# Pandemic infection forecasting
through compartmental model
and learning-based approaches

Marianna Karapitta, Andreas Kasis, Charithea Stylianides, Kleanthis Malialis,
Panayiotis Kolios
The authors are with the KIOS Research and Innovation Center of Excellence,
University of Cyprus, Nicosia, Cyprus (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>kolios.panayiotis@ucy.ac.cy).This work was
supported by the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 739551 (KIOS CoE - TEAMING) and from the
Republic of Cyprus through the Deputy Ministry of Research, Innovation and
Digital Policy. It was also supported by the CIPHIS (Cyprus Innovative Public
Health ICT System) project of the NextGenerationEU programme under the
Republic of Cyprus Recovery and Resilience Plan under grant agreement C1.1l2.

###### Abstract

The emergence and spread of deadly pandemics has repeatedly occurred
throughout history, causing widespread infections and loss of life. The rapid
spread of pandemics have made governments across the world adopt a range of
actions, including non-pharmaceutical measures to contain its impact. However,
the dynamic nature of pandemics makes selecting intervention strategies
challenging. Hence, the development of suitable monitoring and forecasting
tools for tracking infected cases is crucial for designing and implementing
effective measures. Motivated by this, we present a hybrid pandemic infection
forecasting methodology that integrates compartmental model and learning-based
approaches. In particular, we develop a compartmental model that includes
time-varying infection rates, which are the key parameters that determine the
pandemic’s evolution. To identify the time-dependent infection rates, we
establish a hybrid methodology that combines the developed compartmental model
and tools from optimization and neural networks. Specifically, the proposed
methodology estimates the infection rates by fitting the model to available
data, regarding the COVID-19 pandemic in Cyprus, and then predicting their
future values through either a) extrapolation, or b) feeding them to neural
networks. The developed approach exhibits strong accuracy in predicting
infections seven days in advance, achieving low average percentage errors both
using the extrapolation (9.90%) and neural network (5.04%) approaches.

##  I Introduction

According to the World Health Organization (WHO), a pandemic is a public
health emergency of international concern, defined as “an extraordinary event
which is determined to constitute a public health risk to other States through
the international spread of disease and to potentially require a coordinated
international response”. This definition implies a situation that is serious,
unexpected, carries implications for public health beyond the affected state’s
national border, and requires immediate international action [1]. Over the
years there have been numerous pandemics that caused a significant burden on
global economies and public health [2]. Existing evidence suggests that the
likelihood of pandemics has increased during the past century [3, 4].

Among a large number of historical pandemics, one that made humanity to highly
suffer, killing tens of millions of people, is the Cholera pandemic. The first
of seven cholera pandemics emerged in India in 1817 and was caused by the
ingestion of food or water contaminated with the bacterium Vibrio cholerae
[5]. The Acquired Immunodeficiency Syndrome (AIDS) is a pandemic, caused by
the HIV virus and was first identified in 1981. AIDS has claimed 40.1 million
lives so far with global ongoing transmission and remains a major global
public health issue [6]. First identified in 2003, Severe Acute Respiratory
Syndrome (SARS) is believed to have possibly originated from bats and spread
to humans in China [7]. SARS is characterized by respiratory problems, fever
and body aches and is spread through respiratory droplets from coughs and
sneezes. Quarantine efforts proved effective in controlling the virus, and
since then the disease has not reappeared.

At the beginning of December 2019, a “pneumonia of unknown etiology” outbreak
was reported in Wuhan, China [8]. Soon, the cause proved to be a new
coronavirus associated with the Severe Acute Respiratory Syndrome virus (SARS-
CoV), which later has been termed coronavirus disease 19 (COVID-19) [9]. The
rapid spread of the virus to all Chinese provinces and, as of 1st March 2020,
to 58 other countries [10], led the WHO to officially declare it a pandemic.
Since then, the world has been facing an unprecedented human crisis with
governments across the planet taking intervention measures to curtail the
spread of the disease. Some efforts to contain the pandemic included a range
of non-pharmaceutical interventions such as closing schools, banning public
events, suspension of all public transport, self-isolation, and lockdown
policies. Given the large uncertainty regarding the virus transmission,
government intervention policies varied considerably among countries. In
addition, although such interventions may cope with the spread of the virus
[11, 12], they impose a vital disruption to the economic and social structure
globally. The latter has lead studies to consider the trade-offs between the
pandemic effects and the economic impact resulting from the imposed
interventions, and proposing suitable strategies [13, 14].

Modelling and forecasting the disease’s spread is key for the assessment of
its transmissibility, and the design of suitable intervention measures [15].
Predicting the growth of infections may inform governments and healthcare
professionals about an upcoming wave, and enable appropriate intervention
strategies. Thus, accurately predicting infections is critical for effectively
managing a pandemic.

Traditionally, compartmental models have proved to be valuable instruments in
describing infectious disease epidemics and facilitating the characterization
of their asymptotic behaviour and dependence on model parameters. Various
approaches to model and predict the progression of the aforementioned COVID-19
outbreak rely on these models. Such models enable the study of the disease,
within a population divided into compartments, from which they are
traditionally named. An extensive review of compartmental models can be found
in [16]. A classic example of a compartmental 50 model is the SIR model [17,
18]. In the SIR model, the population is split into three compartments:
Susceptible (S) are the individuals who have no immunity and might become
infected if exposed, Infected (I) are the individuals who are currently
infected and can transmit the disease to susceptible individuals, and
Recovered (R) are the individuals who are immune to the disease and do not
transmit the disease to others. The variation of each compartment with time is
described by a set of equations with suitable parameters describing the rates
of transition among states [19]. Extensions of current models may involve the
incorporation of additional compartments, such as Exposed, Diagnosed, Ailing,
Threatened, Extinct, and Vaccinated. An eight-compartment model for COVID-19
is proposed in [20], which separates diagnosed and non-diagnosed infected
individuals as the former are typically isolated and hence less likely to
spread the disease.

For COVID-19, certain parameters within compartmental models represent crucial
information that determines the progression of the pandemic. This lead to
studies considering compartmental models with time-varying parameters [21,
22]. These models enable the study of the pandemic evolution by evaluating
possible influencing factors such as vaccine availability, restrictive
measures by governments, environmental temperature, or changes in virus
features. For example, [21] uses a SIRD (Susceptible, Infected, Recovered,
Deceased) model with time-varying parameters to capture possible changes of
the epidemic behaviour in Italy, due to measures enforced by authorities or
modifications of the epidemic characteristics. Furthermore, [22] uses a SEIR
(Susceptible, Exposed, Infected, Recovered) model with time-dependent
infection rate, recovery rate and reproduction number to predict the spread of
the COVID-19 pandemic in the United States.

The continuous evolution of dominant variants and the impact of the SARS-CoV-2
virus underscore the necessity for precise modeling of pandemic behavior,
including the identification of the most important influencing factors.
Predicting infection rates involves analyzing past data, acknowledging their
highly variable and dynamic nature.

