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This article was published as a part of the Data Science Blogathon

Introduction

According to Haines and Crounch, mathematical modelling is a process in which real-life situations and relations in these situations are expressed by using mathematics. In simpler terminologies, mathematical modelling is the process of describing systems (activities) with mathematics. Mathematical modelling is the process of using mathematics to model real-world processes and occurrences.

Mathematical modelling is used virtually in every sector, in the manufacturing industry mathematical modelling is used to model heat and mass transfer of fluids flow, the transformation of materials, e.t.c. The construction industry is not spared from the beauty of mathematical modelling, mathematical modelling is used to optimize the amount of port in structures, calculating the stress that will be imposed on buildings and how to counterbalance it. You probably must have seen the tallest building in the world either virtually or physically, you will be in awe if you were to see all the mathematical models that were used to model the building.

Burj Khalifa (Tallest Building in the world). Source

                                 .

Football athletes use mathematical modelling to score goals, for football lovers you probably must have seen how Messi, Rolando, and other popular footballers use free kicks to score goals. The free-kick goals can be modelled with mathematics, by modelling the angle of trajectory, the drag e.t.c.

Modelling free-kick with Mathematics. Source 

The astronomy industry heavily relies on mathematical modelling, mathematics is used to model the movement of spacecraft and other orbital objects. Katherine Johnson, a former mathematician at NASA used her mathematical prowess to help put an astronaut into orbit around the earth. Her mathematical skills were also used to deploy a man on the moon.

Photos of Katherine Johnson. Source 

I can continue to list the sacrosanct role of mathematics in our world, but because of time constraints, I will stop here. The reality is that the world can exist without the English Language but the world can’t exist without Mathematics.

This article will walk you through the processes of modelling disease spread with mathematical models. You might be wondering can mathematics really model disease spread? The answer is yes, mathematics is very important in the health sector. According to TheConversation “Mathematical models are used to create a simplified representation of infection spread in a population and to understand how an infection may progress in the future. These predictions can help us effectively use public health resources such as hospital space or a vaccination programme. For example, knowing how many people in a population are likely to become infected can tell hospitals how much space and resources they will need to allocate for treatment.” Source

What it takes to mathematically model any disease 

At the beginning of an epidemic, there exist, people who will be infected, prone to getting infected and those who might recover from the disease or die as a result of the disease. Those who were initially prone to the disease will get infected if they come in contact with infected people and those who will die will originate from the infected people. Mathematicians have been trying to successfully find a way to mathematically model the relationship between those who are prone to be infected, those who are infected and those who will recover from the disease. In 1927, Kermack & McKendrick came up with what is called the Susceptible, Infected and Recovered (SIR) Mathematical model. The SIR model assumes that for any given disease, there exist 3 categories of people those who are Susceptible (Prone to contracting the disease but are yet to be infected), those who are Infected and those who have been Removed(recovered) (either by death or with the aid of drugs). The SIR model has been of help to mathematicians and has made modelling disease spread easy.

To mathematically model any disease using the SIR model, you will need to assume that the population remains constant i.e ( No birth takes place, nobody migrates into the population, no natural death ( with an exception of death from the disease)). The SIR model models diseases by taking into cognizance that, the movement of people from the Susceptible into the Infected state and from the Infected State into the Removed state is defined by some constants. These constants are the tripod that the SIR model sits on, and that is what will be discussed soonest. You will agree with me that, for any disease to spread there must be contact between susceptible people and infected people or person( disease carriers).

 Assuming for a particular epidemic, there exist 1000 Susceptible people and 3 persons that are infected. Take, for instance, every day 1 person gets infected due to the contact between Susceptible and Infected people. You will agree with me that, on the fifth day, 8 people will be infected and the number of susceptible will be 995. We might want to assume that 2 persons or 3 persons get infected, one thing here is that we are just making assumptions that might not be mathematically accurate. Hence the need to use the SIR model to mathematically and accurately model the spread of the disease.

The SIR model models the number of people who are infected by assuming that everyone in the susceptible category has an equal probability of being infected by a constant fraction which is called the contact rate (infection rate). The number of people that are infected is computed by multiplying the contact rate with the number of infected people and the Susceptible after which the population number is used to divide the result i.e (contact rate * S * I)/N. S-Susceptible, I– Infected, and N– Total Population Number.  The contact rate will be a fraction of the population which is computed by analyzing the number of contacts made with infected people per day. The SIR model also models the number of people who will be removed by a certain fraction which is called the recovery rate. The number of people that will be removed is computed by multiplying the recovery rate with the number of infected people i.e recovery rate * infected people.

