RESEARCH GAP IN THE STUDY
hi the literature review, the significant work is done on computational modeling. Here, it is showing that there is a gap in a different area of computational modeling on epidemiology. Previous research investigates the board area of modeling, which needs to be accomplished with so far not explored this matter to improve the computational modeling on epidemic diseases. In recent years, maximum research work on different diseases with mathematical modeling has been present in a different maimer. Current research work is a unique approach to fill the gap with the help of computational modeling on epidemiology. It may be used to reduce the effect of epidemic diseases on humans and creating awareness among human beings to be protected and future medication as well as future predictions. The gapes noted passed up a major opportunity in prior research amid the work for displaying for transmittable illnesses should be filled. An epidemic disease (influenza A H1N1, Swine Flu) study has been presented [5]. There should be consideration of removal rate and the disease-free equilibrium. Discussion of contagious diseases can happen with the help of the SIR model by which a new compartment like the influence of treatment can add. There is a conversion of the SIR model to a new model named SITR model. Occasionally, the fixed population infected by influenza produces bronchitis, the minor infection, and then applies the Sf TIRS Model. Very few transmittable diseases are there in a long history on which mathematical modeling can be applied to develop a model.
PROBLEM IDENTIFICATION
The world in which we live today is a place veiy different from the one that existed 50 years ago. The modem life is full of facilities and technology (everything at one click today) due to which there is a significant change observed in the lifestyle of the people. Change in lifestyle such as advancement in technology and uptake of fast food in excess provides many factors that have a high potential of developing diseases in an individual. Risk factors associated with the environment such as pollution in the air, water, and soil, change in climate, and exposure to chemicals contribute significantly toward the development of the disease. In addition, genetic factors such as a change in the single or multiple base-pair by mutation due to environmental and lifestyle factors are responsible for disease development. Although technology in medicine has reached its height, it helps in finding the cure and methods for the prevention of many diseases. However, the cure for all is very challenging. Therefore, epidemiology came into existence that provides better decision making to improved accuracy in the diagnostic processes.
PROPOSED SOLUTION
The solution to the problem in this chapter has inspired by ethnographies from different computational techniques present cultures. The concept focused on human health-related issues.
9.5.1 DETERMINISTIC MODELS
In order to find the population size as a uniquely determined function of time without randomness but by probability distribution are called deterministic models.
9.5.1.1 SIR MODEL
In the simple deterministic model, we consider that the total population, say N, at any tune “Г is taken to be constant. If a small set of infected persons is introduced into a big population, the fundamental problem is to explain the spread of the infection within the population. It depends on a variety of conditions, which includes the particular disease concerned. We consider a disease in which removal is also included recovery after taking any drug or death or loss of interest. Consider the disease is such that the population can be divided into three different classes: the susceptible, S, who are prone to disease; the invectives, /, who are having the disease and able to transmit it; and the removed class, R. namely, those who are removed from the population by recovery, death, hospitalization or by any other means. The structure of the above model is represented in Figure 9.1.
FIGURE 9.1 Presentation of the SIR model.
Such type of models is called SIR models. Let n be the initial number of susceptible persons in the total population in which there is only one infected person has introduced. As a result, the number of susceptible starts decreasing. At the same time, the number of infected persons increases. We now consider the different classes when mixed uniformly with the condition that every individual has an equal probability of coming in contact with each other. Basic assumptions for the mathematical model are the following.
- • The total number of the population is fixed.
- • The infectious disease is transmitted by direct individual contact.
- • The recovery from an infectious disease will vary.
Adding the above equations, we get —+—+—= о, which implies that
dt dt dt
S +1 + R = N, at tunes with the initial conditions that S (0) > О, I (0) > 0, and R(0) = 0.