Machine learning approaches have been widely employed for this effort,
demonstrating notable effectiveness in the case of COVID-19 [23, 24].
Incremental learning approaches [25, 26, 27, 28], enable the necessary
flexibility to achieve optimized solutions in non-stationary environments. In
[29] and [30], incremental learning of a neural network calculates five
parameters (rate of infection during lockdown, time lockdown begins, rate of
death, rate of recovery) of a SIRVD (Susceptible, Infected, Recovered,
Vaccinated, Deceased) model. Moreover, [29] aims to forecast the monthly
deceased population under different scenarios, while [30] focuses on
predicting the monthly total number of cases, active infections and deaths.
Furthermore, the hybrid approach developed in [31] uses data on the exposed,
infected, recovered and dead population with a combination of compartmental
models and a statistical ARIMA model to identify the most suitable time series
window that optimizes prediction. Then, a data stream and an ensemble of
machine learning algorithms incrementally provide predictions at each time
step [31]. Lastly, there exist works which are solely incremental learning-
based [32, 33, 34].

Our study develops a novel compartmental model designed to align with the
characteristics of the COVID-19 pandemic. The model enables the incorporation
of time-varying infection rates, that constitute the key parameters that
determine the evolution of the pandemic. In addition, it exhibits strong
performance in describing the infected population from our available COVID-19
data regarding the Republic of Cyprus, concerning a population of
approximately one million people, yielding a mean absolute percentage error of
$3.01\%$. The time-varying rates are forecast by a hybrid approach that
combines the proposed compartmental model with machine learning approaches. In
particular, our study utilizes the available data to estimate the dynamic
parameters of the model. Subsequently, predictions of their future values are
obtained through either a) extrapolation, or b) the employment of neural
networks. The neural networks are trained by incremental learning, allowing
the continual adaptation of the time-varying rates to the non-stationary
characteristics of the pandemic. The produced infection rates are later input
into the compartmental model to enable predictions of the infected population.
The developed approach demonstrates high predictive accuracy for future
infections seven days in advance, with low average percentage errors observed
in both extrapolation (9.90$\%$) and neural network (5.04$\%$) approaches.

The contributions of this work are summarized as follows:

  1. (i)

We propose a novel compartmental model that involves time-dependent infection
rates, which are the parameters with the largest impact on the pandemic
progression. The proposed model is able to describe the infected population
with high accuracy, yielding a mean average percentage error of 3.01%.

  2. (ii)

We establish a hybrid approach, that combines a compartmental model with
machine learning, suitable for the identification of the time-varying
infection rates. The developed approach achieves significantly low errors in
predicting future infections seven days in advance, both through the
extrapolation (9.90%) and neural network (5.04%) approaches, which are close
to the model’s ability to describe the data.

##  II Results and Discussion

To study the progression of the COVID-19 pandemic, we developed the
compartmental SIDAREVH model with time-varying infection rates, which
significantly enhance the model’s ability to describe the data associated with
the pandemic evolution. The estimation of the time-varying infection rate
values was achieved through two developed approaches explained in subsections
Model-based infection rates estimation and Learning-based infection rates
prediction and evaluation. The identified parameter values were then utilized
to develop predictions of the infected population, following the approach
described in subsection Infected population prediction and evaluation. All
subsections referenced above are detailed in section Methods. The proposed
approach’s forecasting accuracy was evaluated using the mean percentage error
between predicted values and data (see subsection Infected population
prediction and evaluation). Below, we present the developed SIDAREVH model,
and the main results concerning (i) the estimation of the time-varying
infection rates, associated with the developed SIDAREVH model, following
extrapolation and neural network approaches, and (ii) the accuracy of our
approach in predicting future infections.

###  II-A The SIDAREVH compartmental model

The SIDAREVH compartmental model is a deterministic model developed to
characterize the progression of the COVID-19 pandemic. The model was named
SIDAREVH, after its compartments, into which it partitions the investigated
population. The proposed compartments are: susceptible (S), unvaccinated
infected detected (I), vaccinated infected detected (D), unvaccinated
hospitalized (A), recovered (R), extinct (E), vaccinated susceptible (V), and
vaccinated hospitalized (H). A schematic representation of the SIDAREVH model
is shown in Fig 1.

Figure 1: The SIDAREVH model. Schematic representation of the SIDAREVH model
used to describe the evolution of the COVID-19 pandemic. The model splits the
population into susceptible, vaccinated susceptible, unvaccinated infected
detected, vaccinated infected detected, unvaccinated hospitalized, vaccinated
hospitalized, recovered and extinct. Model parameters $\zeta$, $\beta_{uu}$,
$\beta_{vu}$, $\beta_{vv}$, $\beta_{uv}$, $\gamma_{i}$, $\gamma_{d}$,
$\gamma_{\alpha}$, $\gamma_{h}$, $\xi_{i}$, $\xi_{d}$, $\mu_{\alpha}$,
$\mu_{h}$ indicate the transition rates between the states, where
$\beta_{uu}$, $\beta_{vu}$, $\beta_{vv}$, and $\beta_{uv}$ are considered as
time varying while the remaining as constant.

The SIDAREVH model includes a large number of parameters that describe the
transition rates between its compartments. The parameters with the largest
impact on the evolution of the pandemic are the infection rates $\beta_{uu}$,
$\beta_{vu}$, $\beta_{vv}$, $\beta_{uv}$ which are considered time-varying to
account for the effect of factors such as changes in the virus features,
control measures, and environmental characteristics. These infection rates
(IR) are described in Table I below.

TABLE I: Description of the time-varying infection rates involved in the SIDAREVH model. Parameter | Description
---|---
$\beta_{uu}$ | Unvaccinated to unvaccinated infection rate
$\beta_{vu}$ | Vaccinated to unvaccinated infection rate
$\beta_{vv}$ | Vaccinated to vaccinated infection rate
$\beta_{uv}$ | Unvaccinated to vaccinated infection rate

A full mathematical description and suitable explanations regarding the
validity of the SIDAREVH model are provided in section Description of the
SIDAREVH compartmental model.

###  II-B Infection rates model-based estimation and learning-based prediction

####  II-B1 Model-based estimated infection rates

The results of the estimated time-dependent infection rates
$\beta_{uu},\beta_{vu},\beta_{vv},\beta_{uv}$ involved in the SIDAREVH model,
using seven and fourteen-day estimation windows are shown in Fig 2. It should
be noted that the infection rates presented in Fig 2 were filtered by
replacing outliers with the nearest non-outlier value. Details regarding the
filtering technique can be found in Conclusion.

