries experienced out-of-season RSV outbreaks that arrived months ahead of
the normal RSV season. For example, in the northern hemisphere, France, Spain,
and the United States have observed the RSV waves in the spring and summer,
similar to countries in the southern hemisphere such as Australia and South
Africa, which have also witnessed abnormal out-of-season RSV waves. Until now,
we are still suffering from an abnormal RSV epidemic. Therefore, comprehending
the association of different factors with the timing and intensity of re-
emergent RSV epidemics is crucial for clinical and public health decision-
making in post COVID-19 seasons.

Predicting post COVID-19 RSV transmission with NPI-awareness is a challenge.
Limited research has achieved satisfactory results for this task. In the
literature, the most related work is Baker et al. (2020), where the authors
employed a time-series Susceptible–Infected–Recovered (TSIR) model to simulate
future trajectories of RSV. This approach is based on the standard SIR model
and it assumes that all susceptible cases will be infected once. Its
limitation is that the number of susceptible people and the infectious rate
are difficult to estimate with COVID-19 NPIs, and it also fails to model the
COVID-19 impact on RSV transmission. Therefore, TSIR works more like an
analytical tool than a predictive model.

In this paper, we propose a new deep coupled tensor factorization approach
named DeCom for post COVID-19 RSV prediction with NPI awareness. DeCom
consists of two coupled spatiotemporal tensor factorization units which give
us the ability to model the regular seasonal pattern of the RSV and understand
how COVID-19-induced NPIs affect RSV transmission. In order to demonstrate the
efficacy and high accuracy of DeCom, we provide experimental results utilizing
real RSV datasets and compare them to state-of-the-art machine learning
algorithms.

###  1.1 Organization of the paper.

We begin with a review of related work in Section 2. Section 3 provides a
brief overview of Canonical Polyadic Decomposition (CPD), a crucial element of
our proposed DeCom model. After that, in Section 4, we give a formal
description of the specific RSV forecasting problem. Section 5 then goes into
depth about the proposed DeCom model. The data and experimental setting are
then described in Section 6, and Section 7 discusses the experimental results,
comparing the predictive skill of DeCom with a number of benchmark models.
Finally, Section 8 draws the conclusion.

##  2 Related Work

RSV prediction and environmental drivers. Baker et al. (2019); Pitzer et al.
(2015) have shown that RSV epidemics exhibit limit cycle behavior tuned by
climate-driven seasonality. In most regions in the United States, RSV and
influenza exhibit peak incidence in the winter months when the weather is cold
and dry (Baker et al., 2019; Shaman et al., 2010). There have been attempts to
link climatic variables with RSV seasonality (Paiva et al., 2012; Alonso et
al., 2012; du Prel et al., 2009; Lapeña et al., 2005; Noyola and Mandeville,
2008; Walton et al., 2010; Yusuf et al., 2007). For example, Pitzer et al.
(2015) employed linear models to connect incidence patterns to candidate
environmental drivers of RSV epidemics in US states. Walton et al. (2010) have
constructed Naïve Bayes (NB) classifier models to predict RSV outbreaks in
Salt Lake County, Utah. The given NB models used weather data from 1985 to
2008, considering only variables that are available in real time and can make
forecasts up to 3 weeks in advance with an estimated sensitivity of up to 67%
and estimated specificities as high as 94% to 100%. However, (White et al.,
2005, 2007; Leecaster et al., 2011; Weber et al., 2001) have found RSV
epidemics show strong signatures of nonlinear epidemic dynamics. Few studies
have explored climatic associations in a variety of locations covering a range
of RSV seasonality patterns and climatic regimes (Yusuf et al., 2007).
(Altizer et al., 2006) refined Pitzer et al. (2015)’s linear models to account
for the dynamics of infection and fluctuations in immunity and susceptibility
that can influence the relationship between environmental factors and epidemic
timing .

###  2.1 Transmission dynamics of RSV.

Mathematical models have been used in epidemiology to characterize annual
epidemics of respiratory infections and estimate the likely effectiveness of
intervention strategies. Using ordinary differential equations (ODEs), many
models have been developed in the past to describe the transmission dynamics
of RSV (White et al., 2005, 2007; Leecaster et al., 2011; Weber et al., 2001;
Paynter et al., 2014; Arenas et al., 2009). The Susceptible-Infected-Recovered
(SIR) model (Kermack and McKendrick, 1927), the Susceptible-Exposed- Infected-
Recovered (SEIR) (Cooke and Van Den Driessche, 1996) and their variants, such
as SIS, SIRS, and delayed SIR, are only a few examples of such models. For
instance, Acedo et al. (2010) introduced an age-structured SIRS model, while
Moore et al. (2014) developed a seasonally forced compartmental age-structured
Susceptible-Exposed-Infectious-Recovered-Susceptible (SEIRS) model to fit to
the seasonal curves of positive RSV detections using the Nelder-Mead method.

Most recently, Baker et al. (2019) used a time-series Susceptible-Infected-
Recovered (TSIR) model to estimate future trajectories of RSV. The simple SIR
component of their model assumes that each susceptible case will contract the
infection once. Therefore, this does not represent the actual situation in
which susceptible cases transform into exposed cases first, and then a
proportion of exposed cases become infected.

###  2.2 ML based RSV predictive models.

Modern machine learning approaches have also shown to be beneficial in
addition to mathematical models. Reis et al. (2019) proposed a superensemble
approach to combine three forecasting methods, for RSV prediction in the US at
three different spatial resolutions: national, regional, and state. They used
three methods to generate retrospective forecasts of RSV outbreaks, including:
1) Bayesian Weighted Outbreaks (BWO), 2) the ensemble adjusted Kalman filter
(EAKF) combined with a dynamical susceptible-infected-recovered (SIR) model,
and 3) the null model using historical expectance (Reis and Shaman, 2016).
Gebremedhin et al. (2022) fitted a multivariable logistic regression model to
identify predictors of RSV-positivity (binary outcome) amongst children under
the age of five who were hospitalized and tested for RSV in Western Australia.

Figure 1: Architectures of the DeCom model.

##  3 Preliminaries

In this section, we review the Canonical Polyadic Decomposition (CPD), an
essential step for developing our model. The notation used throughout the
paper has been summarized in Table 1.