SIR Mathematical Model Source

ds/dt = the rate of change of the susceptible over time

dI/dt is the rate of change of infected over time

dR/dt is the rate of change of removed over time

The equation simply states that susceptible people will be reduced over time based on the contact rate (beta), the number of susceptible, the number of infected, and the total population (N). You will notice the presence of the negative sign, this is to show the fraction of people that will be lost from the susceptible category. The fraction of people that are lost from the susceptible category will be added to the infected category, hence the presence of the positive sign in the infected equation. Recall that the removed people originate from the infected category and the number of people that are removed is based on the removal rate multiplied by the number of infected people (gamma * I). Those that are removed will be a loss to the infected people hence the need to subtract the number of removed from the number of infected. The removed people will be gain to the removed category, hence the positive sign for the removed category.

Multiplying both sides with dt will give

dS is the rate of change i.e the difference between the old susceptible and the new susceptible ( Snew– Sold). The number of susceptible, infected, and Recovered for the next day can be modeled by moving the old Susceptible, Infected, and Recovered numbers to the other side of the equation to give.

SIR Model. Image by Author

The above equation can be used to model the number of susceptible, infected, and recovered for the next day. The number of infected people in a day depends on the contact rate(Beta) and the recovery rate (gamma).

Other Types of Mathematical Models Used to Model Diseases

 Apart from the SIR model, several varieties of mathematical models can be used to model diseases. Other models that were derived from the SIR models are the SEIR model, SIRV model, SIRD model e.t.c. The SEIR model models disease based on four-category which are the Susceptible, Exposed (Susceptible people that are exposed to infected people), Infected, and Recovered(Removed). The Susceptible, Infected, Recovered(Removed) and Vaccinated(SIRV) is another type of mathematical model that can be used to model diseases. The focus of this article is on the SIRD or SIID model which is Susceptible, Infected, Removed(Recovered with immunity), and Dead or Susceptible, Infected, Immune, and Dead model. 

The SIID or SIRD model is an extension of an addition of two assumptions which are recovery with immunity and Death. For the rest of this article, I will interchange SIRD for SIID, both refer to the same acronym.

SIRD Model. Source

You will notice that the difference between the SIR and the SIRD model is the addition of the dD/dt which is the death rate per time. The SIID model models the death rate by considering a constant called the mortality rate(mu), which is the rate at which infected people die. The number of people who are dead is based on the product of the mortality rate with the number of Infected people.  You will agree with me that the number of people who were infected and died must be removed from the number of infected people. If we remove the number of dead people, then our rate of change of infection over time will be modified to accommodate the loss due to death, which will give this.

SIRD image showing the mortality rate. Image by chúng tôi that is coloured with yellow is the mortality rate.

Simulating Diseases with SIRD(SIID) (Practical) 

Given the above information that immunity exists and people die as a result of the disease, it means we will use the SIID model to model the disease. Let’s assume the number of people who are infected by the disease is 3, the number of dead and recovered is zero, the infection rate(beta) is 0.5, the recovery rate(gamma) is 0.035 and the death rate(mu) is 0.005. Note that the infection rate, recovery rate, and death rate were gotten from here, but you can try any number.

SIRD Model modified from SIR. Image by Author. 

The Susceptible number for the next day can be computed by using this method

Snew = Sold – (beta * Sold* Iold )/N

Sold = N – Iold = 1000-3 = 997 (i.e the susceptible number for the current day is the difference between the total population and the number of infected people in the current day)

beta = 0.5

Iold  = 3

N = 1000 (The total Population)

Snew = 997 – (0.5 * 997 *3 )/1000

Snew = 997 – 1.4955

Snew = 995.5045

The total number of Susceptible for the next day is approximately 995.5

Let us compute the rest, the next day number of infected can be computed with this method

Inew = Iold + (((beta * Sold * Iold)/N) – (gamma * Iold) – (mu * Iold))

gamma ( recovery rate) = 0.035

mu (death rate) = 0.005

Inew = 3 + ((0.5 * 997 * 3)/1000) – (0.035 * 3) – (0.005 * 3))

Inew = 3 + (1.4955 – 0.105 – 0.015)

Inew = 3 + 1.3755

Inew = 4.3755

The number of people that will be infected the next day is approximately 4.4

Modeling the number of people that would have recovered with immunity the next day, that can be modelled with this equation.