Figure 2: The model-based estimated infection rate (IR) values associated with
the SIDAREVH model. The model-based values of the time-dependent infection
rates, using sliding windows of seven and fourteen days: (A) Unvaccinated to
Unvaccinated IR: $\beta_{uu}$, (B) Vaccinated to Unvaccinated IR:
$\beta_{vu}$, (C) Vaccinated to Vaccinated IR: $\beta_{vv}$, and (D)
Unvaccinated to Vaccinated IR: $\beta_{uv}$.

Higher values of infection rates yield a higher frequency of new infections
within the population. The infection rate of the COVID-19 disease is affected
by various factors, such as the government’s intervention measures [35] and
the new variants introduced [36] during the examined period. In our considered
data, regarding Cyprus, the variants Alpha (B.1.1.7), Delta (B.1.617.2), and
Omicron (B.1.1.529) coexisted [37], being dominant at different periods. In
addition, studies such as [38] and [39] show that transmission rates, and
consequently the spread of the SARS-CoV-2 virus, are significantly correlated
with the environmental temperature and humidity.

Fig 2A depicts the estimated rate at which unvaccinated people infect other
unvaccinated people. This rate takes the highest values among the four rates
in Fig 2, due to the susceptibility of the unvaccinated individuals to the
disease. The $\beta_{uu}$ rates using seven and fourteen-day windows follow
the same shape in the entire investigated time, except the period around
January 2022 when the Omicron variant became dominant and replaced the Delta
variant. In this period, the resulting rate using fourteen-day windows was
changing slower and the filtering technique used to identify and fill the
outliers decreased these values. Fig 2B depicts the estimated rate at which
vaccinated people infect unvaccinated people. The estimated parameters
indicate that, with both seven and fourteen-day windows, the rates remain
nearly constant throughout the entire time period. The rate $\beta_{vv}$ at
which vaccinated people infect other vaccinated people, presented in Fig 2C,
takes the lowest values of all four infection rates, demonstrating the lower
infectivity among immunized population. Finally, the rate at which
unvaccinated people infect vaccinated people, denoted by $\beta_{uv}$ and
presented in Fig 2D, also follows the same shape for the entire time period.
The three rates presented in Figs 2B, C, D, that are associated with
vaccinations, are zero until the beginning of 2021 when the vaccinations
started in Cyprus [40].

The following results demonstrate the model’s ability to accurately describe
the data. In particular, Table II presents the model’s accuracy in terms of
describing the infected population by considering seven-day prediction
windows. For Table II, the unfiltered estimated infection rates
($\beta_{uu},\beta_{uv},\beta_{vu},\beta_{vv}$) were used (these are the
output values from Algorithm 1 presented in section Model-based infection
rates estimation). These estimated infection rate values, obtained by fitting
the model to the data, give a lower bound to the prediction accuracy that the
proposed SIDAREVH model may achieve through the use of seven-day windows. As
Table II shows, the SIDAREVH model may describe our data with a mean absolute
percentage error (MAPE) of 3.01%. In addition, when the largest 5% of the
absolute errors is excluded then the corresponding error drops to 2.63%. This
5% of highest errors largely coincides with periods of abrupt changes in the
pandemic behaviour attributed to external factors, and hence not being
predictable through the available data, such as when a new variant became
dominant or under sudden changes in the intensity of government intervention
measures. Excluding the largest 5% of the errors yields a notable reduction in
the MAPE (3.01% to 2.63%) and a substantial reduction in the associated
standard deviation (S.D.) (2.30% to 1.44%). The corresponding results
concerning windows of fourteen days are presented in Conclusion and
demonstrate higher error values. This lead us to choose seven-day windows,
rather than fourteen-day windows, for the estimation of the infection rates.

TABLE II: SIDAREVH model’s accuracy in describing the infected population data using model-based optimized unfiltered time-varying infection rates with seven-day windows. | MAPE | S.D.
---|---|---
All data | 3.01% | 2.30%
Lowest 95% of errors | 2.63% | 1.44%

####  II-B2 Learning-based predicted infection rates

In this section, we present the results associated with the infection rates
prediction using the approach described in section Learning-based infection
rates prediction and evaluation. The prediction accuracy is evaluated through
the widely used regression forecast metrics Mean Absolute Error (MAE) and Mean
Absolute Percentage Error (MAPE) (see Eqs (5a) and (5b in Methods).
Furthermore, all experiments are run over 10 repetitions, with a Multilayer
Perceptron approach, and the mean and standard deviation (S.D.) of the
relevant metrics are presented.

We have examined the performance of the proposed method using three lookback
window sizes $W\in\\{7,14,30\\}$ (see section Learning-based infection rates
prediction and evaluation). In all cases, the four considered infection rates
were forecast a week ahead. The best performing lookback window size,
presented in Table III below is with a fourteen-day window. It should be
clarified that the windows for the infection rates estimation, which was
selected to be seven-days, is different than the lookback window, which is
fourteen-days. The latter determines how many past values of infection rate
estimates are considered for prediction. For comparison, the results for all
window sizes are shown in Conclusion.

TABLE III: Multilayer Perceptron performance for each infection rate, for a
lookback window size of fourteen days, in terms of mean absolute error, mean
absolute percentage error and associated standard deviations.

| $\beta_{uu}$ | $\beta_{vu}$ | $\beta_{vv}$ | $\beta_{uv}$
---|---|---|---|---
MAE (S.D.) | 0.011 (0.015) | 0.003 (0.008) | 0.003 (0.005) | 0.006 (0.012)
MAPE (S.D.) | 6.346% (7.567%) | 1.483% (4.047%) | 15.403% (17.779%) | 7.996% (13.105%)

The infection rates are described as $\beta_{uu}$: Unvaccinated $\rightarrow$
Unvaccinated IR, $\beta_{vu}$: Vaccinated $\rightarrow$ Unvaccinated IR,
$\beta_{vv}$: Vaccinated $\rightarrow$ Vaccinated IR, $\beta_{uv}$:
Unvaccinated $\rightarrow$ Vaccinated IR.

In addition to Table III, the performance of the predicted infection rates
compared to the estimated ones is depicted in Fig 3, which demonstrates that
the estimated and forecasted values are very close, despite the dynamic nature
of the pandemic. Important to be noted is that the infection rate values are
directly affected by policy changes and weather-related conditions that are
not reflected in the data, and hence some error margin in their forecast is
expected, regardless of the adopted approach.

Figure 3: Model-based estimated versus learning-based predicted infection rate
(IR) values associated with the SIDAREVH model. (A) Unvaccinated to
Unvaccinated IR: $\beta_{uu}$, (B) Vaccinated to Unvaccinated IR:
$\beta_{vu}$, (C) Vaccinated to Vaccinated IR: $\beta_{vv}$, and (D)
Unvaccinated to Vaccinated IR: $\beta_{uv}$.