###  3.1 Canonical Polyadic Decomposition.

The Canonical Polyadic Decomposition (CPD) approximates a 3-way tensor
$\underline{\mathbf{X}}:=[\\![\mathbf{A},\mathbf{B},\mathbf{C}]\\!]\in\mathbb{R}^{L\times
M\times T}$ with a sum of $K$ rank-1 components, where $K$ is the tensor rank,
i.e.,

$\underline{\mathbf{X}}\approx\sum_{k=1}^{K}\mathbf{a}_{k}\circ\mathbf{b}_{k}\circ\mathbf{c}_{k},$
(1)

where $\mathbf{A}=[\mathbf{a}_{1},\dots,\mathbf{a}_{K}]\in\mathbb{R}^{L\times
K}$, $\mathbf{B}=[\mathbf{b}_{1},\dots,\mathbf{b}_{K}]\in\mathbb{R}^{M\times
K}$, $\mathbf{C}=[\mathbf{c}_{1},\dots,\mathbf{c}_{K}]\in\mathbb{R}^{T\times
K}$, and $\circ$ denotes the outer product.

The CPD of a tensor can be expressed in many different ways. For example, we
can unfold
$\mathbf{X}^{(1)}=\left(\mathbf{C}\odot\mathbf{B}\right)\mathbf{A}^{T}$ which
can be seen as forming a matrix using the mode-1 fiber of the tensorUsing
’role symmetry’, the mode-2 and mode-3 matrix unfolding are given by
$\mathbf{X}^{(2)}=\left(\mathbf{C}\odot\mathbf{A}\right)\mathbf{B}^{T}$,
$\mathbf{X}^{(3)}=\left(\mathbf{B}\odot\mathbf{A}\right)\mathbf{C}^{T}$,
respectively. With only $(L+M+T)\times K$ parameters, CPD can parsimoniously
represent tensors of size $L\times M\times T$. The CPD model is a particularly
powerful tool for data analysis due to two key properties: 1) every tensor
admits a CPD of finite rank, indicating that it is universal, and 2) under
moderate conditions, it can also be unique, i.e., it is possible to extract
the true latent factors from the synthesized $\underline{\mathbf{X}}$.

Table 1: Notations.

Notation | Description
---|---
$\underline{\mathbf{X}}\in\mathbb{R}^{L\times M\times T}$ | spatio-temporal tensor
$\mathbf{A}\in\mathbb{R}^{L\times K}$ | location factor matrix
$\mathbf{B}\in\mathbb{R}^{M\times K}$ | signal/feature factor matrix
$\mathbf{C}\in\mathbb{R}^{T\times K}$ | temporal factor matrix
$\underline{\mathbf{X}}^{(i)}$ | mode-$i$ unfolding of $\underline{\mathbf{X}}$
$K$ | No. of hidden components
$L$ | No. of locations
$M$ | No. of features
$T$ | No. of time steps
$\circ$ | outer product
$\odot$ | Khatri-Rao product

##  4 Problem Formulation

Consider a location $l\in L$ where we monitor $M$ features/signals related to
the RSV evolution over time, such as the number of new RSV infections,
temperatures, humidity, the number of new COVID-19 infections, etc. At time
$t\in T$, the value of the $m$-th signal at location $l$ is denoted as
$x_{lmt}$. Given that there are $L$ locations, $M$ features and $T$ time
steps, the dataset can be described as a 3-way spatio-temporal tensor
$\underline{\mathbf{X}}\in\mathbb{R}^{L\times M\times T}$, where
$\underline{\mathbf{X}}(l,m,t)\coloneqq x_{lmt}$. The evolution of all
features and regions from $t=1$ to $t=T$ is represented by the tensor
$\underline{\mathbf{X}}\in\mathbb{R}^{L\times M\times T}$. The goal is to
estimate the RSV cases from time $t=T+1$ to $t=T+T_{0}$ through the prediction
of the frontal slabs $\underline{\mathbf{X}}(:,:,t)$ for $T_{0}$ time-steps in
the future, where in our case $T_{0}$ usually sets to $52$ weeks.

(a) DeCom

(b) DeTensor

(c) LSTM

(d) ARIMA

(e) SARIMA

Figure 2: Comparison of state-level predictive performance.  Table 2: State-
level performance comparison in terms of RMSE and MAE of RSV cases. Results
show 3 different perdition intervals: up to 12 weeks, 24 weeks, and 52 weeks.

Model | RMSE (12 weeks) | MAE (12 weeks) | RMSE (24 weeks) | MAE (24 weeks) | RMSE (52 weeks) | MAE (52 weeks)
---|---|---|---|---|---|---
DeCom | 12.63(4.86) | 11.01(3.53) | 41.26(39.18) | 30.64(17.35) | 86.75(56.71) | 41.67(25.81)
DeTensor | 25.48(29.00) | 16.19(26.15) | 56.69(70.48) | 32.07(49.11) | 103.09(56.74) | 55.83(34.41)
LSTM | 42.72(108.57) | 37.08(91.87) | 54.53(62.40) | 45.00(51.65) | 99.19(122.49) | 71.14(86.91)
ARIMA | 18.47(24.81) | 16.68(24.02) | 78.37(92.44) | 71.20(85.05) | 100.11(115.40) | 69.34(78.36)
SARIMA | 15.91(14.26) | 14.83 (12.14) | 56.24(68.17) | 50.15(61.51) | 101.76(120.21) | 71.57(85.91)

##  5 Proposed Method

DeCom is a post-COVID-19 RSV prediction model with NPI awareness. We are
motivated by the fact that the seasonal RSV transmission pattern was
interrupted as soon as country-specific NPIs were implemented in 2020. Later
on, RSV resurfaces again with the reopening happening once the number of
COVID-19 cases falls below a certain level, signaling the end of the NPI
control period. Therefore, the NPI control has a significant impact on
breaking the normal RSV seasonality.