Rnew = Rold + gamma * Iold

Rold = 0

Rnew = 0 + 0.035 * 3

Rnew = 0 + 0.105

Rnew = 0.105

The number of people who would have recovered with immunity the next day is approximately 0.11

Lastly, modelling the number of people who would be dead the next day, this method can be used which is the application of the last equation

Dnew = Dold + mu * Iold

Dold = 0 + 0.005 *3

Dold = 0 + 0.015

Dold = 0.015

These steps can be repeated to model the number of susceptible, infected, recovered and dead for the next 2 days and more days. What if the steps can be automated, instead of manually computing the numbers. Python Programming language will be used to automate the process and plot the result.

Modelling Disease with Python Programming Prerequisites

To follow along, you will need to have python and preferably Jupyter notebook installed on your system. You can use this link to download anaconda, anaconda comes with a Jupyter notebook and python. You can use this video to familiarize yourself with the Jupyter notebook and how to install it.

Now that you have Jupyter notebook installed, you are good to go. Let us fire down

# importing neccessary libraries import matplotlib.pyplot as plt %matplotlib inline # defining the variables total_population = 1000 total_infected = 3 total_susceptible = total_population - total_infected total_recovered = 0 total_dead = 0 # Number of days to simulate disease simulation_days = 500 # list to store the numbers of recovered people with immunity over time # the first element will be the initial number of people that has recovered with immunity recovered_list = [total_recovered] #list to store the number of dead people over time dead_list = [total_dead] infected_list = [total_infected] susceptible_list = [total_population] infection_rate = 0.5 recovery_rate = 0.035 death_rate = 0.005 #using the range function to simulate for 500 days which is the simulation days for days in range(1,simulation_days): num_infected_daily = (infection_rate * total_infected * susceptible)/total_population # get the susceptible number for next day total_susceptible = total_susceptible - num_infected_daily num_recovered_daily = recovery_rate * total_infected num_dead_daily = death_rate * infected total_infected = total_infected + (num_infected_daily - num_recovered_daily - num_dead_daily) total_recovered = total_recovered + num_recovered_daily total_dead = total_dead + num_dead_daily susceptible_list.append(total_susceptible) # adding to the list of susceptible people infected_list.append(total_infected) recovered_list.append(total_infected) dead_list.append(total_dead)

Now that we have simulated Konvid-18 for 500 days, we can now visualize our result.

Visualizing the result

# Using chúng tôi to plot plt.plot(range(0,simulation_days),susceptible_list,color='blue',label='Susceptible') plt.plot(range(0,simulation_days),infected_list,color='red',label='Infected') plt.plot(range(0,simulation_days),recovered_list,color='green',label='Recovered) plt.plot(range(0,simulation_days),dead_list,color='orange',label = 'Dead') plt.legend() #add the labels to the plot plt.title('Konvid-18 Disease Simulation in JavaGo city') plt.xlabel('Days') plt.ylabel('Total Population') plt.show()

After running the above code, the image below will be displayed.

Visualization Result. Image by Author

Deductions Conclusion 

The article has shown you the importance of mathematical models, how to model diseases with the SIRD model, how to automate the process for days, and how to visualize it. The article introduced you to the SIRD model, there are other mathematical models that you can explore further and dive deeper into like the SEIR, SIS, SIRV e.t.c. The article also didn’t cover the mathematics of deriving the contact ratio, recovery rate, and death rate, you can explore these concepts further. I hope you have realized the importance of mathematics in the healthcare industry.

I created a demo web app for further exploration, the web app was developed with streamlit. You can access the web app with this link and check the source code with this link.

You can connect with me on LinkedIn,

References/More Resources

(3) The MATH of Epidemics

The media shown in this article on SIRD Model are not owned by Analytics Vidhya and are used at the Author’s discretion.

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You're reading Mathematical Modelling: Modelling The Spread Of Diseases With Sird Model

Basic Understanding Of Time Series Modelling With Auto Arimax

This article was published as a part of the Data Science Blogathon.