###  II-C Model and learning-based prediction of infected population

####  II-C1 Model-based prediction of infected population

The prediction results presented below were generated using the estimated
infection rates obtained from Algorithm 1 (see section Model-based infection
rates estimation). The projections refer to the infected population (both
vaccinated and unvaccinated). Table IV presents the model’s seven-day
predictive accuracy, in terms of mean absolute percentage errors, utilizing
the filtered estimated infection rates, illustrated in Fig 2, where the
outlier values were removed. This approach extrapolates the optimized
estimated infection rates, obtained at each day using the previous seven-day
window, to the next seven-day window, as explained in section Infected
population prediction and evaluation. Essentially, using past week’s
information, this approach enables predictions for the infected population for
the subsequent seven days. As shown in Table IV, this results in a mean
absolute percentage error of 9.90% (standard deviation 8.05%), which decreased
to 8.70% (standard deviation 6.22%) when the highest 5% of errors was
excluded.

TABLE IV: SIDAREVH model’s accuracy in describing the infected population data using model-based optimized filtered time-varying infection rates with seven-day windows. | MAPE | S.D.
---|---|---
All data | 9.90% | 8.05%
Lowest 95% of errors | 8.70% | 6.22%

Predictions of the infected population were also generated by utilizing
fourteen-day windows of previous data, through a similar approach. The
corresponding predictive performance and errors are available in Conclusion.
We chose to only present the prediction errors using seven-day windows of
previous data since, as demonstrated in Conclusion, the average daily error of
the predicted windows increases when fourteen-day windows of previous data are
considered, yielding a higher average percentage prediction error. This holds
regardless of the prediction timeframe, i.e. regardless of how many days in
the future we forecast.

####  II-C2 Learning-based prediction of infected population

The prediction results presented below were generated using the infection
rates obtained from Algorithm 2 (see section Learning-based infection rates
prediction and evaluation). The forecasts correspond to seven-day sliding
prediction windows utilizing the infection rates that best describe the
preceding seven-day period. Table V presents the error statistics of the
predictions created using the infection rates resulting from the neural
network approach presented in section Learning-based infection rates
prediction and evaluation. This approach enables a mean absolute percentage
error of 5.04% (standard deviation 7.03%), which drops to 3.74% (standard
deviation 4.27%) when the highest 5% error values are excluded. Hence, this
approach enables a significant improvement in terms of forecasting accuracy,
compared to the results presented in Table IV. They also demonstrate the
strong performance of the developed approach, since the results are close to
those depicted in Table II, concerning the SIDAREVH’s model ability to
describe the data, that can be interpreted as a lower bound to the errors that
may be obtained using this model.

TABLE V: SIDAREVH model’s accuracy in describing the infected population data using learning-based predicted time-varying infection rates with seven-day windows. | MAPE | S.D.
---|---|---
All data | 5.04% | 7.03%
Lowest 95% of errors | 3.74% | 4.27%

####  II-C3 Combined forecasts

A graphical representation of the predicted infected population during the
examined period, is depicted in Fig 4. In particular, Fig 4 illustrates the
average predicted infected population of every seven-day prediction sliding
window, as generated by using the infection rates identified by the two
proposed methodologies, in comparison with the real data. The figure shows
that, in both cases, the predictions are very close to the data, demonstrating
the performance of the established approaches and the suitability of the
developed SIDAREVH model.

Figure 4: Prediction results. Schematic representation of the model and
learning based prediction results compared to the data. The daily values of
prediction refer to the average infected population, considering seven-day
windows.

##  III Materials and methods

This section describes the available data regarding the COVID-19 pandemic in
the Republic of Cyprus and the methodologies employed to study and forecast
its evolution. Within the Data subsection, we present the available data and
explain how they are processed to enable their analysis. The Methods
subsection includes five stages that are interdependent. Firstly, in
subsection Description of the SIDAREVH compartmental model we present the
developed SIDAREVH model and its parameters. Then, in subsection Model-based
infection rates estimation we describe the model-based approach developed to
estimate the time-dependent infection rates included in the SIDAREVH model,
and in subsection Learning-based infection rates prediction and evaluation we
present the learning-based approach to predict the future values of the time-
dependent infection rates, based on the model-based approach’s results. The
parameter values resulting from the two approaches were used for predicting
the infected population in short-time windows and our method to evaluate the
predictive ability of the developed approaches is explained in Infected
population prediction and evaluation. A graphical representation of the
methodology steps is provided in Fig 5. To enhance the readability of the
manuscript, the main symbols used through this section are provided in
Conclusion.

Figure 5: Flowchart of the steps followed in the proposed methodology.

###  III-A Data

The data used in this study were provided by the Cyprus Ministry of Health.
They include anonymous daily COVID-19 confirmed infections and details on
their progression for Cyprus, an EU country with a population of $920000$
[41]. Particularly, for every infection case the available data include its
confirmation date, vaccination status, hospitalization and discharge dates
(when applicable), and recovery or disease dates. Additionally, data about the
daily vaccinations in Cyprus were obtained from [40]. The investigated time
period of the collected data was from March 2020 to March 2022.

To best represent the entire population of the available data, we considered
the following eight groups, such that each individual falls in one of them.
Moreover, these groups are associated to the states of the SIDAREVH
compartmental model described above. The eight groups are given as follows:
susceptible, unvaccinated infected, unvaccinated hospitalized, vaccinated
infected, vaccinated hospitalized, recovered, and deceased. These groups are
sufficient to describe the status of each individual in relation to the
pandemic at each time. For each day, we calculated the number of individuals
that belonged to each of the eight considered groups. Additionally, we
considered that initially, the total population was susceptible. The remaining
groups were initialized at zero and modified based on the confirmed case
progression. Specifically, upon infection, susceptible individuals were
removed from the susceptible group and added to the unvaccinated infected
group. If unvaccinated infected individuals required hospitalization, they
were removed from the unvaccinated infected group and added to the
unvaccinated hospitalized group; if not, they were removed from the
unvaccinated infected group and added to the recovered group. Unvaccinated
hospitalized individuals were removed from the unvaccinated hospitalized group
and added either to the recovered or the deceased group. Susceptible
individuals who got vaccinated were removed from the susceptible and added to
the vaccinated group. Then, the same group transitions pattern for
unvaccinated individuals was followed for the vaccinated individuals.

In Fig 6 the percentage of each group of the population is graphically
demonstrated. The presented group values refer to the time period from
September 2020 to March 2022, which was the period used for the procedures
explained in the following sections. Although our dataset includes data from
March 2020, the number of cases between March and September 2020 was
negligible and therefore not used for analysis purposes.

Figure 6: Groups’ progression over time, as calculated from the available
data. Percentage of individuals within the susceptible, unvaccinated infected,
vaccinated infected, unvaccinated hospitalized, vaccinated hospitalized,
recovered and deceased groups of population, from September 2020 to March
2022.

###  III-B Methods

####  III-B1 Description of the SIDAREVH compartmental model

The developed SIDAREVH model characterizes the progression of the COVID-19
pandemic and includes eight compartments. The proposed compartments were
selected to include all the available data about Cyprus. The dynamics of the
SIDAREVH model are given below.