Figure 1 shows a high-level illustration of DeCom. As one can see, DeCom
requires two spatiotemporal tensors as input, each with three attributes:
location $l$, feature $m$, and time $t$. The first tensor, denoted as
$\underline{\mathbf{X}}_{1}\in\mathbb{R}^{L\times M_{1}\times T_{1}}$, is the
pre-COVID tensor, which includes all feature values that occurred before March
15, 2020, when the seasonal RSV transmission has not been affected by COVID-19
yet. The second tensor $\underline{\mathbf{X}}_{2}\in\mathbb{R}^{L\times
M_{2}\times T-T_{1}}$ denotes the post-COVID tensor, which covers the time
horizon between $T_{1}$ and $T$.

DeCom first performs a nonnegative CPD on $\underline{\mathbf{X}}_{1}$ to
learn the subspaces $\mathbf{A}_{1}\in\mathbb{R}^{L\times K_{1}}$,
$\mathbf{B}_{1}\in\mathbb{R}^{M_{1}\times K_{1}}$, and
$\mathbf{C}_{1}\in\mathbb{R}^{T_{1}\times K_{1}}$ for the typical pre-COVID
RSV seasons, which represent the latent subspaces of location, feature and
time, respectively. Here $K_{1}$ denotes the rank of $\underline{\bf X}_{1}$.
This is done by solving the following problem:

$\begin{split}\min_{\mathbf{A}_{1},\mathbf{B}_{1},\mathbf{C}_{1}}&||\underline{\mathbf{X}}_{1}-[\\![\mathbf{A}_{1},\mathbf{B}_{1},\mathbf{C}_{1}]\\!]||_{F}^{2}\\\
\text{s.t.}&~{}\mathbf{A}_{1}\geq\mathbf{0},~{}\mathbf{B}_{1}\geq\mathbf{0},~{}\mathbf{C}_{1}\geq\mathbf{0}.\end{split}$
(2)

Then it employs the first long short-term memory (LSTM) (Hochreiter and
Schmidhuber, 1997) network $f_{1}(\cdot)$ to capture the normal RSV
seasonality from the temporal subspace $\mathbf{C}_{1}$, such that the future
change of $\mathbf{C}_{1}$ from $t=T_{1}$ to $t=T$ can be estimated as

$\displaystyle\tilde{\mathbf{C}}_{1}=f_{1}(\mathbf{C}_{1})$ (3)

By following the regular seasonality (i.e., without COVID-19), the RSV cases
$\underline{\tilde{\mathbf{X}}}_{1}$ from $t=T_{1}$ to $t=T$ can be
approximated as

$\underline{\tilde{\mathbf{X}}}_{1}=[\\![\mathbf{A}_{1},\mathbf{B}_{1},\tilde{\mathbf{C}}_{1}]\\!]=\sum_{k=1}^{K_{1}}\mathbf{a}_{k,1}\circ\mathbf{b}_{k,1}\circ\tilde{\mathbf{c}}_{k,1}$
(4)

where $\mathbf{a}_{k,1}$, $\mathbf{b}_{k,1}$ and $\tilde{\mathbf{c}}_{k,1}$
denote the $k$-th column of $\mathbf{A}_{1}$, $\mathbf{B}_{1}$ and
$\tilde{\mathbf{C}}_{1}$, respectively.

The effect of COVID-19 NPIs on RSV transmission is usually reflected in the
number of RSV confirmed cases. One straightforward way is to calculate the
difference in RSV cases between a normal RSV season and a season with a
COVID-19 impact. Note that during the COVID-19 outbreak, what we observed is
the number of RSV cases under COVID-19 impact, while that number by assuming
no COVID-19 can only be approximately estimated using (4). Therefore, we first
use $\underline{\tilde{\mathbf{X}}}_{1}$ to generate a residual tensor

$\Delta\underline{\mathbf{X}}_{2}=\underline{\mathbf{X}}_{2}-\underline{\tilde{\mathbf{X}}}_{1}$
(5)

which approximates the perturbations caused by COVID-19 NPIs on a normal RSV
transmission. Then we implement the second CPD on
$\Delta\underline{\mathbf{X}}_{2}$ to learn the residual subspaces
$\mathbf{A}_{2}\in\mathbb{R}^{L\times K_{2}}$,
$\mathbf{B}_{2}\in\mathbb{R}^{M_{2}\times K_{2}}$ and
$\mathbf{C}_{2}\in\mathbb{R}^{(T-T_{1})\times K_{2}}$, where $\mathbf{C}_{2}$
captures how COVID-19 NPIs affect RSV transmission in the latent subspace and
$K_{2}$ is the tensor rank of $\Delta\underline{\mathbf{X}}_{2}$. We train the
second LSTM network $f_{2}(\cdot)$ on $\mathbf{C}_{2}$ to learn the residual
transmission pattern in the latent space $\mathbf{C}_{2}$.

Since the last row of $\mathbf{C}_{2}$ corresponding to time $T$, the residual
transmission pattern from $(T+1)$ to $(T+T_{0})$ is predicted by passing
$\mathbf{C}_{2}$ into $f_{2}(\cdot)$, i.e.,

$\displaystyle\tilde{\mathbf{C}}_{2}=f_{2}(\mathbf{C}_{2}).$ (6)

As a result, the residual tensor from $(T+1)$ to $(T+T_{0})$ is calculated as

$\displaystyle\Delta\underline{\tilde{\mathbf{X}}}_{2}=[\\![\mathbf{A}_{2},\mathbf{B}_{2},f_{2}(\mathbf{C}_{2})]\\!]=\sum_{k=1}^{K_{2}}\mathbf{a}_{k,2}\circ\mathbf{b}_{k,2}\circ\tilde{\mathbf{c}}_{k,2}.$
(7)

It follows from (4) and (5) that the RSV cases from $(T+1)$ to $(T+T_{0})$ is
predicted through

$\begin{split}\hat{\underline{\mathbf{X}}}_{2}\approx[\\![\mathbf{A}_{1},\mathbf{B}_{1},f_{1}(\tilde{\mathbf{C}}_{1})]\\!]+[\\![\mathbf{A}_{2},\mathbf{B}_{2},f_{2}(\mathbf{C}_{2})]\\!]\end{split}$
(8)

where $[\\![\mathbf{A}_{1},\mathbf{B}_{1},f_{1}(\tilde{\mathbf{C}}_{1})]\\!]$
denotes the prediction of RSV cases without COVID-19 impact.

