A MATHEMATICAL MODEL OF COVID-19 PANDEMIC
A Research Proposal
Presented to the
Panelist from
Nueva Vizcaya State University
Bayombong, Nueva Vizcaya
In Partial Fulfillment
Of the Course Requirement in
Research
Gacad, Jennifer R.
Castro, Marjorie M.
CHAPTER I
INTRODUCTION
Background of the Study
Corona Virus Disease 2019 is a disease that was named containing the SARS-CoV2 virus that was first found in Wuhan China. Experts say SARS-CoV-2 originated in bats and
was able to host humans at one of Wuhan’s open-air markets. The people infected in the corona
virus pandemic experience mild respiratory problems. Fever, dry cough, throat infection, and
fatigue are the symptoms of this disease. People may also have the symptoms like nasal
infection, aches, and sore throat. And It was on December 2019 when a 55-year-old Chinese
individual from Hubei province in China became the first ever victim of COVID-19. In the
following weeks, the disease spread widely in China mainland and other countries, which causes
global panic. Governments and institutions began responding to the disease and eventually
formed strict policies and cases control. Scientists and researchers began studying the disease
and start to model and forecast the spread of the disease.
Japan, South Korea and the United States confirmed their first case of COVID-19
approximately three weeks after the reported outbreak in China. From then on, disease
progression within these countries varied quite a bit. While it took 42 days to reach 100
confirmed cases in the U.S., Japan reached that number in 31 days and South Korea in 29 days.
Singapore and France both reported their first confirmed case 23 and 24 days from the outbreak
and both countries reached 100 confirmed cases in 37 and 36 days, respectively. Singapore
promptly instituted an international travel ban around the same time as their first confirmed case,
followed by a very strict screening and quarantine measures. France, in contrast, instituted only
screening measures for international travelers a few days later and a ban on public events around
the same time that its confirmed cases rose to 100. India reported the first confirmed case 29
days after the outbreak around a month after the report of the outbreak around the same time as
Italy, the U.K., Spain and Belgium. Italy was more proactive than India in instituting
international travel controls. India started taking social distancing measures like closing
workplaces and schools before reaching 100 confirmed cases. But public gatherings were still
occurring in Italy, the most crucial of which turned out to be a football match. As of March 29,
2020 India had over 1,000 confirmed cases while Italy had over 97,000 confirmed cases.
In the Philippines, the development of policy making, restrictions and interventions
began when the first case of COVID-19 was reported on January 30, 2020. First local
transmission was confirmed on March 7, 2020. The Philippine government declared enhanced
community quarantine (ECQ) on March 17, 2020 in the entire island of Luzon and other parts of
the county for a month and later extended until April 30 to control the transmission of the
disease. Months since the lockdown, the economy gradually reopened with less stringent
quarantine regulations in each province or region. However, the daily infection cases have not
been significantly reduced yet. The Philippines shifted from a focus on community quarantine
orders towards greater reliance on Minimum Health Standards (MHS) policies, requiring the
proper use of face coverings, physical distancing, and hand hygiene. As of January 15, 2022 the
total cases in the Philippines escalated to 3,238,017.
Here in Nueva Vizcaya the first case reported was a 65-year-old resident of Solano town in
Nueva Vizcaya who died in March 19, was also reported as the 4th case recorded in the entire
Cagayan Valley. As of today Nueva Vizcaya has a total recorded cases of 18,148, 682 total
deaths and recovered cases of 17157. The Provincial government and local units continue to
implement strict crowd control to reduce transmission of the disease specially now that there
were various variation of the disease. This is mainly the reason why mathematical modeling the
spread of the disease is very important that it will assist in decision making by making
projections regarding the disease such as intervention-induced changes in the spread of the
disease and to determine which combination interventions would be most effective for reducing
the outbreak of COVID-19.