$\displaystyle\dot{s}=-\beta_{uu}is-\beta_{vu}ds-\zeta s,$ (1a)
$\displaystyle\dot{i}=\beta_{uu}is+\beta_{vu}ds-\xi_{i}i-\gamma_{i}i,$ (1b)
$\displaystyle\dot{d}=\beta_{vv}iv+\beta_{uv}dv-\xi_{d}d-\gamma_{d}d,$ (1c)
$\displaystyle\dot{a}=\xi_{i}i-\gamma_{\alpha}\alpha-\mu_{\alpha}\alpha,$ (1d)
$\displaystyle\dot{r}=\gamma_{i}i+\gamma_{d}d+\gamma_{\alpha}\alpha+\gamma_{h}h,$
(1e) $\displaystyle\dot{e}=\mu_{\alpha}\alpha+\mu_{h}h,$ (1f)
$\displaystyle\dot{v}=\zeta s-\beta_{vv}iv-\beta_{uv}dv,$ (1g)
$\displaystyle\dot{h}=\xi_{d}d-\gamma_{h}h-\mu_{h}h,$ (1h) $\displaystyle
s(0)=s_{0},i(0)=i_{0},d(0)=d_{0},\alpha(0)=\alpha_{0},$ (1i) $\displaystyle
r(0)=r_{0},e(0)=e_{0},v(0)=v_{0},h(0)=h_{0},$

where $s,i,d,\alpha,r,e,v,h\in[0,1]$ are the states of the system, describing
the population percentage of susceptible, unvaccinated infected detected,
vaccinated infected detected, unvaccinated hospitalized, recovered, extinct,
vaccinated susceptible, and vaccinated hospitalized, respectively. The initial
values for $s,i,d,\alpha,r,e,v,h$ are denoted by
$s_{0},i_{0},d_{0},\alpha_{0},r_{0},e_{0},v_{0},h_{0}\in[0,1]$ respectively.

The SIDAREVH model parameters are described in Table VI. We assume that all
parameter values are known and constant, except the four infection rates
$\beta_{uu}$, $\beta_{vu}$, $\beta_{vv}$, and $\beta_{uv}$. These are the key
parameters that govern the pandemic dynamics, since they determine the effect
of the included bilinear terms in Eq (1), and hence have the largest impact on
the progression of the COVID-19 pandemic. Due to their association with the
non-linear dynamics and the effect of influencing factors on them, such as
government intervention measures and changing dominant variants, they are
assumed to be time-varying. Estimations of the infection rates and predictions
on their future values are made using the approaches described in sections
Model-based infection rates estimation and Learning-based infection rates
prediction and evaluation respectively. The constant parameters involved in
the SIDAREVH model have a smaller impact on the pandemic infections. They were
obtained from existing literature studies and are illustrated in Conclusion.

TABLE VI: Description of the parameters involved in the SIDAREVH model. Parameter | Description
---|---
$\zeta$ | Rate of vaccination of susceptible individuals
$\beta_{uu}$ | Time-dependent infection rate at which unvaccinated people infect other unvaccinated people
$\beta_{vu}$ | Time-dependent infection rate at which vaccinated people infect unvaccinated people
$\beta_{vv}$ | Time-dependent infection rate at which vaccinated people infect other vaccinated people
$\beta_{uv}$ | Time-dependent infection rate at which unvaccinated people infect vaccinated people
$\gamma_{i}$ | Recovery rate for unvaccinated infected detected individuals
$\gamma_{d}$ | Recovery rate for vaccinated infected detected individuals
$\gamma_{\alpha}$ | Recovery rate for unvaccinated hospitalized individuals
$\gamma_{h}$ | Recovery rate for vaccinated hospitalized individuals
$\xi_{i}$ | Rate at which unvaccinated infected detected individuals become hospitalized
$\xi_{d}$ | Rate at which vaccinated infected detected individuals become hospitalized
$\mu_{\alpha}$ | Rate at which unvaccinated hospitalized individuals decease
$\mu_{h}$ | Rate at which vaccinated hospitalized individuals decease

The SIDAREVH model is based on the following assumptions:
_(i)_ The considered population is constant, i.e. births and deaths not
attributed to COVID-19 are not considered.
_(ii)_ The considered population is isolated and imported cases are not
included.
_(iii)_ Infected individuals become first hospitalized before they decease.
_(iv)_ Vaccinations are performed on susceptible individuals only.
_(v)_ Recovered individuals are no longer susceptible to the disease because
of acquired immunity.
_(vi)_ Vaccinated individuals cannot be susceptible again.
The constant population assumption suggests that the model states satisfy at
all times the condition $s+i+d+a+r+e+v+h=1$. The above assumptions constitute
theoretical requirements for compartmental models.

######  Remark 1.

It is important to note that, although some of these assumptions may be only
partly satisfied in practice, this does not hinder the applicability of our
SIDAREVH model. The latter is justified by Table II in section Results and
Discussion, which shows that the SIDAREVH compartmental model is able to
describe the infected population with an average percentage error of $3.01\%$.
The suitability of the considered assumptions also follows from the fact that
this study aims to obtain predictions on a relatively short time horizon,
using the available data at the time of the prediction. Although relaxing some
of the above assumptions, e.g. by introducing re-susceptibility of the
recovered population, could enable a better characterization of the overall
pandemic progression [42], this would only have a negligible effect on the
forecasting accuracy on such short time windows.

####  III-B2 Model-based infection rates estimation

This section describes the methodology employed for the estimation of the
infection rates $\beta_{uu}$, $\beta_{vu}$, $\beta_{vv}$, $\beta_{uv}$
associated with the proposed SIDAREVH model. These four parameters are
associated with the bilinear terms within the dynamics of Eq (1), and
constitute the parameters with the largest impact on the evolution of the
infected cases in the pandemic. They are assumed to be time-varying since they
are affected by different factors, such as the government intervention
measures, and environmental and virus features. Estimates of the infection
rates were produced by fitting the SIDAREVH model to the observed time series,
depicted in Fig 6. The procedure was made through sliding windows of seven or
fourteen days, which moved by one day per increment, between September 2020
and March 2022.

To estimate the parameter values, we aimed to minimize the optimization cost
function:

$\displaystyle
C_{j}(n,i,d,\hat{i}_{j},\hat{d}_{j})=\sum_{t=j+1}^{j+n}[(i(t)-\hat{i}_{j}(t))^{2}+$
(2) $\displaystyle(d(t)-\hat{d}_{j}(t))^{2}],$

where $C_{j}$ describes the deviations between the actual and predicted
infected cases at day $j$, for a time window of size $n$, where
$n\in\\{7,14\\}$. Moreover, $i(t)$ and $d(t)$ denote the unvaccinated infected
and vaccinated infected states from the data at time $t$ respectively, and
$\hat{i}_{j}(t)$ and $\hat{d}_{j}(t)$ are the estimated unvaccinated infected
and predicted vaccinated infected states for time $t$ using the knowledge
available at time $j$ respectively.