######  Remark 1.

The advantages of our method are two-fold: 1) It allows us to incorporate data
from other sources to improve predictive power.2) It can easily be generalized
to predict other respiratory disease outbreaks under COVID-19 impact, such as
influenza.

##  6 Experimental Setup

Data. The dataset used in the experiments is a US state-level RSV dataset that
contains weekly RSV cases for 52 US states, including Puerto Rico and
Washington, D.C., between 2015 to 2022. It consists of three data sources: 1)
RSV features from a large-scale medical claims database, where we extracted
the state-level RSV confirmed cases and RSV related ICD-10 diagnosis features
such as B79.4 (RSV), J21.0 (acute bronchiolitis due to RSV), J20.5 (acute
bronchitis due to RSV), and J21.1 (pneumonia due to RSV) (W.H.O., 2004). 2)
COVID-19 related features from Johns Hopkins University (Dong et al., 2020).
3) Climate features from the Atmospheric Research Reanalysis Dataset include
humidity, precipitation, and temperature (Kalnay et al., 1996). These features
have been found to be significantly associated with the mean timing of the
onset of the RSV epidemic (Baker et al., 2019).

###  6.1 Evaluation pipeline.

The training data ranges from 2015-11-06 to 2021-04-16, whereas the testing
data ranges from 2021-04-23 to 2022-04-15. It is worth mentioning that there
was no RSV pandemic in 2020, despite testing data indicated an out-of-season
RSV outbreak across all states, with peak levels exceeding prior maxima in
most states. The purpose of utilizing such a training-testing split is to see
if an algorithm can learn the NPI influence on RSV transmission and forecast
the time and strength of the next RSV wave without having observed an RSV
surge in the previous year. The experimental design does, in fact, provide a
highly difficult setting for all machine learning algorithms. This way, we are
able to verify if DeCom’s NPI awareness mechanism can properly predict RSV
outbreaks under the effect of COVID-19.

(a) DeCom

(b) DeTensor

(c) LSTM

(d) ARIMA

(e) SARIMA

Figure 3: Country-level predictive performance for (a) DeCom, (b) DeTensor,
(c) LSTMs, (d) ARIMA, and (e) SARIMA. DeCom can accurately predict the peak
time ($\pm 1$ week) and generate forecasts with significantly larger values in
magnitude.

###  6.2 Evaluation metrics and baselines.

We report the predictive performance on the US state-level and country-level
RSV data, where the latter is aggregated from 52 states, to evaluate the
performance of our model. We focus on estimating RSV peak dates and weekly
confirmed cases. We compute the root mean squared error (RMSE) and mean
absolute error (MAE) as the performance metric.

$\mathrm{RMSE}=\sqrt{\frac{1}{T}\sum_{t}(X_{t}-\hat{X}_{t})^{2}}$ (9)
$\mathrm{MAE}=\frac{1}{T}\sum_{t}|X_{t}-\hat{X}_{t}|$ (10)

We compare the predictive performance of DeCom with deep tensor factorization
model (DeTensor) (Kargas et al., 2021), LSTM (Hochreiter and Schmidhuber,
1997), ARIMA, and Seasonal ARIMA (SARIMA) (Hyndman and Athanasopoulos, 2018)
models trained for each state.

### DeTensor.

Tensor factorization model without residual connection. Unlike proposed DeCom
model, DeTensor performs nonnegative CPD only once to learn the latent
subspace representing a mixture of both normal RSV seasonality and the
influence due to COVID-19. By comparing DeTensor, we are able to assess the
effectiveness of using the second tensor in our model for NPI awareness.

### LSTM.

The LSTM is a special Recurrent Neural Network (RNN), and it is capable of
learning long-term dependencies in time series data. For every state in the
dataset, we train its unique LSTM model using historical features, including
COVID-19 features, for each state as input, and independently predict the RSV
evolution for each state.

### ARIMA and SARIMA.

ARIMA, or AutoRegressive Integrated Moving Average is a statistical method
widely used to identify a suite of different standard temporal structures in
time series data. SARIMA is an extension to ARIMA that supports the modeling
of the seasonal component of the time series. The ARIMA and SARIMA models were
implemented in Python using the statmodel package Seabold and Perktold (2010),
and the optimal parameters were found using a grid search on $(p,q,d)$. We use
$s=48$ steps per circle for SARIMA.

######  Remark 2.

We note that there are standard epidemiological models such as SIR-based
methods for seasonal respiratory disease transmission (Baker et al., 2020).
The standard way is to embed a cosine function with a perios of 52 weeks to
capture the seasonality of RSV. To implement NPI, the SIR-based approaches
manually select a reduction ratio and multiply it with a pre-defined
transmission rate, which may not be able to capture heterogeneities in NPIs
across locations and different time windows (Baker et al., 2020). Therefore,
the SIR-based models that predict the arrival time of the next infectious peak
strongly rely on the previously learned phase shift, the selection of
transmission rate as well as when NPIs are terminated, where the latter is
usually unpredictable. However, data-driven models like DeCom and other deep
learning models do not have such a limitation since the effect of NPIs will be
reflected on the COVID-19 infections and will be learnt by the model. This is
the major reason that we do not include such epidemiological methods for
comparison.

##  7 Experimental Results

We first evaluate the state-level RSV prediction performance of DeCom,
DeTensor, LSTM, ARIMA, and SARIMA. Figure 2 shows the outcomes of state-level
RSV prediction, with Florida, Virginia, Maryland, and New York among the
chosen states. The reason those four states have been selected is that they
span from the south to the north and feature a shift in the transmission of
RSV outbreaks. Therefore, we can test if an algorithm can capture the
transmission difference caused by geometry differences.

Table 2 compares the state-level prediction performance of different models
over various time intervals (12, 24, and 52 weeks, respectively). We provide
the mean value of RMSE and MAE as well as their standard deviation (in
parenthesis). DeCom has a significantly lower RMSE and MAE when focusing on
the most difficult challenge of predicting RSV cases up to 52 weeks.
Meanwhile, DeCom outperforms other models by a wide margin in 24-week
predictions and performs best for forecasts up to 12 weeks. However, the
performance of DeTensor, ARIMA, and SARIMA becomes more competitive as the
problem gets easier, i.e., the prediction interval gets smaller.