A mathematical model is an imaginary micro world consisting of entities behaving
according to precisely specified rules. A mathematical model for the spread of an infectious
disease in a population of hosts describes the transmission of the pathogen among hosts
depending among patterns of contacts among infectious and susceptible individuals and the
period from being infected to becoming infectious. Models are developed to provide insights and
make predictions about the pandemic to plan effective control strategies and policies. Infectious
disease models are an integral part of public health decision making, and have been crucial tools
throughout the COVID-19 pandemic. In early 2020, models have estimated the extent of
COVID-19 in Wuhan and regional patterns of importations, made projections of potential
epidemics and healthcare needs, and provided short-term predictions of case numbers in newly
unfolding outbreaks. As countries considered control and mitigation measures, models helped
assess likely effects of proposed interventions and determine ‘counterfactuals’, that is, expected
epidemic trajectories if an intervention was not implemented. Models have played an important
role in policy development to address the COVID-19 outbreak from its emergence in China to
the current global pandemic.
Statement of the Problem
This study aims to establish a COVID-19 mathematical model that has numerical and
approximation solution to calculate the current situation report from real life data of COVID19 cases in Bayombong, Nueva Vizcaya to predict the future report from the current COVID19 data to provide decision support for pandemic prediction and prevention. Specifically, this
study aims to seek answers for the following:
a. What is the maximum number of infective at any time?
b. How many people will catch the disease?
c. When will growth of infective stop?
d. Do strict protocols and intervention working?
e. When is the peak of infection?
Significance of the Study
The result of the study would be significant to the following:
The government:
The findings of this study will help the government in developing policies, control and
mitigation measures in addressing the COVID-19 pandemic.
The school administrators:
The findings of this study will help the school administrators to develop strategies on
handling education in line with the government disease control interventions while at the
same time providing quality education even amidst pandemic.
The teachers:
The findings of this study would help teachers to come up with a long term solutions
basing from the predictions of the study in making the learning experience of the students
safe and quality at the same time.
The parents:
The findings of this study will help parents in guiding and managing the households that
every member of the family maintains safety measures and follows protocols.
Everyone in the Community:
The findings of this study will help everyone as a source of information that will serve as
a guide in facing COVID-19 risks every day. With the findings, people can develop their
own repertoire of safety measures and practices to prevent them from catching the
disease.
Future researchers:
The result of the study would provide possible information for parallel studies to be
conducted.
Scope and Delimitation of the Study
This study aims to establish a COVID-19 mathematical model that has numerical and
approximation solution to calculate the current situation report from real life data of COVID-19
cases in Bayombong, Nueva Vizcaya to predict the future report from the current COVID-19
data to provide decision support for pandemic prediction and prevention.
Theoretical Framework
Collect COVID-19 current data from Municipal Health Office/ Provincial Health Office
Find Parameter Estimation through Susceptible-Infectious-Recovered Model
Create a new mathematical model from SIR Model and use the current COVID-19 data
Analyze stability, sensitivity and bifurcation
Use analytical method of the developed model in the study
Use numerical methods for simulation
Definition of terms
The following terms are defined operationally/conceptually for a better understanding of
the study.
COVID-19 – abbreviated term of Corona Virus disease 2019 an infectious disease caused by the
SARS-CoV-2 virus
Infective – term use to address a population of infected people
Intervention – action taken to improve COVID-19 situation
Mathematical Model – a representation in mathematical terms of the behavior of COVID-19
Recovered – return to normal state of health or strength
Susceptible – likely or liable to be infected by the COVID-19
Spread- a happening to which a disease is continuously transmitting from one person to another
CHAPTER II
REVIEW OF RELATED LITERATURE AND STUDIES
Chapter II
Review of Related Literature and Studies
Mathematical models are useful to understand the behavior of an infection when it enters
a community and investigate under which conditions it will be wiped out or continued.
Currently, COVID-19 is of great concern to researches, governments, and all people because of
the high rate of the infection spread and the significant number of deaths that occurred. In
December 2019, coronavirus first reported in Wuhan, China, is an infectious disease caused by a
newly discovered coronavirus. The virus that causes COVID-19 is mainly transmitted through
droplets generated when an infected person coughs, sneezes, or exhales. These droplets are too
heavy to hang in the air and quickly fall on floors or surfaces. Coronavirus-confirmed cases
reached nearly four million in 187 countries, and approximately 295,000 people have lost their
lives due to this virus.