Algorithm 1 describes the procedure for estimating the four infection rates
$\beta_{uu}$, $\beta_{vu}$, $\beta_{vv}$, $\beta_{uv}$. The aim of this
algorithm is to obtain the time-dependent parameter values that minimize the
cost defined by Eq (2). The targeted infection rates determine the estimated
values $\hat{i}_{j}$ and $\hat{d}_{j}$, and are optimized with aim to minimize
the associated cost, given by Eq (2). Starting from selecting the examined
window $z_{j}=\\{j-n+1,...,j\\}$, initial values for the four infection rates
were given, and an initial cost $C_{j}(n,i,d,\hat{i}_{j},\hat{d}_{j})$ was
calculated. Subsequently, each of the four rates was increased and decreased
by a constant percentage defining the trial parameters set $P_{k}$, that
contains the total eight modified rates, where $k$ is the number of
iterations. A new cost was calculated for each set of rates and then, the
minimum cost $C_{j}^{\prime}(n,i,d,\hat{i}_{j},\hat{d}_{j})$ of the total
eight costs was found. If the minimum cost resulted in a decreased cost
compared to the previous iteration, the considered modified infection rate was
kept and used for the subsequent iterations, creating the new set of trial
parameters $P^{\prime}_{k}$, until the set of parameters that minimizes the
cost of the window $z_{j}$ was found. The same procedure was applied for all
the sliding windows $z_{j}$ of the examined period. Finally, four sets of
optimal infection rates were obtained, that contained the parameters’ time-
dependent values that best describe each considered $n$-day window. These four
sets are denoted by $\vec{\beta}_{uu},\vec{\beta}_{vu},\vec{\beta}_{vv}$, and
$\vec{\beta}_{uv}$, with their values corresponding to the optimal time-
depended infection rates $\beta_{uu},\beta_{vu},\beta_{vv}$, and $\beta_{uv}$
respectively for all times.

Algorithm 1 Optimal infection rates estimation

1:Inputs: $s,i,d,a,r,e,v,h$, daily vaccinations

2:Output: $\vec{\beta}_{uu},\vec{\beta}_{vu},\vec{\beta}_{vv}$,
$\vec{\beta}_{uv}$

3:Initialization: $n=7$ or $n=14$

4:for each time step $j$ do

5: Select window $z_{j}=\\{j-n+1,...,j\\}$

6: Initialize parameters $\beta_{uu}$, $\beta_{vu}$, $\beta_{vv}$,
$\beta_{uv}$

7: Calculate the initial cost $C_{j}(n,i,d,\hat{i}_{j},\hat{d}_{j})$

8: $flag=0$

9: while $flag=0$ do

10: Create the trial parameters set $P_{k}$

11: Calculate a cost using every parameter from $P_{k}$ set

12: Find the minimum of all costs
$C_{j}^{\prime}(n,i,d,\hat{i}_{j},\hat{d}_{j})$

13: if
$C_{j}^{\prime}(n,i,d,\hat{i}_{j},\hat{d}_{j})<C_{j}(n,i,d,\hat{i}_{j},\hat{d}_{j})$
then

14:
$C_{j}(n,i,d,\hat{i}_{j},\hat{d}_{j})=C_{j}^{\prime}(n,i,d,\hat{i}_{j},\hat{d}_{j})$

15: Keep the modified infection rate and create new set of trial parameters
$P_{k}^{\prime}$

16: else

17: $flag=1$

18: $\vec{\beta}_{uu}(j)=\beta_{uu}$

19: $\vec{\beta}_{vu}(j)=\beta_{vu}$

20: $\vec{\beta}_{vv}(j)=\beta_{vv}$

21: $\vec{\beta}_{uv}(j)=\beta_{uv}$

####  III-B3 Learning-based infection rates prediction and evaluation

We consider the set of the four positive infection rates sets
$\mathcal{B}=\\{\vec{\beta}_{uu},\vec{\beta}_{vu},\vec{\beta}_{vv},\vec{\beta}_{uv}\\}$
as generated by Algorithm 1. Without any loss of generality, let
$\vec{\beta}\in\mathcal{B}$ be any of the aforementioned infection rate sets.
Each $\vec{\beta}\in\mathbb{R}^{T},T\in[1,\infty)$ constitutes a univariate
time series, defined as $\vec{\beta}=\\{\beta^{j}\\}^{T}_{j=1}$, where $j$ is
the current day, and $\beta^{j}$ is the infection rate at day $j$. To address
the temporal correlations encountered in the time-series data, we use a
sliding or “lookback” window, such that,
$\grave{\beta}^{j}=\\{\beta^{j},\beta^{j-1},...,\beta^{j-W+1}\\}\in\mathbb{R}^{W}$
where $W\in\mathbb{N}$ is the window size. Our proposed method was evaluated
across three window sizes ($W\in\\{7,14,30\\}$), with the optimal performance
consistently observed with a fourteen-day window. For comparison, the results
for all window sizes are shown in Conclusion.

Let $\mathcal{F}=\\{f_{uu},f_{vu},f_{vv},f_{uv}\\}$ be a set of four
regression models, each corresponding to a time series
$\vec{\beta}\in\mathbb{R}^{T}$ used for forecasting. At each day $j\geq
W+n-1$, the aim is to forecast the corresponding infection rate $D$ days
later, by using $\grave{\beta}^{j}$. Without any loss of generality, let
$f_{\theta}\in\mathbb{F}$ be a regression model (e.g., a neural network)
parameterised by ${\theta}$, defined as
$f_{\theta}:\mathbb{R^{W}}\rightarrow\mathbb{R}$, such that,
$\hat{\beta}^{j+D}=f_{\theta}(\grave{\beta}^{j})$.

At any day $j\geq W+D$, the loss function used between a prediction
$\hat{\beta}^{j}=f_{\theta}(\grave{\beta}^{j-D})$ and ground truth $\beta^{j}$
is the Mean Squared Error (MSE) defined as:

$Q^{j}=(\hat{\beta}^{j}-\beta^{j})^{2}.$ (3)

The model gradually adapts without complete re-training using incremental
learning, that is,
$f_{\theta}^{j}=f_{\theta}^{j-1}.train(\grave{\beta}^{j-D},\beta^{j})$. It is
continually updated using the online Stochastic Gradient Descent approach (or
any Gradient Descent-based algorithm) where each model parameter
$\theta_{i}\in\theta$ is updated according to the following formula

$\theta^{j}_{i}\leftarrow\theta^{j-1}_{i}-\alpha\frac{\partial{Q^{j}}}{\theta_{i}},$
(4)

where $\frac{\partial{Q^{j}}}{\theta_{i}}$ is the partial derivative with
respect to $\theta_{i}$, $\theta^{j}_{i}$ denotes the update in the jth
iteration of $\theta_{i}$, and $\alpha$ is the learning rate. The pseudocode
of the proposed method is shown in Algorithm 2.