Table 3: Country-level performance comparison in terms of peak time (weeks) estimates, RMSE and MAE of RSV cases.  Model | Peak Diff. | RMSE | MAE
---|---|---|---
DeCom | 1 | 2461 | 1891
DeTensor | 3 | 3175 | 2485
LSTM | 4 | 4555 | 3711
ARIMA | 8 | 4523 | 3490
SARIMA | 5 | 2688 | 2102

The results clearly demonstrate that, in terms of peak time and intensity
estimations, DeCom is superior to the other four approaches. In Maryland, for
instance, the estimates of DeCom are aligned with the ground truth, whereas
those of DeTensor, LSTM, ARIMA, and SARIMA have shifted left or right of the
true peak.

We also evaluate the effectiveness of DeCom using RSV predictions at the US
country-level. Specifically, we first use all five models to estimate the RSV
cases in 52 US states and then accumulate the state-level cases to the
country-level. The RSV estimates are plotted in Figure 3 and the performance
metrics are summarized in Table 3. When the difference between the actual peak
and the predicted peak is less than $\pm 1$ week, DeCom can accurately predict
the peak time. Furthermore, when compared to other baseline models, DeCom
forecasts RSV cases much closer to the ground truth, allowing it to capture
the intensity more accurately. Overall, DeCom delivers 23% lower RMSE and 24%
lower MAE than DeTensor, 46% lower RMSE and 49% lower MAE compared to state-
wise LSTMs, 46% lower RMSE and MAE compared to ARIMA, and approximately $10\%$
improvement over SARIMA.

##  8 Conclusion

In this paper, we introduce DeCom, a deep coupled tensor factorization machine
that captures both the seasonality of RSV and the influence of COVID-19 NPIs
on RSV transmission. Computer results showcase that DeCom outperforms several
time-series prediction baselines in predicting the timing and magnitude of RSV
outbreaks under the influence of COVID-19. As a consequence, we believe that
DeCom can be an effective model to forecast and assess post COVID-19 RSV
outbreaks.

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# Investigating the classical problem of pursuit, in two modes

A.H. Arshadi Kalameh  Department of Physics, Isfahan University of Technology,
Isfahan, Iran  K. Bayati Komitaki  Department of Physics, Isfahan University
of Technology, Isfahan, Iran  R. Sharifian  Department of Physics, Sharif
University of Technology, Tehran, Iran  M.M. Eftekhari  Department of Physics,
Isfahan University of Technology, Isfahan, Iran

(4 September 2023)

###### Abstract

The pursuit problem is a historical issue of the application of mathematics in
physics, which has been discussed for centuries since the time of Leonardo Da
Vinci, and its applications are wide ranging from military and industrial to
recreational, but its place of interest is nowhere but nature and inspiration
from the way of migration of birds and hunting of archer fish. The pursuit
problem involves one or more pursuers trying to catch a target that is moving
in a certain direction. In this article, we delve into two modes of movement:
movement on a straight line and movement on a curve. Our primary focus is on
the latter. Within the context of movement on a straight line, we explore two
methods and compare their respective results. Furthermore, we investigate the
movement of two particles chasing each other and extend these findings to N
particles that are chasing each other in pairs. By leveraging these two modes
of movement, we present a novel relationship for two-particle and N-particle
systems in pursuit. Lastly, we analyze the movement of moths around a lamp and
evaluate their motion in relation to two-particle and N-particle systems in
pursuit. The results of this analysis are carefully examined.

††preprint: AIP/123-QED

##  I Introduction

###  I.1 History

The pursuit problem was apparently first stated by Leonardo da Vinci. In this
problem, the cat is chasing the mouse while the speed of both of them is
constant and the mouse is moving in a straight line guha1994leonardo .In 1732,
a problem called (bugs), which was known as the dog, mouse, ant or beetles
problem, was investigated by the French scientist Pierre Bouguer. These
problems originate from the mathematics of pursuit curvesmarshall2003pursuit .
In 1877, Edward Lucas posed the three dog problem. It is stated in this
problem: The dogs are first placed at the three vertices of the equilateral
triangle, then each one follows the other. What is the curve that describes
their motion? Three years later, Henri Brocard showed that the dog’s path is a
logarithmic spiral and the dogs eventually meet at a common point, this common
point is called Brocard point of a trianglemarshall2003pursuit . For N bugs
that are initially located on the vertices of regular N-polygon and follow
each other at a constant speed, their trajectory is a logarithmic spiral and
finally they all meet at the center of the polygonbernhart1959polygons .

###  I.2 Applications

It has been suggested in kaden2022maneuvers  that air transportation will
reduce its emissions in the near future. One proposed method to achieve this
is by using the upwash of the wake vortex generated by a leading aircraft.
This concept is inspired by migratory birds; however, the main challenge is
overcoming the difficulties associated with pursuing the leading aircraft,
which can be resolved by implementing a generalized solution of the chase
problem. Multiple unmanned aerial vehicles (UAVs) or mobile robots, as
discussed in hattenberger2007autonomous ; galzi2006uav , require a generalized
solution of the chase problem, similar to that discussed in kaden2022maneuvers
.

There is a wide range of applications for controlling multiple UAVs, such as
exploration or surveillance, military or civilian scenarios, as noted in
hattenberger2007autonomous ; mungan2005classic . Additionally,
shaffer2009football describes a new strategy, called the interception
strategy, for American football players to catch the ball carrier. This
strategy was commonly used when the sky was overcast shown in
FIGshaffer2009football . 1.

Figure 1:  New strategy

Houseflies, like migratory birds, tend to follow the leader and choose their
path based on the angula velocity of the chasing fly mungan2005classic ;
land1974chasing . The cinematography of the chase problem was used to measure
the path taken by teleost fish and archer fish when catching their food
lanchester1975pursuit ; rossel2002predicting ; wohl2006hunting . Furthermore,
the initial angle and distance strongly influence the route selected by
teleost fish and archer fish to avoid collisions or choose the best path. This
was noted in fajen2003behavioral . Lastly, cinematography is required to
target a quarry when only a partial location is known, as well as to control
impact direction or limit the minimum radius curve (which maximizes
acceleration) mungan2005classic . A framework for controlling the formation of
a group of robots was developed, and one critical parameter for this framework
is the leader-follower graph wang1991navigation ; das2002vision ;
sattigeri2003adaptive .