Johns Hopkins University, focused the largest cases occurred at the US. Noting that more
than 77,000 deaths happened, it also has the world’s highest death. Researchers have been
tracking the spread of the virus, have mobilized to speed innovative diagnostics, and are working
on a number of vaccines to protect against COVID-19. Cao et al. studied the clinical features of
corona virus and discussed the short-term outcomes of 18 patients and 102 patients with COVID19 in intensive care units. Corona viruses are typically transmitted from person to person through
respiratory droplets and close contact. The majority of the transmission is happening through
respiratory droplets that we may inhale from close contact with one another. A modified SIR
epidemic model is presented in to project the actual number of infected cases and the specific
burdens on isolation wards and intensive care units. Nesteruk developed an SIR (susceptible,
infected, and recovered) epidemic model and discussed statistically the parameters used in the
proposed model and showed how to control this infection
Similarly, W. Ming, J. V. Huang, and C. J. P. Zhang applied a modified SIR model to
project the actual number of infected cases and the specific burdens on isolation wards and
intensive care units (ICU), given the scenarios of different diagnosis rates as well as different
public health intervention efficacy. Our estimates suggest, assuming 50% diagnosis rate if no
public health interventions were implemented, that the actual number of infected cases could be
much higher than the reported, with estimated 88,075 cases (as of 31st January, 2020), and
projected burdens on isolation wards and ICU would be 34,786 and 9,346 respectively The
estimated burdens on healthcare system could be largely reduced if at least 70% efficacy of
public health intervention is achieved.
I. Nesteruk, states the development of an epidemic outbreak caused by coronavirus
COVID-19 (the previous name was 2019-nCoV). Since long-term data are available only for
mainland China, we will try to predict the number of corona virus victim’s V (number of persons
who caught the infection and got sick) only in this area. The first estimations of V(t) exponential
growth versus time t, typical for the initial stages of every epidemic have been done. For longtime predictions, more complicated mathematical models are necessary. For example, a
susceptible-exposed-infectious-recovered (SEIR) model was used in. Nevertheless, complicated
models need more effort for unknown parameters identification. This procedure may be
especially difficult if reliable data are limited. The simple mathematical model was used to
predict the characteristics of the epidemic caused by corona virus in mainland China. The
numbers of infected, susceptible, and removed persons versus time were predicted and compared
with the new data obtained after February 10, 2020, when the calculations were completed.
Unfortunately, many cases have not been included in the official counts and have appeared on
February 12 only. It makes the predictions reported on February 10, 2020, no longer relevant.
Further research should focus on updating the predictions with the use of corrected data and
more complicated mathematical models.
Furthermore M. Batista was used the SIR model is used for the estimation of the final
size of the coronavirus epidemic. The current prediction is that the size of the epidemic will be
about 85 000 cases. One of the common questions regarding an epidemic is its final size. To
answer this question various models are used: analytical (Danby 1985, Brauer 2019a, b, Murray
2002), stochastic (Miller 2012), and phenomenological (Fisman D 2014, Pell et al. 2018). In this
note, we attempt to estimate the final epidemic size using the phenomenological logistic growth
model (Pell et al. 2018, Chowell G 2014) and the classic susceptible infected-recovered (SIR)
model (Hethcote 2000). With both the models, we obtain a series of daily predictions. The final
sizes are then predicted using iterated Shanks transformation (Shanks 1955, Bender and Orszag
1999). The data used for the calculations are taken from worldmeters0F 1. Before proceeding,
we note that the final size of the epidemic in its early stage was discussed by Wu et al. (Wu,
Leung, and Leung 2020) using the susceptible-exposed infected-resistant model, by Xiong and
Yan (Xiong and Yan 2020) using the exposed infected-resistant model, by Nesteruk (Nesteruk
2020) using the SIR model, and by Anastassopoulou et al. (Anastassopoulou et al. 2020) using
the SIR/death model. These early predictions range from 65000 to a million cases. Roosa et al
recently gave short term forecasts of the epidemic (Roosa et al. 2020).
R. E. Mickens explains the most numerical integration schemes for the Lotka–Volterra
system have the property that the computed solutions spiral when in fact the actual solutions are
periodic corresponding to closed curves. We show that a direct application of the nonstandard
methods of Mickens allows the construction of a finite-difference scheme that is dynamically
consistent with the differential equations. Numerical results are given to support this result.