Introduction

Data Science associates with a huge variety of problems in our daily life. One major problem we see every day include examining a situation over time. Time series forecast is extensively used in various scenarios like sales, weather, prices, etc…, where the underlying values of concern are a range of data points estimated over a period of time. This article strives to provide the essential structure of some of the algorithms for solving these classes of problems. We will explore various methods for time series forecasts. We all would have heard about ARIMA models used in modern time series forecasts. In this article, we will thoroughly go through an understanding of ARIMA and how the Auto ARIMAX model can be used on a stock market dataset to forecast results.

Understanding ARIMA and Auto ARIMAX

Traditionally, everyone uses ARIMA when it comes to time series prediction. It stands for ‘Auto-Regressive Integrated Moving Average’, a set of models that defines a given time series based on its initial values, lags, and lagged forecast errors, so that equation is used to forecast forecasted values.

We have ‘non-seasonal time series that manifests patterns and is not a stochastic white noise that can be molded with ARIMA models.

An ARIMA model is delineated by three terms: p, d, q where,

p is a particular order of the AR term

q is a specific order of the MA term

d is the number of differences wanted to make the time series stationary

If a time series has seasonal patterns, then you require to add seasonal terms, and it converts to SARIMA, which stands for ‘Seasonal ARIMA’.

The ‘Auto Regressive’ in ARIMA indicates a linear regression model that employs its lags as predictors. Linear regression models work best if the predictors are not correlated and remain independent of each other. We want to make them stationary, and the standard approach is to differentiate them. This means subtracting the initial value from the current value. Concerning how complex the series gets, more than one difference may be required.

Hence, the value of d is the merest number of differences necessitated to address the series stationary. In case we already have a stationary time series, we proceed with d as zero.

”Auto Regressive” (AR) term is indicated by ”p”. This relates to the number of lags of Y to be adopted as predictors. ”Moving Average” (MA) term is associated with “q”. This relates to the number of lagged prediction errors that should conform to the ARIMA Model.

An Auto-Regressive (AR only) model has Yt that depends exclusively on its lags. Such, Yt is a function of the ‘lags of Yt’.

Furthermore, a Moving Average (MA only) model has Yt that depends particularly on the lagged forecast errors.

The time series differencing in an ARIMA model is differenced at least once to make sure it is stationary and we combine the AR and MA terms. Hence, The equation becomes:

We have continued operating through the method of manually fitting various models and determining which one is best. Therefore, we transpire to automate this process. It uses the data and fits several models in a different order before associating the characteristics. Nevertheless, the processing rate increases considerably when we seek to fit the complicated models. This is how we move for Auto-ARIMA models.

Implementation of Auto ARIMAX:

We will now look at a model called ‘auto-arima’, which is an auto_arima module from the pmdarima package. We can use pip install to install our module.

!pip install pmdarima

The dataset applied is stock market data of the Nifty-50 index of NSE (National Stock Exchange) India across the last twenty years. The well-known VWAP (Volume Weighted Average Price) is the target variable to foretell. VWAP is a trading benchmark used by tradesmen that supply the average price the stock has traded during the day, based on volume and price.

df.set_index(“Date”, drop=False, inplace=True) df.head()

df.VWAP.plot(figsize=(14, 7))

Almost all time series problems will ought external characteristics or internal feature engineering to improve the model.

We add some essential features like lag values of available numeric features widely accepted for time series problems. Considering we need to foretell the stock price for a day, we cannot use the feature values of the same day since they will be unavailable at actual inference time. We require to use statistics like the mean, the standard deviation of their lagged values. The three sets of lagged values are used, one previous day, one looking back seven days and another looking back 30 days as a proxy for last week and last month metrics.

During boosting models, it is very beneficial to attach DateTime features like an hour, day, month, as appropriate to implement the model knowledge about the time element in the data. For time sequence models, it is not explicitly expected to pass this information, but we could do so, and we will discuss in this article so that all models are analysed on the exact identical set of features.

The data is split in both train and test along with its features.

train: We have  26th May 2008 to 31st December 2023 data.

valid: We have  1st January 2023 to 31st December 2023 data.

The most suitable ARIMA model is ARIMA(2, 0, 1) which holds the lowest AIC.