##  II STRAIGHT LINE CHASING PROBLEM

For many years, the issue of chasing with a straight line has been of great
importance and has been extensively studied. This problem involves examining
two main types of straight lines: those that are parallel to the horizontal or
vertical axis, and those that are angled with respect to these axes.

The first type of straight line path is a special case of the second, and both
types are crucial to understanding this problem and finding potential
solutions. To begin solving this problem, we can utilize the equation of the
curve’s slope for the first type of straight line path (as shown in FIG. 2).
This approach was presented in mungan2005classic  and can be followed by
proceeding with the rest of the necessary work.

Figure 2: First type Figure 3: Second type

$Vt-x=(H-y)\frac{v_{x}}{v_{y}}\\\ $ (1)

By adopting this approach, they were able to derive various equations,
including the equation for the distance between two objects in motion at a
given moment, the equation for the vertical component of velocity, and the
pursuit curve.Additionally, the equations can be used to determine the
collision point.

$z=1-\frac{y}{H}$ (2) $v_{y}=\frac{2v}{z^{\frac{-V}{v}}+z^{\frac{V}{v}}}$ (3)

$L=\frac{H}{2}(z^{1-\frac{V}{v}}+z^{1+\frac{V}{v}})$ (4)
$\frac{x}{H}=(\frac{z^{2}-1}{4})-\ln(\sqrt{z})$ (5)

As discussed in mungan2005classic , the problem was solved in Palmaccio  by
defining the slope (follower). However, the difference this time was that the
slope was expressed as the derivative of y with respect to x. Using method
Palmaccio , it is also possible to identify the collision location.
Essentially, the fundamental concepts were the same in both methods, and the
solution could be obtained.

The two methods discussed above have been utilized to resolve the problem.
Another approach is presented in chashchina2007dog . In this method, rather
than examining the slopes, location vectors are initially assigned to both
movers. Since the pursuer is constantly moving towards the pursued, this
yields the following equation:

$\vec{r}_{1}-\vec{r}_{2}=k(t)\vec{V}$ (6)

where:

$k(0)=\frac{L}{V}$ (7) $k(T)=0$ (8)

By utilizing the equations mentioned above, it is possible to determine the
arrival time of both movers, as well as the pursuit curve (which was also
obtained in Palmaccio ). Additional information regarding this can be found in
chashchina2007dog .

We will now explore the techniques for solving the second type of problems. In
guha1994leonardo , one of the methods to resolve this problem is presented.
The scenario involves a cat chasing a mouse that moves along a straight line,
making an angle of $\alpha$ with the horizon line. The cat moves at a speed of
kv (where k > 1), while the mouse moves at a speed of v. M is the intersection
point of the tangent line at point C on the line IT, creating the angle
$\beta$ (FIG. 3).

The solution to the problem can be summarized by the following three key
equations:

$\frac{dx}{dt}=kvcos\beta$ (9) $\frac{dy}{dt}=v$ (10)
$\frac{dz}{dt}=vcos\beta-kv$ (11)

As evident, the solution approach in this case is comparable to the previous
two methods of the first type, except for the fact that the angle’s impact
needs to be taken into account since the path is not parallel to the horizon
line. The problem is ultimately resolved by utilizing the aforementioned
equations. For instance, the arrival point of the two mobiles can be
determined as follows:

$X_{T}=y_{0}\frac{cos\alpha(k+sin\alpha)}{k^{2}-1}$ (12)
$Y_{T}=y_{0}(1+\frac{k+sin\alpha}{k^{2}-1}sin\alpha)$ (13)

Moreover, this method derives the pursuit differential equation (refer to
guha1994leonardo  for more information).

Next, we will examine an alternative method introduced in barton2000pursuit .
The approaches outlined in barton2000pursuit  are considered general methods
for resolving all pursuit problems, even if the motion functions are non-
linear and intricate. The following equations govern the pursuer and pursued
coordinate systems, where (x, y) represents the former and (X, Y) represents
the latter:

$X=x+\lambda\dot{x}$ (14) $Y=y+\lambda\dot{y}$ (15)
$\dot{x}^{2}+\dot{y}^{2}=c^{2}(\dot{X}^{2}+\dot{Y}^{2})$ (16)

The equations are rewritten below, taking into account the angle $\alpha$
formed with reference to the horizontal axis:

$x+\lambda\dot{x}=tcos\alpha$ (17) $y+\lambda\dot{y}=y_{0}+tsin\alpha$ (18)
$\dot{x}^{2}+\dot{y}^{2}=c^{2}$ (19)

The collision position determined from the equations is in agreement with the
collision location stated in method guha1994leonardo . It can be asserted that
among all these techniques, barton2000pursuit  is the most efficient. This is
because it offers a general principle and can analyze more intricate movements
with greater ease.

##  III ROTATIONAL MOVEMENT

###  III.1 What is the problem?

Consider a scenario where a fox with a speed of v is chasing a rabbit with the
same speed. The rabbit moves at a path where its velocity makes an angle of
$\alpha$ with respect to the line connecting the two entities. If their
initial distance is l, where and when will they meet?Montazeri

###  III.2 Why this problem?

The determination of the rabbit and fox’s arrival time and position is not a
primary focus of this article. The solution can be found in the same book
Montazeri  and in similar studies chashchina2007dog ; barton2000pursuit ;
Irodov ; ptak1996dog . What is significant in this context is the behavior and
movement pattern of the hunter and prey in such systems. Furthermore, the
study of this behavior in complex systems is of great importance
marshall2003pursuit ; marshall2004formations . Most of the previous
investigations chashchina2007dog ; ptak1996dog have solely considered the
convergence point of the two entities.

###  III.3 Solution

Figure 4: Motion’s diagram of fox and rabbit in Cartesian coordinate system As
we see rabbit’s velocity makes an angle of $\alpha$ with respect to the line
connecting the two entities(L′).