Analysis has been the most dominant part of mathematics, and also differential equation
is the heart of analysis. A major difficulty in the study of differential equations is in general, the
lack of exact analytical solutions that cannot be solved by a straight forward formula. One way to
proceed is to use numerical integration techniques to obtain useful information on the possible
solution behaviors. A popular and important one is based on the use of finite differences to
construct discrete models of the differential equations of interest. Almost all of the standard
procedures yields schemes which are convergent with restriction on the step size. The
preservation of the qualitative properties of the considered differential equation with respect to
these schemes is of great interest in finite difference methods of solving differential equations.
The major consequence of this result is that such scheme does not allow numerical instabilities to
occur. Mickens gave a novel approach for developing new finite difference schemes for
differential equations.
He concludes an essential feature of nonstandard finite difference schemes for differential
equations is the precise manner in which the discretization of derivatives is made. We
demonstrate, for differential equations modeling systems where the solutions satisfy a positivity
condition, that procedures can be formulated to calculate the so-called denominator functions
that appear in the discrete derivatives. These procedures are applied to a number of both ordinary
and partial model differential equations to illustrate their use. © 2006 Wiley Periodicals, Inc.
Number Methods Partial Differential as 2007.
The need often arises to analyze the dynamics of a system in terms of a discrete
formulation. This can occur by using an a priori discrete model of the system or by discretizing a
continuous model. For the latter case, the continuous model is represented by differential
equations and the discrete forms come from the requirement to numerically integrate these
equations. The concept of “dynamic consistency” plays an essential role in the construction of
such discrete models which usually are expressed as finite difference equations. We define this
concept and illustrate its application to the construction of nonstandard finite difference schemes.
(R. E. Mickens)
On the other hand, N.R. Sasmita states the outbreak of coronavirus disease (COVID-19)
has spread to nearly every country around the world. With a high density population, Indonesia
is predicted to have a high number of infectious persons and, consequentially, suffer over a
longer time period. The first positive cases of COVID-19 in Indonesia were confirmed with two
cases in Java Island on March 2, 2020, then spreading to other islands. As of the first week of
April, there are about 3000 cases reported. However, there is skepticism about whether the true
number of COVID-19 cases in the community is higher than reported, due to inadequate testing
and under detecting.
Without well prepared interventions, the number of COVID-19 cases will grow
exponentially. Though delayed, the Indonesian government has taken numerous measures to
control the community spread of COVID-19. Control measures, such as large-scale social
restriction policies in specific provinces, a campaign for frequent hand washing and face mask
use, and rapid testing suspected cases, have been implemented across Indonesia. Indonesia
benefits from being a country of a thousand islands, as this geography may delay disease
transmission from the main epicenter on Java Island to surrounding islands.
Considering limited knowledge of COVID-19 in Indonesia, where the number of cases
has been fast-growing since the onset of the epidemic, this study proposes a deterministic
mathematical model based on the susceptible (S), exposed (E), infectious (I), recovered (R)(SEIR) model. Selected control variables were simulated, representing measures that have been
and will be implemented by the Indonesian government to detect and reduce COVID-19
transmission. These measures include large-scale social restriction (u1), contact tracing (u2), mass
rapid antibody testing (u3), case detection and treatment (u4), and the wearing of face masks (u5).
Using a mathematical model as an approach to solving problems can help in explaining current
phenomena. A mathematical model may help us to understand patterns in disease outbreak,
especially that of COVID-19, and lead to a more public health informed policy making process.
Applying optimal control to a mathematical model can predict, forecast, estimate, or
choose the best scenario to eliminate a disease in a dynamical system, based on epidemiological
characteristics. In the phenomenon of the COVID-19 outbreak, there are lags among the real ongoing spread of infection, the case detection and report and the response action. If the case load
could be properly predicted, the government would have the ability to fine tune their reaction,
such as adjustment of the intensity of social distancing, preparation of medical resources to cope
with the case load, and planning for socio-economic recovery at proper time points.