Let S(t) denote the number of persons who can be infected from any given set of population, called susceptible class. The standard epidemic model was first invented in 1927 by Kennack and McKendrick [5] and has played a significant role in mathematical epidemiology. We use probability to simulate the variability of the total number of the infected population in the compartmental model. The newly infected population caused by the infected population at each stage is obtained by the numerical value of the probability of an event, while the infection rate is not fixed. Therefore, here, we have two cases to investigate the status of the disease in society.
Case 1. If the infection rate is higher than one or Case 2. If the infection rate is less than one.
Case 1:
Probability of event |
1 |
2 |
3 |
4 |
5 |
Number of the infected population |
0 |
3 |
4 |
7 |
8 |
The expected rate of infection =-*0+-*3 + -*4 + -*7 + -*8 = — = 4.4 ^{r} 5 5 5 5 5 5
Which is higher than one, so the epidemic will spread-out in the population? Case 2:
Probability of event |
1 |
2 |
3 |
4 |
5 |
Number of the infected population |
3 |
i |
0 |
0 |
0 |
The expected rate of infection =-*3+-*l+-*0 + -*0 + -*0 = — = 0.8 ^ 5 5 5 5 5 5
which is less than 1, so the epidemic will die-out in the population. Different types of epidemiological improved SIR model are as follows:
Stability Analysis for the System of Differential Equations (SIR) Model Jacobian matrix of the governing equation (SIR model) is given as
-pi- X -ps 0
Now, Det (J-/J)= PI pS-a-X 0 =0
0 a ju-X
i.e., Я, < 0, A_{2}< 0, andA_{2}< 0; if -(pi - pS + a) > ■yjipi - PS + a)^{2} - 4pal.
Since all the eigenvalues are negative, then the given model is steady (stable), otherwise nonsteady (unstable).
FIGURE 9.2 The graphical relationship of SIR.
In the above graph, the susceptible class (5) shows that the persons, who are prone to get any communicable disease, go on decreasing as they are getting infected when coming in contact with infectives. The infected persons (7) show that the number of infective goes on increasing and then become stable as at the same time people are getting recovered with some vaccinations and getting into recovered class. It can be seen from the graph that the recovered class (R) is increasing but with less rate as it includes death cases also.
In a social network, the user’s dynamicity is a significant feature, which can impact the user's behaviors. The graph shows that the number of possible authors is decreasing concerning time and the range of authors whose posts lose infectivity to others on a subject is increasing concerning time. The infected class is growing initially, but it is decreasing.
9.5.1.2 S1TR MODEL
The compartment model SITR is a set of differential equations, which are designed for the susceptible to infection, infection to treatment, and treatment to complete recovery (see Figure 9.3).
FIGURE 9.3 Presentation of the SITR model.
Susceptible class, S: susceptible to infection, p transmission rate; Infective class, /: infection /, у rate of selection treatment, d death rate due to infection; Treatment class, T: treatment for infection, a removal rate from infection due to treatment; Removed class, R: complete removal of infection. Table 9.1 represents the estimation of values for the parameters involved in SITR Model.
Parameters |
Values [Reference] |
P infection rate |
1.28 per year |
Y treatment rate |
0.70 per year |
d death rate |
0.20 per year, [51] |
о removal rate due to treatment |
0.10 per year [51] |
Inchoate value of S_{0} |
1 estimated |
Inchoate value of I_{0} |
0.01 estimated |
Inchoate value of T_{0} |
0.50 estimated |
Inchoate value of R_{0} |
0.20 estimated |
Figure 9.4 represents the status of mathematical epidemiology. The result of the above model is verified by the Routh-Hurwitz criterion and provides the future prediction of reverting it [55].
FIGURE 9.4 Graphical representation of the SITR model.
9.5.1.3 SIJI_{2}RS MODEL
In this model, infected individuals become infectious immediately, as shown in Figure 9.5.
FIGURE 9.5 Presentation of the SIjTL.RS model.