Conclusion:

In this article, we explored details regarding ARIMA and understood how auto ARIMAX was applied to a time series dataset. We implemented the model and got a score of about 147.086 as RMSE and 104.019 as MAE as the final result.

Reference:

About Me: I am a Research Student interested in the field of Deep Learning and Natural Language Processing and currently pursuing post-graduation in Artificial Intelligence.

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Data Modelling Growth Projections In 2023

Data Modelling as a Key Industry Growth Driver

According to Business Standard, India’s software products sector was able to amass over $10 billion in revenue in 2023-2024. This shows that it is a valuable asset to the country’s growth and sets a good foundation for leveraging new and emerging technologies. As a crucial part of any software project, data modelling will significantly impact business growth as it can help garner information that can provide an edge over competitors.

Moreover, the digitisation of data has become increasingly popular thanks to the COVID-19 pandemic, which has prompted more and more companies to invest in data modeling and easy-to-use tools on the market. Data modelling will play an even bigger role in a wide array of industries such as retail, healthcare, finance, supply chain, education, hospitality, agriculture, and many more.

Data Modelling to Aid Booming Startup Ecosystem

In a recent podcast, MongoDB Vice President Sachin Chawla highlighted how India is currently enjoying a robust startup system. More than $38 billion in funding flowed into Indian startups last year — over three times more than the $11.1 billion total funding received in 2023. As a fast-growing company, MongoDB has also provided data platforms, enterprise software and support, and resources to rising startups such as Ultrahuman, Flexa, Bliinx, and many more.

Startups, most especially, will benefit from investing in data modelling tools earlier on in their business evolution. Not only will it aid them in collecting, updating, storing, and analysing information, but it will also identify their key business concepts and map them out to available and prospective data. An effective data modelling plan also improves system performance while saving money, which improves the chances for startups to succeed.

Data Scientists Here To Stay

The rise of AutoML (Automated Machine Learning) was brought about by the rapid digitisation of data in recent years, as previously mentioned. AutoML streamlines data collection, data preparation, deployment, as well as modelling tasks that can be quite time-consuming and repetitive without automation. Because of this, many have begun to question whether or not AutoML will eventually pose a challenge to data science jobs.

Contrary to this, the US Bureau of Labor Statistics predicts that the field of data modelling will grow by 8% over the next 10 years. There is no expected shortage of job opportunities for data scientists, especially data modellers. Moreover, the expertise that data scientists possess will remain far superior to AutoML features, specialists or those training in this field can expect an abundance of opportunities in the coming year.

Data Modelling to Address Societal Challenges

Thanks to improvements in technology, data modelling can even be leveraged to help address modern societal challenges occurring today. Writer and strategist Amy Lynn Smith highlights how the UN Refugee Agency (UNHCR) and the UNHCR’s Innovation Service and UN Global Pulse (UNGP) use data and analytics to get real-time feedback on how well policy responses are working. We can expect data modelling to contribute not only to businesses and enterprises but also to societal development as a whole.

Conclusion

Data modelling has become such an exciting field. As a key industry growth driver, a major contributor to information technology, and a necessity to most businesses, data modelling promises industry players a strong job outlook and will likely continue to evolve at a rapid pace in the near future.

Coronavirus Analysis: Will Social Distancing Help Prevent The Spread?

Introduction

We are in the midst of a global crisis. The coronavirus, or COVID-19, has officially been declared a pandemic and it is wreaking havoc across the globe. Countries are getting shut down, economies are severely affected and the stock market is crashing to the ground.

Given that we now have a bit of time on our hands, I wanted to use this to understand if there’s a correlation between working from home and the spread of the coronavirus. As a member of the data scientist community, I go with logical quantitative estimates rather than qualitative estimates. This triggered me to do some research on this topic.

My aim here is to answer why we need to strictly follow social distancing to control the spread of Coronavirus with the support of available data.

I have analyzed the disease outbreak using the data hosted by John Hopkins University which is updated on an hourly basis. The data source contains three files – Total Confirmed Cases, Deaths and Recoveries. From this data (till March 22, 2023), I plotted how the cases grew over time from the starting day of outbreak i.e. from the day when the first case is reported.

Here’s What We’ll Look At:

Top 10 Affected Countries

Current Coronavirus Situation in India

Is Social Distancing the Right Measure to Stop the Spread?