$\dot{\theta}=\frac{d\theta}{dt}=\frac{vsin\alpha}{L-(1-v)cos\alpha}$ (20)
$\bigtriangleup v(\hat{v})=vcosd\theta-v=0$ (21) $\lim_{\bigtriangleup t\to
0}\bigtriangleup v(\hat{\theta})=dv(\hat{\theta})=vsind\theta=vd\theta$ (22)
$dv=\sqrt{dv(\hat{\theta})+dv(\hat{v})}\xRightarrow{dv(\hat{v})=0}dv=dv(\hat{\theta})$
(23)

Using Eqs. (20), (21), (22) and (23) then we can write:

$L^{\prime}=L-(1-v)t\cos{\alpha}$ (24)
$\begin{split}d\theta=(\frac{v\sin{\alpha}}{L-(1-v)t\cos{\alpha}})dt\\\
\xRightarrow{dv=vd\theta}dv=(\frac{v\sin{\alpha}}{L-(1-v)t\cos{\alpha}})vdt\end{split}$
(25) $dv(\hat{x})=dv\cos{\theta}$ (26) $dv(\hat{y})=-dv\sin{\theta}$ (27)
$\sin{\theta}=\frac{dx}{\sqrt{(dx^{2}+dy^{2})}}\xRightarrow{\frac{1}{dt}}\sin{\theta}=\frac{\frac{dx}{dt}}{(\frac{\sqrt{(dx^{2}+dy^{2})}}{dt})}=\frac{v(x)}{v}$
(28)
$\cos{\theta}=\frac{dy}{\sqrt{(dx^{2}+dy^{2})}}\xRightarrow{\frac{1}{dt}}\cos{\theta}=\frac{\frac{dy}{dt}}{(\frac{\sqrt{(dx^{2}+dy^{2})}}{dt})}=\frac{v(y)}{v}$
(29)

By merging Eqs. (25), (26), (27), (28), and (29), it is feasible to formulate
two coupled differential equations for velocity in two directions.

$dv(\hat{x})=(\frac{vv(\hat{y})\sin{\alpha}}{L-v(1-\cos{\alpha})t})dt$ (30)
$dv(\hat{y})=-(\frac{vv(\hat{x})\sin{\alpha}}{L-v(1-\cos{\alpha})t})dt$ (31)

##  IV RESULT

The approach we have employed thus far is somewhat similar to the method
presented in Montazeri  for determining the endpoint and we can determine the
end point without solving Eqs. (30) and (31). However, our primary focus is to
examine the motion parameters, such as velocity and location, along the
direction of motion. To achieve this, we can solve the coupled differential
equations of the system using software tools like Mathematica. Solving with
Mathematica:

$\begin{split}v_{x}(t)=C_{1}\cos{[\frac{\sin{\alpha}\ln{(L+(-1+\cos{\alpha})tV)}}{-1+\cos{\alpha}}]}+\\\
C_{2}\sin{[\frac{\sin{\alpha}\ln{(L+(-1+\cos{\alpha})tV)}}{-1+\cos{\alpha}}]}\end{split}$
(32)

$\begin{split}v_{y}(t)=C_{2}\cos{[\frac{\sin{\alpha}\ln{(L+(-1+\cos{\alpha})tV)}}{-1+\cos{\alpha}}]}-\\\
C_{1}\sin{[\frac{\sin{\alpha}\ln{(L+(-1+\cos{\alpha})tV)}}{-1+\cos{\alpha}}]}\end{split}$
(33)

Well, we can take the integral of these equations to give us the coordinate
equation for motion over time:

$x(t)=B_{1}+{(L+(-1+\cos{\alpha})tV)}\times\\\
(\frac{((-1+\cos{\alpha})C_{1}-\sin{\alpha}C_{2})\cos{[\frac{\sin{\alpha}\ln{[L+(-1+\cos{\alpha)tV}]}}{-1+\cos{\alpha}}]}}{{(1-2\cos{\alpha}+\cos{\alpha}^{2}+\sin{\alpha}^{2})V}}+\\\
\frac{(\sin{\alpha}C_{1}+(-1+\cos{\alpha})C_{2})\sin{[\frac{\sin{\alpha}\ln{[L+(-1+\cos{\alpha})tV]}}{-1+\cos{\alpha}}]}}{{(1-2\cos{\alpha}+\cos{\alpha}^{2}+\sin{\alpha}^{2})V}})\\\
$ (34)
${y(t)=B_{2}+{(L+(-1+\cos{\alpha})tV)((\sin{\alpha}C_{1}+(-1+\cos{\alpha})}}\times\\\
\frac{C_{2}\cos{[\frac{\sin{\alpha}\ln{[L+(-1+\cos{\alpha)tV}]}}{-1+\cos{\alpha}}]}}{(1-2\cos{\alpha}+\cos{\alpha}^{2}+\sin{\alpha}^{2})V}+\\\
\frac{(C_{1}-\cos{\alpha}C_{1}+\sin{\alpha}C_{2})\sin{[\frac{\sin{\alpha}\ln{[L+(-1+\cos{\alpha})tV]}}{-1+\cos{\alpha}}]}}{(1-2\cos{\alpha}+\cos{\alpha}^{2}+\sin{\alpha}^{2})V})$
(35)

Note that the coefficients $C_{1}$,$C_{2}$,$B_{1}$ and $B_{2}$ are established
based on the initial conditions. We have flexibility in selecting the
coordinates by rotating them, and we opt for the coordinates that satisfy the
following initial conditions:

$v_{x}(0)=0,v_{y}(0)=v,x(0)=0,y(0)=0\\\
v_{x}(t)=\frac{-v\cos{[\frac{[\sin{\alpha}\ln{[L+(-1+\cos{\alpha})tv]}}{-1+\cos{\alpha}}]}\sin{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]}}{\cos{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}}+\sin{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}}}+\\\
\frac{v\cos{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]}\sin{[\frac{\sin{\alpha}\ln{[L+(-1+\cos{\alpha})tV]}}{-1+\cos{\alpha}}]}}{\cos{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}}+\sin{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}}}$
(36)

$v_{y}(t)=\frac{(v\cos{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]}\cos{[\frac{\sin{\alpha}\ln{[L+(-1+\cos{\alpha})tv]}}{-1+\cos{\alpha}}]}}{(\cos{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}}+\sin{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}})}+\\\
\frac{v\sin{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]}\sin{[\frac{\sin{\alpha}\ln{[L+(-1+\cos{\alpha})]tv}}{-1+\cos{\alpha}}]})}{(\cos{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}}+\sin{[\frac{\sin{\alpha}\ln{[L]}}{-1+\cos{\alpha}}]^{2}})}$
(37)