The governing differential equations of the model are as follows:
TABLE 9.2 Estimation of Parameters Involve in SI,TI,RS Model
Sr. No. |
Variables/Parameter |
Values [Reference no.] |
1 |
S “Susceptible class” |
399 per year |
2 |
f “Infective class 1” |
23 per year |
3 |
/,“Infective class 2” |
1 per year |
4 |
T “Treatment class” |
23 per year |
5 |
R “Recovered class” |
0.00125 per year |
6 |
pi “Transmission rate 1” |
0.012531 per year |
7 |
p2"Transmission rate 2” |
0.006 per year |
8 |
0 “Rate to secondary infection” |
0.0625 per year |
9 |
d 1 “death rate 1” |
0 per year |
10 |
d2 “death rate 2” |
0 per year, [52] |
11 |
yl “Recovery rate 1” |
0.2 pear year |
12 |
y2 “Recovery rate 2” |
0.09 per year, [52] |
The model provides the information to the human population that they should be more educated to collect more data and attend health awareness camp of the infectious disease transmission and medication for in human population (Table 9.2). In addition, the practitioner can give better attention to successful treatment. Figure 9.6 shows the graphical solution of the SIJhRS Model.
FIGURE 9.6 Graphical solution of the SI.TI.R model.
9.5.1.4 SlJIJtV
Figure 9.7 gives the vaccination model.
FIGURE 9.7 Presentation of the SIjT^RV model.
The governing differential equations of the model are as follows:
This model is helpful to predict the growth of infectious disease and the effect of the vaccination model. Our investigation also provides a considerable role in the correlation between mathematical modeling and dynamical aspects of specific epidemic diseases.
9.5.1.5 S1TRS
Figure 9.8 gives the resusceptible model.
FIGURE 9.8 Presentation of the SITRS model.
The governing differential equations of the model are as follows:
Table 9.3 represents parameter estimations for the parameters involve in SITRS model. The graph shown in Figure 9.9 represents the behavior of susceptible to influenza (5), infected with influenza (7), treatment for influenza (7), and completely recovered from influenza (R). The graph represents that the susceptible rate (S) decreases concerning time and removal (R) from the disease due to treatment. Graphically solutions prove that the variables are not asymptotically stable. Simulation results are showing the trajectories and behavior of SITRS model.
TABLE 9.3 Parameter Estimations for the Parameters Involve in SITRS Model
Parameters and Variables |
Values with Reference |
p transmission rate |
1.30 per year |
у infective is selected for treatment |
0.50 per year |
(I death rate |
0.20 per year |
a removal rate from the treatment |
0.10 per year [51] |
ц losing immunity |
0.027 per year |
Initial value of S_{Q} |
1 [51] |
Initial value of/_{0} |
0.01 [51] |
Initial value of T_{0} |
0.5 [51] |
Initial value of R_{0} |
0.2 [51] |
FIGURE 9.9 Graphical solution of the SITRS model for influenza.
Result and Discussion
- • The number of new infection increases in the population because the basic reproduction number is positive and greater than one.
- • The force of infection is also increased in the population, and the number of the suspected class will decrease with concerning tune.
- • The numerical data are verified and analyzed by MATLAB Graph analysis.
This work represents simultaneous differential equations which are formed for the Susceptible Class (5), Infective Class (/), Recovered Class (R), Infected with Influenza Vims (7_{;}), Recovered from Infection (7), Infected with secondary pneumonia (/,), Transmission rate of Influenza (p_{t}), Recovery Rate of Infection (p), Rate at which an individual loses susceptibility (a), Rate of Re-susceptibility (8), Transmission Rate of Infection (p), Excess death rate due to Infection (d), Transmission Rate of Bacterial Infection (p,), Recovery Rate of Bacterial Infection (y,), Excess Death Rate due to Bacterial Infection (<-/,), and vaccination rate (v) represents the parameters and variables used/ involved in SITRS Model. In the mathematical model, each compartment can compute the number of effective people due to infectious disease at any time, hr mathematical models, it is also essential to keep a list of variables, which can be changed according to situations of the model. Each variable has the properties to express the epidemic situation in the human population.