Hui-Jia Li et al., explains about the public health event, the COVID-19 pandemic has
triggered massive crises in public health systems and economy. The COVID-19 has fleetly
spread to most countries of the world. By the end of May 2021, SARS-CoV-2 has infected 180
million people and caused over three million deaths worldwide. To control the outbreak of
COVID-19, travel restrictions, economic lockdowns, and border controls have been taken by
many countries. The ongoing COVID-19 pandemic motivates the scientific community to
contribute to infectious disease modeling, epidemiological study and outbreak prediction. That
is, understanding the mechanism of COVID-19 pandemic spreading, exploiting the infection
prevention and mitigation, and evaluating policy implementation, are important research
questions for academics and policymakers.
To exploit this urgent question, mathematical models have been widely used to analyze
the characteristics, impacts and emergency responsefor COVID-19. Mathematical modeling is a
kind of mathematical structure that is made up of various concepts and formulas in mathematics,
used for generally or approximately expressing the characteristics or quantitative
interdependence of a certain system of things.
This special issue has collected a series of studies on epidemic data analysis and
supporting decision-making. Those studies fit mathematical models for COVID-19 to analyze
the characters of the COVID-19 pandemic in different regions, for instance, Azanza Ricardo et
al. investigate the outbreak of COVID-19 in Mexico with different scenarios. While Miao et
al. use mathematical modeling to study options for business reopening during the COVID-19
pandemic. As a commonly used epidemiologic model, the Susceptible, Infectious, and/or
Recovered (SIR) model, the reproductive number of the SIR model is discussed to assess
COVID-19 spread (Silveira et al.; Espinosa et al.; Schlickeiser et al.). Also, a mathematical
model is developed to study what role social heterogeneity plays in the formation of complex
infection propagation patterns (Maltsev et al.). Moreover, statistical analysis has been used to
characterize the COVID-19 pandemic and predict the spread trend (Wang et al.; Zhao et al.; Liu
et al.). Mathematical models and statistical analysis play important roles in public health
emergencies, using these models to analyze and develop strategies for the COVID-19 pandemic
is also an essential mission. For example, fast epidemic recognition of the epidemic as soon as it
appeared, optimal supply and allocation of medical emergency resources (Wang et al.), and
assessing the epidemiological consequences of an emergency evacuation (Butail et al.).
Model estimation also plays an important role in this special issue. Wu et al. propose a
generalized-growth model to present the evolution of the number of the total confirmed cases
over time. The model provides a simple phenomenological approach, with potential implications
for forecasting of the pandemic trend. Zhang et al. quantify the incubation period, transmission
rate from close contact to infection, and the properties of multiple-generation transmission from
a detailed database in mainland China. Tiwari et al. sought to provide a prediction of the
epidemic peak and to evaluate the impact of lockdown on the epidemic peak shift in
India. Peker-Dobie et al. argue that the peak of infected individuals coincides with the inflection
point of removed individuals. Recent COVID-19 data and the records for Spanish flu and SARS
epidemics confirm this observation. Using multistage models, the authors provide an explanation
for this time shift. Chae et al. estimate the parameters and the initial infections from fitted values,
and quantify the infection rate, the basic reproduction number, and the initial number of infected
individuals for a number of countries. Kröger and Schlickeiser argue that the Gauss model is the
simplest analytically tractable model that allows us to quantitatively forecast the time evolution
of infections and fatalities during a pandemic wave and provide relationships between peak time
and width, the transient behavior of doubling times, and reproduction factors. Zhuang et al. built
a model to estimate the total number of COVID-19 cases in Wuhan, based on the number of
cases detected outside Wuhan city in China, with the assumption that cases exported from
Wuhan were less likely underreported in other cities/regions. Total cases are determined by the
maximum log likelihood estimation and Akaike Information Criterion (AIC) weight.
This special issue aims to understand the impact of COVID-19 spreading on public
health, society, and economics, and provides efficient reference values for economic
redevelopment and social stability. So far, the authors have fitted mathematical models for
COVID-19, studied the connectedness of the global COVID-19 network across countries,
captured the virus transmission among cities, analyzed the dynamic characteristics of the
COVID-19 pandemic, assessed the impact of COVID-19 spreading, and predicted the spread
trends of COVID-19, which alleviates the public panic caused by COVID-19, grasps the trend of
COVID-19 transmission, and assists the government or policymakers to make efficient decisions
according to the spreading situation of the COVID-19 pandemic.