Learning From the 1918 Flu

Global Epidemic and Mobility Model (GLEAM) Estimate on China with Travel Ban

Estimating the Impact of Social Distancing in the US in the Coming Days

Model Virus Transmission with Social Distancing in the US

Top 10 Affected Countries

The graphs clearly show how the outbreaks grew exponentially after crossing the ‘outbreak’ threshold. This triggers a strong signal to every country about the intensity of the situation. If not taken seriously, the coronavirus cases can compound quickly and the growth is almost exponential so even a small number of cases could balloon into a full-blown outbreak very fast.

For instance, Italy took 23 days to cross 100 cases and just 13 more days to cross 5000 cases and is now at 53,578. Likewise, the US seems to be following the same pattern. It took 41 days for the US to cross 100 cases and just took 14 days to cross 6000 cases and now stands at 25,489. Currently, the US appears to have an even worse trajectory than Italy.

Current Coronavirus Situation in India

The world’s second-largest populated country India took 44 days to cross 100 cases and is now at over 400 cases. The virus spread growth post 100 days as compared to other top countries (in terms of virus transmission) in India seems to be in better shape.

Is it because of the early and serious implementation of social distance measures? India reported its first restriction of a travel ban on international arriving passengers starting from Mar 13, 2023, when it had 82 confirmed cases.

Is India not testing enough cases? Is that where we are missing the exact spread growth? The assumption is the disease has still not spread in the community. The country tested 826 samples collected from patients suffering from an acute respiratory disease from 50 government hospitals across India between 1 and 15 March.

Existing labs in India are able to provide results in six hours and each lab has the capacity to test 90 samples a day. The country is planning to increase its capacity to test 8,000 samples with the regular process and 1400 with rapid testing labs. So, to clearly analyze disease spread and model the Indian scenario, we need to wait for a couple of weeks.

Is social distancing the right measure to stop the spread?

Preliminary analysis suggests that the key influencing factor for the rise in cases across the globe is disease spread. Those with the virus can unknowingly infect others before symptoms appear, some as soon as two days after infection. Patients are able to spread the infection until they recover.

According to the “Estimating the generation interval for COVID-19 based on symptom onset data” report, the proportion of pre-symptomatic transmission was 48% (95% CI 32-67%) for Singapore and 62% (95% CI 50-76%) for Tianjin, China. 

There is one very simple thing we can do that works in reducing spread – social distancing. The idea is to reduce person-to-person contact in order to make spreading the disease less likely. This could ensure that there are sufficient resources available for a sick population, which in turn will help improve survival rates.

Learnings from the 1918 Flu Pandemic

The below chart shows the impact of social distancing in 1918 for the flu in the US. For example, a city like St. Louis took measures 6 days before Pittsburgh and had less than half the deaths per citizen. On average, taking measures 20 days earlier halved the death rate:

Source

Global Epidemic and Mobility Model (GLEAM) Estimate on China with Travel Ban

The GLEAM model generates an ensemble of possible epidemic scenarios described by the:

Number of newly generated infections

Times of disease arrival in each subpopulation, and

Number of traveling infection carriers

The below chart shows the impact that the Wuhan travel ban had on delaying the epidemic. The bubble sizes show the number of daily cases. The top line shows the cases if nothing is done. The two other blocks show decreasing transmission rates. If the transmission rate goes down by 25% (through Social Distancing), it flattens the curve and delays the peak by a whole 14 weeks. Lower the transition rate by 50%, and you can’t see the epidemic even starting within a quarter.

Source

There are many companies like Google, Microsoft, Verizon and others that are encouraging social distancing policies. As per this link, 790+ companies are currently encouraging social distancing.

Impact of Social Distance on the US in the Coming Days – Statistical Simulation Estimate

Susceptible:

Individual is able to become infected

Exposed:

Individual has been infected with a virus, but due to the virus incubation period, is not yet infectious

Infectious:

Individual is infected with a virus and is capable of transmitting the virus to others

Recovered:

Individual is either no longer infectious or “removed” from the population

Source

In the SEIR model, the population is classified into one of the compartments mentioned in the above figure. Compartmental models are governed by a system of differential equations that track the population as a function of time, stratifying it into different groups based on risk or infection status.

The independent variable used in the model is time, measured in days. The dependent variable of interest is a fraction of the total population in each of the four compartments. 