We set the initial conditions as follows (note that the specific choice of
initial conditions does not impact the overall outcome of our movement
analysis):

v=2 (m/s) ,L=20(m), $\alpha$ = $\pi$/3

Figure 5: graph of the Fox’s speed over time Figure 6: A graph of the rabbit’s
speed over time

FIG. 5 and FIG. 6 depict the initial phenomenon that occurs in this system,
where the changes in speed begin to increase with the passage of time. In the
twentieth second, these changes, including acceleration and a sharp increase
in speed with a sign change, become more prominent. To analyze this event, we
need to examine the system in a relative device. In a relativistic device, the
length of the connecting line decreases (radius of gyration), leading to an
increase in angular velocity changes. At the point where the two moving
entities meet, this length tends to zero, causing our rotational acceleration
to tend towards infinity. Furthermore, the time required to complete a round
tends to zero, resulting in an infinite acceleration over time, and an
increase in the speed of changing the speed sign due to the reduction in
rotation time. (the two moving parts rotate together)

It is worth noting that we have formulated the equations of motion for the
fox, and the equations of motion for the rabbit are identical to those of the
fox, except for the boundary conditions.

Figure 7: Fox and rabbit movement path diagram in Cartesian coordinates

FIG. 7, we observe that the shape of the rabbit’s movement path is similar to
that of the fox, except that the initial conditions in the equations show that
we have shifted the fox’s diagram by the initial distance L and rotated it to
the right by $\alpha$. This indicates that we can convert the fox diagram into
a rabbit diagram, as mentioned earlier.

Figure 8: When the $\alpha$ = 60∘ the initial conditions of our two-particle
system exhibit behavior that is similar to that of a six-particle system.
Additionally, this behavior displays the same symmetry as the figure opposite
to it.

Another conclusion that we can draw from FIG. 7 is that the rabbit moves like
a fox that is chasing a rabbit in the direction of its initial speed, while
being located at a distance of L. If we continue this reasoning, we can assume
that the rabbit behaves like a fox that is pursuing another rabbit. Thus, the
system of two particles following each other behaves like a system of N
particles following each other (FIG. 8).

For situations where $\alpha$ is a divisor of 360, our N-particle system
becomes a closed system, where the N bodies follow the first body due to the
constraint of movement in the connecting line. The interesting thing is that
this N-particle system behaves like a two-particle system. Therefore, we can
use the resulting relationship for an N-particle system for a two-particle
system.

For a closed N-particle system, we know that the shape symmetry system
preserves its initial shape. This symmetry implies that we can write the rates
of motion in the polar device to the center of this closed system.As the
system is symmetrical, the point of convergence of the particles is certainly
at its center.

Figure 9: In a generalized two-particle to N-particle system, the angle
between the center of rotation and each vertex of the N polygon that is con-
structed is equal to beta. This relationship holds true regardless of the
number of parti-cles involved in the system, allowing for a gen-eralized
approach to the problem. Figure 10: we can observe the triangle that is formed
by connecting $N_{i}$,$N_{i+1}$ and the center of rotation. By analyzing the
motion of the particle in two directions of polar coordinates. We can
generilized this method for every paticle.

$2\beta=\pi-\alpha\xRightarrow{\frac{1}{2}}\beta=\frac{\pi}{2}-\frac{\alpha}{2}$
(38)
$2R\sin{\frac{\alpha}{2}}=l\xrightarrow{}R=\frac{l}{2\sin{\frac{\alpha}{2}}}$
(39)

$v=\sqrt{(\dot{r})^{2}+(r\dot{\theta})^{2}}$ (40)
$\dot{r}=-v\cos{\beta}=-v\sin{\frac{\alpha}{2}}$ (41)

$r=R-v\sin{\frac{\alpha}{2}}t\xrightarrow{}r=\frac{l}{2\sin{\frac{\alpha}{2}}}-vt\sin{\frac{\alpha}{2}}$
(42)

Using Eqs. (38), (39), (40),(41) and (42) then we can write:

$v=\sqrt{(-v\sin{\frac{\alpha}{2}})^{2}+((\frac{l}{2\sin{\frac{\alpha}{2}}}-v\sin{\frac{\alpha}{2}}t)\dot{\theta})^{2}}$
(43)

$\dot{\theta}=\frac{v\cos{\frac{\alpha}{2}}}{(\frac{l}{2\sin{\frac{\alpha}{2}}}-v\sin{\frac{\alpha}{2}}t)}$
(44)

If we solve this differential equation

$-\frac{\theta}{\cot{\frac{\alpha}{2}}}=\ln{(\frac{l}{2\sin{\frac{\alpha}{2}}}-v\sin{\frac{\alpha}{2}}t)}+c$
(45)

c is a constant determined by the initial conditions

Using Eqs. (42) and (45) we can write :

$r(\theta)=r_{0}e^{-\frac{\theta}{\cot{\frac{\alpha}{2}}}}$ (46)

Eq. (46) is applicable to all the systems we have analyzed so far, including
the two-particle system in pursuit and N-particle systems’ motion behavior in
polar coordinates. Eq. (46) can also be observed in moth insects. In the past,
when the air was dark due to the absence of lamps and lighting in modern
cities, moths used the moon for orientation and movement, keeping the angle
between the direction of their movement and the distance vector between the
moth and the moon constant. This was possible because the moon was considered
to be at an infinite distanceWinNT . However, with the increase in brightness
in cities and the prevalence of lamps, the light from lamps became brighter
than that of the moon, causing moths to choose lamps as a source of movement.
By approaching the lamp, the moth tries to keep the angle between the vector
connecting to the light source and the direction of movement constant,
resulting in a rotational movement similar to the system of two tracked
particles. It can be argued that the motion of the two-particle system in
pursuit and the N-particle system in pursuit, previously investigated, moves
with a constant angle between the line connecting the particle to the center
of rotation and the velocity vector. This movement is similar to the movement
of moths.

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