Algorithm 2 Incremental learning for infection rates prediction

1:Input: “Lookback” window size $W$, Day ahead to predict $D$

2:Wait $W$ days to fill window.

3:Create model $f^{W}.init()$ $\triangleright$ $j=W$

4:Observe instance
$\grave{\beta}^{W}=\\{\beta^{W},\beta^{W-1},...,\beta^{1}\\}$

5:Predict $\hat{\beta}^{W+1}=f^{W}.predict(\grave{\beta}^{W})$

6:for each day $j\in[W+1,W+D-1)$ do $\triangleright$ Only predictions these
days

7: Observe ground truth $\beta^{j}$

8: Create instance
$\grave{\beta}^{j}=\\{\beta^{j},\beta^{j-1},..,\beta^{j-W+1}\\}$

9: Predict $\hat{\beta}^{j+D}=f^{W}.predict(\grave{\beta}^{j})$
$\triangleright$ $f^{W}$ hasn’t been updated yet

10:for each day $j\in[W+D,\infty)$ do $\triangleright$ Predictions and
training

11: Observe ground truth $\beta^{j}$

12: Incremental training
$f^{j}=f^{j-1}.train((\grave{\beta}^{j-D},\beta^{j}))$ $\triangleright$ As in
Eq (4)

13: Create instance
$\grave{\beta}^{j}=\\{\beta^{j},\beta^{j-1},..,\beta^{j-W+1}\\}$

14: Predict $\hat{\beta}^{j+D}=f^{j}.predict(\grave{\beta}^{j})$

Evaluation methodology. All experiments are evaluated according to the widely
used regression forecast metrics Mean Absolute Error (MAE) and Mean Absolute
Percentage Error (MAPE), as defined in Eqs (5a) and (5b) respectively.

$\displaystyle MAE$
$\displaystyle=\frac{1}{n}\sum_{t=W+D}^{T}(\hat{\beta}^{t}-\beta^{t}),$ (5a)
$\displaystyle MAPE$
$\displaystyle=\frac{1}{n}\sum_{t=W+D}^{T}\frac{|\beta^{t}-\hat{\beta}^{t}|}{|\beta^{t}|}\times
100\%,$ (5b)

where $n=T-W-D+1$ is the total number of predictions made.

Furthermore, all experiments are run over 10 repetitions where we present the
mean and standard deviation of the relevant metrics.

Data preprocessing. The infection rates are normalized in the range $[0,1]$ by
dividing them with the largest value of the time series. Outliers (typically,
very small values, e.g., less than 0.1) have been removed through visual
inspection of data plots. Linear interpolation is used to fill in missing
values (e.g., due to outlier removal) within a window $\grave{\beta}^{t}$,
while the First Observation Carried Backwards and the Last Observation Carried
Forwards methods are used for any missing values in the beginning or end of
the window, respectively.

Learning models. Each learning model (one for each infection rate) is a
Multilayer Perceptron (MLP) model, which is a standard feed-forward fully-
connected neural network. Most of the hyper-parameters of the MLPs have the
same values, while small differences exist in the rest. All hyper-parameter
values are reported in Conclusion.

Role of the window size $W$. We have examined the performance of the proposed
method using three window sizes $W\in\\{7,14,30\\}$. In all cases, we forecast
each rate $\hat{\beta}^{j+D}$ a week ahead, i.e., $D=6$. The best performing
choice is with a fourteen-day window. The results are presented in Table III,
while the results for all window sizes are shown in Conclusion.

Performance. In addition to Table III, the performance of the predicted
infection rates compared to the original ones is depicted in Fig 3. The models
appear to perform well, while the fact that some instances are not captured
might be attributed to the dynamic, non-stationary nature of the data.

####  III-B4 Infected population prediction and evaluation

The parameterization of the SIDAREVH model enabled predictions of the
vaccinated and unvaccinated infected population. Predictions were made by
extrapolating the infection rates identified firstly in section Model-based
infection rates estimation by fitting the model to data, and then using the
infection rates predictions through the neural network approach presented in
section Learning-based infection rates prediction and evaluation. Projections
were generated from September 2020 to March 2022, in sliding windows of size
$n$, using the information from the corresponding previous windows of size
$n$, where $n\in\\{7,14\\}$, all moving by one day per increment. The
information used for the prediction included the daily vaccinated population
and the values of the four infection rates
$\beta_{uu},\beta_{vu},\beta_{vv},\beta_{uv}$ that describe the previous
$n$-day window. The initial values of the eight states of the model, described
by Eq (1), for every predicted sliding window, were the percentage population
of Fig 6, on the day before every window of prediction.

The SIDAREVH model’s predictive accuracy was validated by the commonly
employed metric Mean Absolute Percentage Error (MAPE), which for day $j$ is
defined as

$\displaystyle
MAPE_{j}(n,i,d,\hat{i}_{j},\hat{d}_{j})=\frac{1}{n}\sum_{t=j+1}^{j+n}|\frac{[\hat{i}_{j}(t)+\hat{d}_{j}(t)]}{i(t)+d(t)}-1|$
(6) $\displaystyle\times 100\%.$

The MAPE metric is commonly employed in medical research, contributing to the
comprehension of the error distributions [43, 44, 45]. $MAPE_{j}$ describes
the average percentage error at day $j$, that appears between the data and the
predicted infected cases of a time window of size $n$. Terms $i(t)$ and $d(t)$
denote the unvaccinated and vaccinated infected states from the data at time
$t$, and terms $\hat{i}_{j}(t)$ and $\hat{d}_{j}(t)$ are the predicted
unvaccinated and vaccinated infected states at time $t$ using the knowledge
available at time $j$, respectively. The values of $i$ and $d$ are assumed
non-negative at all times. Percentage prediction errors are calculated for
every sliding window of the examined period, resulting in a set of percentage
errors for all the predicted windows.

##  IV Conclusion

This study developed a novel compartmental model to characterize the
progression of pandemics. The proposed SIDAREVH model includes time-varying
infection rates, which are the key parameters with the largest impact on the
pandemic evolution. We first made estimations of the infection rate values by
fitting the available data, regarding the COVID-19 pandemic in Cyprus, to the
proposed model and by extrapolating them, we created seven-day infection
projections with a mean absolute percentage error of 9.90%. Then, we used
neural regression models trained by incremental learning, to predict the
infection rates as they were generated by the model-based estimations, and we
observed an improvement in the predictive accuracy of our model, with a mean
absolute percentage error of 5.04%. The resulting high prediction accuracy
showcases the reliability of the proposed approach, which is also demonstrated
by the small difference between the resulting forecasting error of 5.04% and
the corresponding error that our model can theoretically achieve on the
considered dataset, which was estimated to be 3.01%. Therefore, the proposed
hybrid approach, combining compartmental mathematical models and neural
networks, consists a reliable forecasting method for the progression of the
infected cases. The proposed approach can timely notify governments and
healthcare professionals for a forthcoming crisis, and aid in the design of
proper intervention measures that will contain the pandemic’s impact, protect
the public community and minimize the resulting economic effects.