Z Liu found the outbreak of coronavirus disease 2019 (COVID-19) has been presenting a
major threat to public health. The first COVID-19 case was reported on Dec 8, 2019. To curb the
spread of the virus, Chinese health authorities have taken the strictest massive anti-epidemic
actions since Jan 2020, including mass isolation, social distancing and community containment.
Moreover, the government has implemented traffic restrictions across the whole country with
massive reduction in public transportation capacity. As the epidemic situation remains fraught in
China, key epidemiological questions, such as the effectiveness of implemented strategies for
disease control, remain to be fully investigated.
The government has been increasingly investing medical resources in the treatment of
patients with COVID-19. On February 5, three cabin hospitals and two other makeshift hospitals
successively started to treat infected patients. By March 9, 346 medical teams and 42,600
medical professionals have been dispatched from other provinces across the country to combat
the epidemic in Hubei province. It is reported that 7512 designated hospitals and related fever
clinic are mobilized nationwide. However, the nationwide mobilization of medical resources
could severely disturb local routine medical service. According to the 2018 National Report on
the Services, Quality and Safety in Medical Care System, currently about 2 million patients each
year travel across regions to seek medical care in China, among which 43% of all the crossregional cases are concentrated in Beijing, Shanghai and Guangzhou (842 thousand cases in
total). These three metropolises play a pivotal role in the healthcare system in China, providing
high-quality medical service for patients in China. Notably, over thousands of medical
professionals from medical centers in these metropolises have been dispatched to Wuhan and
other cities in Hubei province to fight COVID-19. As the full resumption of normal healthcare
services in the metropolises marks the complete restoration of healthcare system in China from
the epidemic of COVID-19, providing estimation on the number of affected patients and
prediction of restoration of routine medical service is urgently needed to facilitate preparedness
of the healthcare system.
In this study, we provided an estimation of the epidemic trend of COVID-19 in Wuhan
and representative metropolises in China and forecast the time point when the routine medical
service would recover from the epidemic. Furthermore, we utilized data on population migration
to construct an improved mathematical model to measure the impact of traffic restrictions on the
migrant patients, providing estimation of operational pressure for metropolitan medical service
after the end of the epidemic.
Moreover, Daniel Deborah O says that several modelling studies have been done to
analyze the spread of COVID-19 pandemic, ranging from stochastic modelling to
mathematical modelling. In SEIRU model involving the susceptible, the exposed, the infected,
the quarantined and the recovered individuals were considered. It was predicted that there is a
chance of a decline in secondary infections when all precautionary measures are observed
globally. In a mathematical model was developed to describe a Bat-Hosts-Reservoir-People
transmission network model for simulating the potential transmission from infection source to
human, which was simplified to Reservoir-People transmission network model. The
reproduction number was calculated from the Reservoir-People model using the nextgeneration matrix to measure the transmissibility of the virus in Wuhan, China. The value
of R 0 was estimated to be 2.30 from the environmental reservoir to humans and 3.58 from
humans to humans. In a compartmental model involving the susceptible, the exposed, the
infected, the recovered and the concentration of the pathogen in the environmental reservoir
was considered. The reproduction number R 0 was calculated using the next-generation matrix
to determine the transmissibility of the virus from the exposed, the infected and the pathogens
in the environmental reservoir to the susceptible in Wuhan, China. The value of R 0 was
estimated to be 4.25 covering 1.5 from indirect transmission route and 2.7 from direct
transmission route.
However, model considering the quarantined individuals, confirmed cases, nonlinear
forces of infection in the form of saturated incidence rates in humans and the impact of the
indirect transmission route from the virus in the environmental reservoir to humans were not
considered in the published models. Hence, we present a mathematical model for the pandemic
COVID-19 to describe the several transmission pathways including the disease-induced rates
for humans, nonlinear forces of infection in the form of saturated incidence rates in humans,
direct and indirect transmission routes. The direct transmission routes describe the
transmission from the exposed individuals to the susceptible individuals and from the infected
individuals to the susceptible individuals, while the indirect transmission routes come from the
interaction of the susceptible individuals with these pathogens in the environmental reservoir.