9.5.1.6 STOCHASTIC MODEL
If we conduct a large number of experiments or cany out a significant number of surveys under the same initial conditions, we find that the population size at a time t is not uniquely determined and keeps on fluctuating randomly. It means that we require probability distributions are called stochastic models.
Statistical methods in various sorts of epidemiological studies are shown through examples with real or imaginary (“Active”) data. Essential measures of recurrence and impact will be presented. Distinctive relapse models will be introduced as instances of sophisticated systematic strategies.
Basic Equation: ^- = -p_{l},^f_{J}(n)+£_{i}P„_jf_{J}(>i-j).
j*о j*о
Multiplying by x’ summing for all n, and using the definition of probability generating function, we obtain
n-0
Here,p_{n} is the probability of n susceptible persons andf. (n) is the probability that the number changes to n + j in the time interval t + At. n and ./-integer.
9.5.1.6.1 The Stochastic SIR Model
This model consists of susceptible, infected, and removed as a population.
Example: In this model, the compartment of individuals of three classes are considered, which are refer to as class S (susceptible), class I
- (infective), and removed (either the individuals recovered or died). The cause of infection’s probability during the time interval t, t + At, where the removal of one individual from class S and addition of that individual to class I is BSI/N At + o(At). Consider у to be the rate of recovery of infected individuals; then, the probability for the state of recovery where one individual from class / has been removed and added to class R during the time interval of [t, t+ At] is у I At + о (At). The cause of infection’s and recovery probabilities are ((S,_{+Af}, I_{I+&I})-(S,, /,) = (-1,1)) = fiS,I,/N At +o(At). P((S,_{+il}, I_{l+A1})-(S_{r}, I,) = (0,-1)) = у It At +o(At) with the complementary probability P((S_{t+il}, I_{t+}J-(S„ I,) = (0,0)) = l-(j3S,/N+y)I,^+ о (At) in the time interval [53].
- 9.5.1.6.2 Discrete-Time Stochastic Models
It is assumed that stochastic variables depict the number of susceptible and infective. In a simple case, the number at t depends only on the numbers at the previous time. So susceptible maybe, considering controlled population.
9.5.1.6.3 Continuous-Time Stochastic Models
To build a continuous-time stochastic model, this considers the epidemics (infection) and recoveries on a continuous time scale. It is a continuous-time stochastic (Markov chain) for frequency-dependent transmission. The probabilities of susceptible approaches to infective are
From the standpoint of an infective, each can infect susceptible in time At as per rate transmission rate /?, where .s is the number of susceptible at a time t, and in the perspective of susceptible, each may be infected in time At as per rate y, where i is the number of infective at time t. This approach is not widely used as computationally, where population increases, which is the case with surveillance data.
9.5.2 GAUSSIAN DISTRIBUTIONS
The standard Gaussian distribution has the probability density centered at 0. The probability density at any value г (positive or negative) is given by
0.3989 e — r .
. 2
Example: If the mean and standard deviation of general Gaussian distribution are 100 and 20, respectively, what ranges of values correspond to probabilities of 0.90 and 0.95, respectively?
Similarly, when x has a Gaussian distribution with mean ft and standard
deviation a. then i = ^ ^ j will have a standard Gaussian distribution.
This fact can be used to get the probability for a range of values of .r using tables of r.
The probability density per unit of .r when x has a Gaussian distribution
, ,, . . . 0.3989 Г — (x — и V
with mean ц and standard deviation a is-exp — -
<7 [2( <7 JJ
9.5.3 APPLICATIONS OF SIR MODELS
The SIR model can be applied to any disease, whether acute or chronic with the condition that the mass is prone to get that disease and then can spread the disease. Some common applications are as follows.
