AI, NLG, and Machine Learning

The Role of Gaussian Distribution in Cascading Effect of Lasers and the Current Pandemic COVID-19

The usage of machine learning and artificial intelligence (AI) is ubiquitous in the digital world and paved the way to realization of smart products, smart medicine, and diagnosis of diseases.

By Dr. Deepak Kallepalli
August 14, 2020

This article was co-authored by Dr. Anil Kumar Munipalli.

Introduction

The connection between artificial intelligence at macro level and the Gaussian distribution of the features or training parameters at grassroots level is ubiquitous. To start with, most of the natural and physical processes are random (stochastic) in nature and are studied using random variables. The behavior or trend associated with such processes follows a Gaussian distribution that arises from the popularly known “central limit theorem and the law of large numbers.” [1] In AI-related problems, for example, voice recognition in speech analysis, each training parameter, when estimated over a large dataset, follows a Gaussian distribution. The sum and product of many such training parameters influencing the net result also becomes a Gaussian distribution, which is characterized by its mean (µ) and variance (σ). The understanding of these two measures are key in AI-related problems.

In this article, we present two cascading effects: (i) a physical process leading to generation of an intense laser, which we call material pandemic, and (ii) a biological process, which is the spread of current pandemic COVID-19. Subsequently, we emphasize the connection between these processes to highlight the role of Gaussian distribution in each case and demonstrate it through a simulation. We conclude how these simulations help us in understanding the role of social distancing.

Understanding of material and COVID-19 outbreak

Albert Einstein first imagined and proposed the material pandemic process in the form of stimulated emission in 1917 that led to the invention of lasers in the 1960s. His model consists of a closed atomic system, in which an atom can be excited with a photon whose energy matches the energy difference between the ground state (E1) and an excited state (E2) of the atomic system. The excited atom can lose its energy by either (i) emitting a photon, known as spontaneous emission, or (ii) interacting with other photons, leading to a cascading effect, known as stimulated emission.

Epidemics and pandemics also spread in a cascading manner like the laser. In both the cases, the extent of contact between the system of interest (people or material) with foreign objects (virus or photons) plays a key role in the outburst. In this article, we draw similarities between the laser system and the current pandemic. We also provide details on how we can use machine learning techniques (supervised) to understand the details of pandemic and the role of social distancing.

Similarities

First, let us compare the mathematics governing the equations for both the laser system (equations L1-L3) and the epidemic SIR model (susceptible, infectives, and removed/recovered) (equations P1-P3) (references 1-5). From Eq. (L1), there are N1 atoms in the initial normal state of Energy (E1) that interact with photon flux (ρ) and hence, the population (N1) declines as (N1ρ). Eq. (L2) and Eq. (L3) depict the possible stimulated (unusual) and spontaneous (normal) emission channels through which the excited atoms (N2) excrete their energy in the form of photons before transitioning to the normal state E1.

The lasing action or outburst depends on (i) the balance between the coefficients (absorption (B12), spontaneous emission (A21), and stimulated emission (B21)), which in turn depend on the properties of material, and (ii) the number of photons interacting with the material under the condition N2 > N1 (gain medium), in which the stimulated emission process generates a multiplicative number of photons in each step. Hence, this process, “Light Amplification by Stimulated Emission of Radiation,” is popularly known as LASER.

In the same way, the spread of current pandemic COVID-19 can be explained through Eq. (P1) – Eq. (P3), that resemble laser equations Eq. (L1) – Eq. (L3). Here, the total population (in a city or a country) is divided into susceptible, infective, and recovered/removed. The dynamics between these populations is explained through SIR model (reference 5). The variation of susceptible population (dS/dt) can be written as multiplication of the susceptible (S) and infective (I) populations. This can be understood in terms of susceptible population meeting infectives, like the stimulated emission process of the atomic system in contact with the number of photons. It is noteworthy to mention here that the generation of an intense laser (spatial intensity profile) and spread of COVID-19 (infective population shown in orange curves in Figure 1 and Figure 2) follow the Gaussian distribution trend.