In addition, the epidemic was primarily determined in Wuhan, China, in Dec 2019,
through the majority near the beginning holder’s organism accounted into the town. On the
whole, globally send abroad holders accounted time have the past of journey to Wuhan
(2019nCoV-2019 Data Working Group). In the premature phases for a novel transmittable
illness epidemic, this is critical to recognize the spread progress for the contamination. Camacho
et al. (Camacho et al., 2015) modified the evaluation in spread after a while could make available
profound knowledge toward the epidemics condition. Funk et al. (Funk et al., 2017) and Riley
et al. (Riley et al., 2003) identified whether outburst manage methods are possessing a
quantifiable consequence. Aforementioned analysis be capable of notifying prophecies regarding
probable potential enlargement (Viboud et al. (Viboud et al., 2018)), assist approximation danger
to previous nationals (Cooper et al. (Cooper et al., 2006)) and lead the intend of substitute
organization of events (Kucharski et al. (Kucharski et al., 2015). Nevertheless, there are quite a
lot of issues to that investigates, predominantly in the genuine occasion. This might be an
impediment to indication manifestation ensuing starting the hatching stage also an obstacle to
authentication of holders consequence on or after disclosure and evidence capability (Aylward
et al. (Aylward et al., 2014)).
Mathematical Modelling move towards can description to those impediments in addition to
ambiguity with unambiguously integrating delayed effecting as of the accepted the past of
diseases and disclosure developments (Nishiura et al. (Nishiura et al., 2009). Moreover, human
being data foundations might be prejudiced, unfinished, or only imprison convinced
characteristics to the outburst kinematics. Data amalgamation methods, this vigorous to manifold
information resources instead of sole information, set might facilitate additional healthy
inference of the fundamental flow of spread with loud facts (Birrell et al. (Birrell et al., 2018)
and Baguelin et al. (Baguelin et al., 2013)). Wu and McGoogan (Wu & McGoogan, 2020)
established that eighty-one percentage of holders are of kind indication (with no pneumonitis or
merely placid pneumonitis), fourteen percentage be a cruel holder through complexity in
respiration, also five percentage be crucial with a resuscitator failure, septicemia, numerous
organ dysfunction or breakdown. Recently, Krishna and Prakash (Krishna & Prakash, 2020)
have described enlarging a phase-based mathematical modelling to specify the transferability of
COVID 19 disease. (M.V Krishna)
CHAPTER III
METHODOLOGY
Data
The time series case data of the COVID-19 will be extracted from the Provincial Health
Office (PHO) of Nueva Vizcaya from January 1, 2020 to December 31, 2021. All cases are
laboratory confirmed following the case definition by the PHO situation report. Clinical
diagnosis of suspected individuals will was used as criterion for confirmed cases since January
2020. The confirmed case is defined as the individual whose real-time transcription polymerase
reaction (RT-PCR) result turned out to be positive.
The SIR Model
The SIR model is a representation that divides a population with respect to a disease’s impact on
an individual over time. An individual can be categorized as susceptible (S (t)), infected (I (t)), or
removed (R(t), dead or cured), denoted by S, I and R respectively, along an independent variable,
time. One of the most common SIR models is the classic Kermack–McKendrick Model for
contagious diseases in a closed population over time, which illustrates rapid changes in the
number of infected patients during epidemics. It is assumed that there is a fixed homogeneous
population size, random population mixing, instantaneous incubation period, and acute onset of
disease. The model variables can be represented as fractions:
s = Ошибка!
where s is a fractional representation of the number of susceptible individuals (S) over a selected
population (N) over time.
i= Ошибка!
where i is a fractional representation of the number of infected individuals (I) over a selected
population (N) over time.
r=Ошибка!
and r is a fractional representation of the number of removed individuals (R) which include the
recovered and deceased individuals over a selected population (N) over time. Overall, these
equations must add to 1:
s+i+r=1
The Susceptible Equation:
Ошибка! = -βsi
(1)
where β represents the infection rate, the probability per day that an I-person can infect a S
person, assuming the absence of social distancing.
The Infected Equation:
Ошибка!= βsi – γi
(2)
Ошибка!= γi
(3)
The Removed Equation:
where the effective γ represents the removal rate (encompassing both the recovered and deceased
individuals), the probability per day that an I-person transitions into an R-person (becoming
noninfectious permanently).