9.5.3.1 HUMAN/ANIMAL RELATED: EPIDEMICS
The SIR model is formulated on any disease, which flows through any medium when a susceptible person meets with an infected person and possibly becomes infected, which includes the parameters like genetics or contagious.
S(t) represents the number of feasible people or animal, who can be in contact with infected person/animal and prone to be infected at any time t.
I(t) represents a wide range of people/animal who has the disease and likely to spread the infection to other persons/animals.
R(t) represents a range of people/animal that has recovered from that disease and has lost power to infect other persons/animals.
Some common diseases are as follows:
Hypertension, a complex issue, described by the dimension of systolic and diastolic blood pressure more than 140 and 90 nun Hg, respectively [6]. It is uncovered to be the most significant risk factor for individuals who have influenced the way of life. Reports from World Health Organization (WHO), Global Burden of Disease, and Noncommunicable Disease Risk Factor Collaboration demonstrate the high predominance of hypertension in India and detailed more than 1 million individuals that are experiencing this ailment (disease) [7]. India has turned into the third driving position of causing demise in India, adding to about 10.8% of all passing in India. 29% of the whole stroke and 24% of all heart attacks are connected with hypertension [8].
Diabetes, a metabolic disorder characterized by the increased level of glucose levels (more than 180 mg/dL) in the blood plasma, causing a condition of hyperglycemia [9-11]. According to the WHO, it has been estimated that sound 422 million adults have diabetes, and around 1.6 million deaths are directly associated with diabetes each year [12].
Alzheimer’s disease, a type of dementia, is a neurodegenerative disorder, characterized by the accumulation of beta-amyloid protein (plaques) and the tau proteins (neurofibrillary tangles) [ 13 ]. It is a stepwise progressive disorder affecting the memory, thinking, and behavior of an individual, resulted from neuronal cell death [14], contributing 60-70% of all dementia cases [15]. It has been revised that around 5.7 million Americans have Alzheimer, and this number can be increased to 13.8 million. Deaths caused due to Alzheimer’s have increased by 123% [16].
The parasite of protozoa causes Malaria, a common, parasitic, and life- threatening disease that belongs to the genus Plasmodium [16]. It remains one of the world’s most devastating infectious diseases found in the tropical and subtropical regions of the world. Plasmodium falciparum, plasmodium vivax, and plasmodium malaria are the primary parasites responsible for causing malaria and sometimes plasmodium knowlesi in human infections, but mortality rates are usually low. It is transmitted by female infected Anopheles mosquito [17]. Plasmodium falciparum is known to be a fatal parasite that is responsible for most of the mortality rates [18]. The WHO has reported about 219 million cases of malaria in 2018, but somehow less than 2010, where about 239 million cases were reported [19]. The frequency of occurrence of Malaria is more in the African region that is about 92%,
5% in the southeast Asian area, and about 2% in the Eastern Mediterranean region [29].
Aedes mosquitoes transmit Dengue, a vector-bome viral disease. Dengue virus belongs to the genus Flavivims and family Flaviviridae and consists of DEN 1, DEN 2, DEN 3, and DEN 4 serotypes [22,23]. About 100 countries are suffering from Dengue, including the USA and Europe [24]. It has become a significant health challenge in the tropical and subtropical regions. An increase hi the gr owth rate of population, lack of mosquitoes repellant, global wanning, water and ah pollution, lack of awareness, and health facilities cause a 30-fold increase hi Dengue infection worldwide [25,26]. hi addition, it has been reported that there may be 40% upsurge in the spread of Dengue infection by 2080 due to change in climate [27].
Contagious diseases are as follows:
Swine Flu, also known as Swine-Influenza A (El IN 1) flu is the infectious and respiratory illness of the pigs, spread by strains of Influenza a virus [28]. The WHO reported the pandemic nature of the H1N1 vims hi 2009 that affected most of the world’s population. In 1918, approximately 50 million people w'ere infected with H1N1 Influenza virus around the world, with 50-100 million people deaths reported [30]. Around 43-89 million cases were reported in 2009 by the Centre for Disease and Control and Prevention and about 1799 deaths in 178 countries worldwide [31]. In 2019, 10,000 cases of swine flu have been reported across India with 774 deaths [30].