The data is simulated using the Euler method i.e. at any given point (t, y), the method will calculate dx/dt. The sequence of x-values like x0, x1, x2, x3 and so on are generated using this method. Starting from a given x0 and computing each rise as slope x run:

xn = xn-1 + slopen-1 Delta_t

where  Delta_t  is a suitably small step size in the time domain.

For the SEIR model, the dependent variables are s, e, I and r. Now, the four Eulers of the form:

SEIR models ordinal differential equations:

Here, N =S+E+I+R. N is a constraint that indicates there are no birth/migration effects in the model; the population is fixed from beginning to end. 

For SIR equations, the final Euler formulas will be:

The following parameters are required to simulate the scenario:

Beta is the inverse of the incubation period (1/incubation days(5.2 days)) 

Alpha is the average contact (infection) rate in the population – 2.2

Gamma is the inverse of the mean infectious period (1/infectious days(2 days))

US Population: 331,002,651 (Mar 22, 2023 Estimate)

Model-Virus Transmission without Social Distancing in the US

Without social distancing, the base model suggests 18% of the US population will be infected with the disease after 40 days from the first exposure, which clearly triggers the warning signal.

Model-Virus Transmission with Social Distance in the US

Adding ρ (encounter rate) to the model to capture the social distancing effect. The value of ρ ranges from 0 to 1, where 0 indicates everyone is locked down and quarantined while 1 is equivalent to our base case.

Considering a scenario of cutting the encounter rates by 50% (through social distancing policies) clearly shows the virus transmission in the above graph has come down to approximately 3%. We can generate different scenarios by modifying all configurable parameters like ρ, incubation period and other things.

Final Thoughts

Coronavirus cases’ exponential growth shows us that we need to strictly follow social distancing measures to protect ourselves and others. Early signs presented in Graph 1 are showing that spread has come to control in China and South Korea. China experienced a period of exponential increases in COVID19 cases but that seems to be leveled out.

China started taking severe restrictive lockdowns and quarantines on its cities starting from Jan 23, 2023. Despite these extreme measures, it took almost about 30 days and an additional 80,000 cases before the curve flattened out. That’s the cost of delaying or not following social distance measures.

About the Author

Bala Gangadhara Thilak Adiboina

I am currently working as a data scientist with a leading US Telecom Company. I am a hardcore data science guy who loves to solve every problem using data science. I am currently pursuing my Ph.D. from IIM Ranchi in the data science space.

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BU’s Shared Computing Cluster: Results Hundreds of Times Faster Helping astronomers study cold atmosphere exoplanets

Paul Dalba used data collected by NASA’s Cassini Spacecraft to develop a model of Saturn’s atmosphere. Photo courtesy of Dalba

To most amateur stargazers, Saturn, with its majestic belt of rings, is the most beautiful planet in our solar system. But to astronomers like Paul Dalba, Saturn is much more than just looks. Because it is a cold giant planet, a large planet that is more distant from the sun than Earth is, Saturn can offer insights into the study of exoplanets, those planets that orbit stars beyond our solar system. Scientists’ big hope for exoplanetary research is that it will lead to the discovery of an Earth-like planet that can support life.

“Although we have not yet discovered an exoplanet with properties exactly matching Saturn’s,” says Dalba (GRS’18), a PhD astronomy student at Boston University’s Graduate School of Arts & Sciences, “a closer examination of Saturn holds important consequences for future studies of giant, cold exoplanets.”

Astronomers have many ways to study these distant worlds. One of them, known as transmission spectroscopy, attempts to analyze their atmospheres by examining starlight that has passed through them. The differences between that light and light that has not passed through an exoplanet’s atmosphere can tell researchers about the density and the chemical makeup of the planet’s atmosphere. By studying Saturn as if it were an exoplanet, Dalba and his team hope to learn if transmission spectroscopy could be applied in the study of cold atmosphere exoplanets.

Working with data collected by NASA’s Cassini Spacecraft, which has been orbiting Saturn and its many moons since 2004, Dalba and his team used ray tracing—a complex process of calculating the properties and direction of a ray of light as it travels from one point to another—to develop a model of Saturn’s atmosphere. Ray tracing works because in empty space, light travels in a straight line, but in an atmosphere, its path is curved by atmospheric refraction. To describe this deviance from a straight line, researchers must solve differential equations for each of the thousands of rays of light passing through Saturn’s atmosphere. Dalba set out to do that by running the algorithms on his MacBook Pro.