The factors that mitigate the performance of the lasers, referred to as losses, arise from misalignment of cavity mirrors and processes such as spontaneous emission and absorption. This can be understood as social distancing, quality of health services, and immunity levels of population in the context of pandemic. In addition, both these systems are closed; the number of atoms in the laser system and the population in epidemic systems (no births and deaths) are fixed. There are losses in both these systems. In the first case, the loss due to spontaneous emission and recovered populations are accounted for in the equations Eq. (L2) and Eq. (P2).

Simulation

Here, we show a diagram using the SIR model. We consider a population of one million (susceptible) population with one infected. Figures 1 and 2 show trends for contact ratios (β) of 0.4 and 0.6 with a recovery rate (γ) of 0.1 (reference 6). From these plots, we observe that the increase in contact ratio (i) increases infectives (Figure 2) and (ii) early rise in infective population (20 days for β = 0.6 compared to 40 days for β = 0.4).

In both the cases, we see the infective curves (orange) reaching maxima (mean of Gaussian) at half of the susceptible population (blue), which is referred to as population inversion in lasers (threshold condition). The epidemic outburst is inevitable once this threshold condition is arrived. The only way to prevent reaching the threshold condition is to either (i) increase losses in the system (the second term γI in the Eq. (P2)) and/or (ii) reduce the contact ratio (γ) (in the Eq. (P1)). The latter is popularly known as social distancing. By reducing the contact ratio, we are reducing the mean (µ) of Gaussian and increasing the variance (σ). This is nothing but “flattening” the curve or distribution. Thus, in this article, we highlighted the importance of Gaussian distribution and understanding of their measures in several natural and physical processes.

The pandemic problem comes under the supervised learning model, as the datasets over a long period are known. From these datasets, using the gradient descent algorithm, we can obtain a hypothesis function by finding out the parameters of β and γ. To obtain correct values, we need to consider large datasets, as these give precise values (reference 7) and show Gaussian distribution trend. The sum and the product of many influencing training parameters once again become Gaussian. Thus, in this article, we demonstrated that by monitoring β (contact ratio) over time and efforts in reducing it flatten the curve; it is the only solution to end the current pandemic.

Conclusions

In the current context of COVID-19, we attempted to demonstrate the common connection between the spread of pandemic with a similar known process, i.e., the generation of an intense laser. “Prevention is better than cure,” as quoted by many medical professionals. The only way to prevent is to follow social distancing that leads to flattening the curve, as seen in simulations.

It is the power of the digital world that brings laser physicists, data scientists, and medical professionals together to understand and study the problem with their expertise and come up with solutions to prevent the pandemic. The beauty of AI lies in understanding and studying the behavioral patterns (Gaussian) of these training parameters. For example, recent studies on mortality rate due to COVID -19 for various countries, like China, USA, Italy, and UK, follow the same Gaussian distribution (reference 8).

Thanks to the artificial intelligence technology that has the power in not only understanding the spread but also making smart products that can prevent it. For example, (i) a voice bot connected to Internet of Things (IoT) guiding people to maintain social distancing, (ii) activating UV disinfectant robots to decontaminate offices and quarantine centers, (iii) treating pandemic patients through contactless healthcare bots, and (iv) detection and prevention through smart sensors and medicine.

References

1) https://discover.bot/bot-talk/law-of-large-numbers-ai/
2) Siegman, A. E., Lasers, University Science Books, Mill Valley, California (1986).
3) Yariv, Amnon, Quantum Electronics (3rd ed.), Wiley, (1989). ISBN 0-4716-0997-8.
4) O. Svelto, Principles of Lasers, Plenum Press, New York (1998).
5) https://tomrocksmaths.com/2020/03/18/oxford-mathematician-explains-sir-disease-model-for-covid-19-coronavirus/
6) Kermack, W. O., and A. G. McKendrick, “A contribution to the mathematical theory of epidemics.” Proc. R. Soc. London, Ser. A 115, 700 (1927).
7) https://github.com/klndeepak/SIR-model/blob/master/SIR%20model.ipynb
8) Lee, T-W., J. E. Park, and David Hung, “Gaussian Statistics and Data-Assimilated Model of Mortality due to COVID-19: China, USA, Italy, Spain, UK, Iran, and the World Total.” medRxiv (2020). (https://covid-19.conacyt.mx/jspui/bitstream/1000/3657/1/1104786.pdf)

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