The ratio of S-persons transitioning into I-persons is the ratio of β to γ, referred to as the
Reproduction Number: Ro
Ro= Ошибка!
The higher the value of Ro, the more transmittable the disease is; the infection rate eclipses the
removal rate.
In these equations, the parameters β (the infection rate) and γ (the recovery or removal
rate of infectives) are constants: β controls the transition between S and I, equation (1), while γ
controls the transition between I and R, equation (3). From a dimensional point of view,
assigning no units to S, I, R, and N the parameters β and γ have units of inverse of time
(measured typically in days, weeks or months in epidemiological records). Notice that equation
(1) expresses the interaction between S and I (at time t) as the product SI and that a fraction of
this product are the individuals that at time t becomes infected and removed from S (which,
because of the negative sign in equation (1), decreases as time increases from zero). This
interaction in the form of the product SI makes difficult to determine the parameter β from
observed epidemiological data. On the other hand, from equation (3), the inverse of the
parameter (γ) gives a measure of the time spent by individuals in the infectious state.
Consequently, by carefully observing the development of an infectious disease, the parameter γ
can be estimated relatively precisely by epidemiologists from epidemiological records (as the
inverse of the recovered or infectious period). For the initial conditions, at time t = 0, we have
S (t = 0) = S0 > 0, I(t = 0) = I0 > 0, and R(t = 0) = R0 = 0. From equation (4), this yields
S0 + I0 = N.
Equation (2) can be written in the form:
Ошибка! = γ ( Ошибка!S − 1)I
(5)
where ρ = Ошибка! is sometimes called the relative removal rate
For an epidemic to occur, the number of infected individuals needs to increase from the initial
number of infected individuals I0. This condition will happen if at time zero, S0 > Sc = ρ. That
is, ρ represents a critical value for an epidemic to occur and the SIR model reveals a threshold
phenomenon. Another important epidemiological parameter which defines how quickly the
infectious disease spreads is the basic reproduction rate (Ro) of the infection, defined as:
Ro = Ошибка!S0 = Ошибка!
(6)
This parameter measures the number of secondary infections produced by one primary infection
in a wholly susceptible population. For instance, if Ro = k, then, before recovering, 5 one
infected individual will likely infect k more individuals, each one of which will, in turn, infect k
more individuals and so on. If more than one secondary infections is produced from the primary
infection, then Ro > 1 which gives S0 > ρ and, obviously, an epidemic ensues.
Assumptions
1. Once recovered, individuals become immunized and cannot be infected again.
2. During the disease no people enter or leave the population.
3. SIR model could be observed at the start of an infective disease, time at which no
action has been taken to control the disease and locally individuals has no restriction
in coming into contact with one another.
4. SIR model also assume that as soon as a susceptible individual catch the disease is
infective right away.
5. Ro only decreases or stays constant. It does not increase.
Variable
Description
N
Total population
S
Susceptible Individuals
I
Infectious Individuals
R
Recovered Individuals
t
Time
Parameter
Description
β
Infection rate
γ
Recovery rate
Ro
Reproduction rate
Parameter Estimation
At the onset of infection, almost entire population is susceptible: S~N, therefore I(t)
grows exponentially.
Ошибка!≈ μI
where μ = β – γ
Estimation of μ can be done on a log plot and using least squares to fit best line fit.
Estimation of γ:
Suppose I(t)= Io is constant, then Ошибка!= γIo
The solution will be, considering infected individuals with no more infection taking place, how
do individuals recover if additionally no one is recovered then Ro=0, if it takes T days to
recover, then R(T)=Io then γt=1 so, γ≈ Ошибка!where T is recovery period.
Eg. Estimates for T≈ 2-5 weeks: γ≈ Ошибка!or Ошибка!
Estimation of γ directly from data:
Ошибка!= γI , approximate the derivative
𝑅(𝑡 + 𝑎) − 𝑅(𝑡)
= γI
𝑎
where a is the number of days, set a=1 we get a rough estimate:
γ≈
𝑅(𝑡+1)− 𝑅(𝑡)
𝐼(𝑡)
Now we have γ and μ we get, β= γ + μ.
After estimation of parameters data will be plugged in a software program having codes
that estimates data.
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