Chickenpo.x, a highly contagious disease, causes infection hi more than 90% of the people that are not vaccinated during their lifetime [32]. It is transmitted by Varicella Zoster Vims, signalized by a pruritic vesicular eruption associated with fever and malaise [33]. Varicella (chickenpox) is more prominent hi temperate regions as compared to tropical regions. The WHO estimated the burden of chickenpox to about 140 million cases with 4.2 million severe complications and 4200 deaths in 2014 [34]. The varicella-zoster-virus- infected 16 cases per 1000 people in developed countries annually [34]. The more burden of disease has been reported in children with 90% of cases [35].
Whooping Cough, also called Pertussis, is an acute illness of the respiratory system. Bordetella pertussis is responsible for causing whooping cough [32]. In 2008, the WHO reported 16 million cases, of which 90% w'ere in developing countries with 195,000 deaths [33]. In 2009, 40% of the cases have been noticed in the USA. Children of age gr oup between 7 and 10 years suffered from whooping cough from an estimate of around 9%, 13%, 23.5%, and 23% in 2003, 2007, 2008, and 2009, respectively [34].
9.5.32 STANDARD EPIDEMIC SIR MODEL IN WEB FORUM
The SIR model is formulated on an idea or topic diffusion on social networks as social media has become one of the easiest ways to share any information within a flying quick. It has influenced many streams such as business, politics, and marketing. SIR is the standard epidemic model, which was first invented in 1927 by W. O. Kennack and A. G. Mckendrick and has played a significant role in mathematical epidemiology. In the web forum context:
S(t) represents the number of feasible authors, who might have a hobby in an issue;
I(t) represents a wide range of authors who write down posts on any issue throughout the identical duration;
R(t) represents a range of recovered authors whose posts not affecting others on a subject.
The infection rate, a, indicates how many possible authors will be infected per contact between an infective and a susceptible. The recovery rate, p, indicates how many infective authors per infective recover during a unit tune.
9.5.3.2.1 Some Other Simple Examples of SIR Models through Social Media Are Rumour/Fake News Spreading on WhatsApp
Fake news in India is a growing problem. The practice of the use of social media platforms such as WhatsApp to popularize false information is leading in a dangerous direction. Two main aspects are urging the fake news singe: fust, the trend of declining smartphone prices over the last couple of years, and, second, the fall in Internet data prices. Therefore, the problem of fake news is not going to leave soon, and knowing the truth is not going to be easy. We heard fake news at the time of demonetization of currency notes in India dining November 2017. Lots of fake news, videos, and photos on the conflicts of Hindus and Muslims are shared through WhatsApp.
Viral marketing
Product information can be popularizing through social networks. This gives a positive impact on the customer, and as we see, lots of people are purchasing things through online shopping sites. It is gaining more fans, and these fans are spreading information and more, and more people are getting involved in eveiy social network.
Audience applause
The widespread example is audience applause. We can see when sitting in a group of people and one person starts to clap and immediately the person sitting in the crowd starts clapping. This is the psychology of the persons in the crowd that they change their behavior quickly in response to others.
The spread of personal computer virus
Nowadays, with the increasing rate of IT (information technology) development, the security of networks is a significant issue in our daily routine. A computer vims is one of the threatening aspects nowadays. It has a high impact on the computer world. The vims attacks the computer system and damages the software.
Academic spreading
Here, the susceptible are research scholars, infected are the research scientists, and the carriers are journals and conferences. There is immigration into research scholars from the educational stream whose strength is assumed to be proportional to the strength of research scholars' class. The increase in research journals and conferences depends on the number of research scientists.