## Description of symbols used.

Symbol | Description
---|---
$s$ | Susceptible state of SIDAREVH model
$i$ | Infected state of SIDAREVH model
$d$ | Vaccinated infected state of SIDAREVH model
$a$ | Hospitalized state of SIDAREVH model
$r$ | Recovered state of SIDAREVH model
$e$ | Extinct state of SIDAREVH model
$v$ | Vaccinated susceptible state of SIDAREVH model
$h$ | Vaccinated hospitalized state of SIDAREVH model
$n$ | Model-based window size
$j$ | Index describing time
$C_{j}$ | Cost between data and prediction on day $j$
$\hat{i}_{j}$ | Predicted infected state on day $j$
$\hat{d}_{j}$ | Predicted vaccinated infected state on day $j$
$MAPE_{j}$ | Mean Absolute Percentage Error on day $j$
$z_{j}$ | Examined window of size $n$
$k$ | Index describing number of iteration
$P_{k}$ | Trial parameters set at iteration $k$
$\vec{\beta}_{uu}$ | Optimal unvaccinated to unvaccinated infection rate
$\vec{\beta}_{vu}$ | Optimal vaccinated to unvaccinated infection rate
$\vec{\beta}_{vv}$ | Optimal vaccinated to vaccinated infection rate
$\vec{\beta}_{uv}$ | Optimal unvaccinated to vaccinated infection rate
$W$ | Learning-based window size
$\grave{\beta}^{j}$ | Lookback window of size $W$
$\beta^{j}$ | Ground truth infection rate at day $j$
$f_{\theta}$ | A regression model parameterised by ${\theta}$
$\theta$ | Learning-based model parameter
$D$ | Days of learning-based prediction of infection rates
$\hat{\beta}^{j}$ | Predicted infection rate on day $j$
$Q^{j}$ | Mean Square Error on day $j$

## The values of the constant parameters involved in the SIDAREVH model.

Symbol | Value | Justification
---|---|---
$\gamma_{\alpha},\gamma_{h}$ | $1/12.4=0.081$ | [46]
$\gamma_{i},\gamma_{d}$ | $1/14=0.071$ | [47]
$\xi_{i}$ | $0.0053$ |  [48], [49]
$\xi_{d}$ | $0.000265$ |  [50], [51]
$\mu_{\alpha}$ | $0.0085$ | [48]
$\mu_{h}$ | $0.0085$ |  [50], [51], [52]

## MLP learning parameters for each infection rate $\beta_{uu}$, $\beta_{vu}$,
$\beta_{vv}$, $\beta_{uv}$.

| $\beta_{uu}$ | $\beta_{vu}$ | $\beta_{vv}$ | $\beta_{uv}$
---|---|---|---|---
Hidden layers | [64, 8] | [32, 128, 8] | [128, 64] | [32, 8, 8]
Learning Rate | 0.01 | 0.01 | 0.001 | 0.01
L2 Regularization | 0.0001 | 0.001 | 0.1 | 0.01
Optimizer | Adam
Weight initialization | He Normal
Hidden activations | Leaky ReLU
Output activation | ReLU
Loss function | MSE
Num. of epochs | 5
Mini-batch size | 1

## Information about the filtering technique used in the model-based
estimation of infection rates involved in SIDAREVH model.

The infection rates $\beta_{uu}$, $\beta_{vu}$, $\beta_{vv}$, $\beta_{uv}$
included in the SIDAREVH model, and estimated by fitting the model to the
available data were filtered by a function of MATLAB, that detects and removes
the outliers from data. Outliers are defined as elements more than three
scaled MAD (Median Absolute Deviation) from the median. The scaled MAD for a
vector of parameter values $B$ is defined as $MAD=c\times
median(|B-median(B)|),$ where $c=-1/(\sqrt{2}\times erfcinv(3/2))$, where
$erfcinv$ is the inverse complementary error function, and the error function
is twice the integral of the Gaussian distribution with 0 mean and variance of
1/2.

## MLP performance for each infection rate $\beta_{uu}$, $\beta_{vu}$,
$\beta_{vv}$, $\beta_{uv}$.

Table 1 presents the MAPE (S.D.) for each of the four rates, per examined
window size.

TABLE 1: MAPE for each infection rate per window size (S.D.). Window size | $\beta_{uu}$ | $\beta_{vu}$ | $\beta_{vv}$ | $\beta_{uv}$
---|---|---|---|---
7 | 6.39% (7.24%) | 1.70% (4.98%) | 12.54% (20.05%) | 10.66% (12.18%)
14 | 6.35% (7.57%) | 1.48% (4.05%) | 15.40% (17.78%) | 8.00% (13.10%)
30 | 6.40% (7.17%) | 1.71% (5.43%) | 17.40% (18.66%) | 10.35% (14.93%)

## SIDAREVH model’s predictive accuracy for fourteen-day windows.

The error results presented below, concern predictions of the infected
population using the model-based infection rates identified at each day of the
examined period, using the previous fourteen-day windows data, to the next
fourteen-day windows. The model-based estimation made by fitting the available
data of Cyprus in the proposed SIDAREVH model. The first row of the tables
below refers to the mean and the standard deviation of all predicted sliding
windows errors, while the second row excludes the 5% of the highest error
values.

Table 1 presents the ability of the SIDAREVH model to describe the data
accurately. Specifically, the error results describe the accuracy of
forecasting the infected population, when considering fourteen-day prediction
windows, and by using the unfiltered model-based estimated infection rates.
These results can be interpreted as the lower bound that the proposed SIDAREVH
model may achieve when considering prediction windows of fourteen days on the
considered dataset.

TABLE 1: SIDAREVH model’s accuracy in describing the infected population data using model-based optimized unfiltered time-varying infection rates with fourteen-day windows. | MAPE | S.D.
---|---|---
All data | 5.05% | 3.81%
Lowest 95% of errors | 4.39% | 2.52%

Table 2 presents the errors of SIDAREVH model’s evaluation, while the
forecasting procedure used the filtered model-based estimated infection rates
with the outliers removed for fourteen-day windows.

TABLE 2: SIDAREVH model’s accuracy in describing the infected population data using model-based optimized filtered time-varying infection rates with fourteen-day windows. | MAPE | S.D.
---|---|---
All data | 26.95% | 23.69%
Lowest 95% of errors | 23.31% | 17.98%

## Daily MPE of prediction windows.

The MPE values associated with the number of days of prediction windows, when
using seven and fourteen days of previous data.

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