“It wasn’t practical,” he says. “It was clear that it was going to take about a month.”

Dalba filled out what he says is a fairly simple application for use of the supercomputer, and he has since logged about 20,000 computer hours on the cluster. Instead of tracing one ray at a time, he found that he could use hundreds of computers to get his results hundreds of times faster.

“Things that would have taken a month can now be done overnight,” he says.

The cluster enabled precise measurements of the degree that light is bent as it passes through Saturn’s atmosphere, indicating the density of the atmosphere and the amount of energy lost as it passes through the atmosphere. The findings of Dalba and his team were published in the December 1, 2024, issue of The Astrophysical Journal.

What did Dalba learn about Saturn’s atmosphere? It’s mainly methane, followed by acetylene, ethane, and other hydrocarbons. More important, he was able to answer the question of whether transmission spectroscopy could be applied in the study of cold atmosphere exoplanets.

The answer is yes.

Types Of The Shareholder With Explanation

Introduction of Shareholder

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Generally, any person becomes a company member by acquiring shares of the same. Now, what does a share mean? A company divides its capital into shares of various denominations. For example, if a company has a capital of Rs. 100,000/-, then it can be divided into 1000 shares of Rs. 100/- each (1000*100=100,000).

Types of Shareholder

There are two major types of shareholders: equity shareholders and preference shareholders. Both of them have their own specific rights and obligations towards the company.

1. Equity Shareholders

Equity Shareholders refer to those shareholders who actively participate in the important decisions of the company and also bear a greater risk as compared to other kinds of shareholders, which entails greater profits in case a company gains and suffers losses if the company does not fare well in business in a particular period.

Equity shareholders enjoy the following benefits:

They get the right to participate in the major decisions taken by the company through voting.

Preference shareholders have their liability limited to the value of the shares they hold.

They possess the entitlement to receive dividends and a share of the profits generated by the company in which they have invested.

2. Preference Shareholders

Preference shareholders have limited decision-making powers in the company, but they hold a preferential right over the profits earned by the company, surpassing that of equity shareholders.

They get preference at the time of payment of dividends.

During the liquidation of a company, the dues of preference shareholders are given priority over those of other shareholders.

In the case of cumulative preference shares, if a company fails to pay dividends in a specific year, the unpaid dividends accumulate and are carried forward to subsequent years.

The sub-types of Preference shareholders are mentioned below:

Convertible and Non-Convertible Preference Shareholders: Convertible Preference shareholders have an option/right to convert their shares into equity shares after a certain period of fulfilling certain terms and conditions. However, Non-Convertible shareholders don’t possess any such rights.

Redeemable and Irredeemable Preference Shareholders: A company must pay back the capital in case of Redeemable Preference shares, which consequently results in the discontinuation of payment of dividends(preferential right on such shares). In the case of Irredeemable Preference shares, the company pays dividend till it continues to exist and does not pay back the capital to such shareholders.

Cumulative and Non-Cumulative Preference Shareholders: In a case where a company is not able to pay dividends to preference shareholders in case of lack of funds, it gets accumulated for the next financial year in case of Cumulative Preference Shareholders. On the contrary, Non-Cumulative Preference shareholders lose their right to receive dividends if a company fails to pay the same in a particular financial year.

Participating and Non-Participating Preference Shareholders: Participating preference shareholders possess an additional right to participate in the decisions of the management of the company. In contrast, non-participating preference shareholders do not possess any such rights.

From the above discussion, we can conclude that there are two types of shareholders: Equity and Preference. Acquiring any of them has its own pros and cons. Equity shareholders get a right to participate in the decisions of the company and have the power to govern the business and eventually change its course of it. The lop side of it is that they have to bear a higher risk and can also go on without even earning a penny if the company suffers losses in a particular financial year. On the other side, Preference shareholders play safe. They don’t get the right to participate in important business decisions. However, they get a definite percentage of profits no matter how the company fared in terms of profits and growth in a particular financial year.

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This is a guide to Shareholder Types. Here we also discuss the introduction and types of the shareholder, which include equity and preference shareholders. You may also have a look at the following articles to learn more –

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