# A Study on Mathematical and Computational Models in the Context of COVID-19

**Dr. Meera Joshi**

## Introduction

The outburst of the COVID-19 pandemic was reported for the first time in December 2019 in the city of Wuhan in China. The COVID-19 crisis, with no cure for the virus, has affected the whole world. Every facet of human life has changed in a shorter span of time than one could imagine. Challenges are hurled at us every day in aspects of the economy, health, and social sectors of society. There is a critical need to control the spread of the pandemic by taking various measures and decisions. In a severe pandemic such as COVID-19, to make a proper longterm economic and medical plan, it is very important to understand the magnitude and propagation of the disease. It is imperative to develop a scientific process which presents the progression of the pandemic and identifies the spread pattern, in order to make decisions related to corrective measures and to distribute resources in an optimal manner. The only procedure which is very economical, apart from being highly scientific in nature, is constructing a mathematical model consisting of equations which is analogous to reality. Such a model provides an indispensable framework based on facts and figures, leading to a concrete and practical solution for any type of situation, even one such as the present pandemic scenario of COVID-19. Mathematical models are created in order to meet the abstract, precise, and significant needs of a complex problem. The goal of mathematical modeling is to deduce the structural and functional properties related to understanding the issues of handling difficult real-life situations. The core activity involved in building this type of model is making a set of basic assumptions about the process by identifying the vital features which impact the phenomenon, and without causing any modifications to the real situation, so that the actual essence of it is preserved. The construction of the model must be refined and vigorous to achieve an evocative conclusion. Though models have limitations, we can rely upon them, as the predictions that emerge from solving a mathematical model are immensely accurate in terms of precision to real situations.

### Classification of Mathematical Models

A mathematical model is a brief comprehension of real situations in life. While studying mathematical models, it is essential to know the classification of the various types of models. Classification of models is required to decide some of the characteristics of their structure. Depending on the type of prediction of outcome, mathematical models are classified as deterministic models, and non-deterministic or stochastic models. In deterministic models, randomness is not considered in the preparation of the mathematical formulation of future circumstances, and the model will generate the same results from given initial conditions every time. A non-deter- ministic or stochastic model has an inbuilt randomness to it and different outputs can be acquired for the same problem (France and Thornley, 1984). In this chapter, some classical and important types of mathematical frameworks in the context of COVID-19 are studied.

### Features of Mathematical Models

The objectives for construction of a mathematical model must be defined clearly and a complete understanding of the system to be modeled is a prerequisite for the creation of the framework describing the model. The framework of the model is a system of equations which govern the system. These equations are solved to obtain meaningful inferences in terms of providing a solution to the real-life problem. The four important phases of mathematical models are construction, studying, validating, and implementation of the models.

During construction of a mathematical model, the analysis of the system is very crucial. It is the fundamental requirement to examine the classification of the model, in the sense of whether it is deterministic or stochastic, to acquire the desired results. It is essential to analyze the interactions in the system logically for meaningful application of models. If it is not possible to establish relationships from logical analysis, the data has to be acquired in order to fit equations to it.

Further, the obtained equations need to be studied to infer results in the perspective of the actual problem. A model can be studied in two ways: as a quantitative model or as a qualitative model. In the case of COVID-19 both perspectives are important, as the quantitative model helps in attaining clarity on the magnitude of the pandemicâ€™s impact in terms of numbers, and the qualitative model helps to answer how and why the disease is spreading. Quantitative behavior is applicable to a case-specific model estimating spread in an area and qualitative behavior is generally common for models like the cause for spread. Therefore, one has to construct a model which can handle both quantitative and qualitative features with the maximum possible accuracy. To confirm the efficiency of the model, it is required to estimate the outcome of the model depending on the variation of parameters in the model, i.e. by performing sensitivity analysis. The solution to equations can be found by analytical method or by numerical method. Numerical methods are applied when analytical methods cannot be used. In the case of numerical methods, computer time has to be monitored, and we need to ensure that the models w'ork by consuming minimum computer time with a negligible compromise on precision.

After studying and ensuring the performance of the model, its validation has to be approved by verifying it with experimental data. During verification of the model, the focus has to be on checking whether the assumptions made while moving from a verbal to a mathematical model are correct. If any assumption is not required, it has to be discarded. The structure of the model must be confirmed and, if necessary, it must be changed to make the model effective. The model must be verified for predicting the results for some data which was not used in estimating the model parameters. The efficiency of the model can be tested based on many aspects but most importantly on the ability of prediction and the requirements for computing.

The final stage of the implementation of the model is based on how accurately it is functioning both qualitatively and quantitatively in comparison to other models. A model can be used depending upon the needs of the user, whether it delivers the task assigned, and its effectiveness in giving the results which the user is interested in as a meaningful and genuine output.

Now we shall proceed further with the study of specific models for COVID-19 and understand the various features of those models. In Section 10.3, a study of various mathematical models for COVID-19 is discussed. In Section 10.3.1, an overview of the basic Susceptible-Infected-Recovered (SIR) model is presented, which is followed by the extensions of the SIR model. To make the study meaningful, in Section 10.3.1, the SIR model is extended by considering aspects like birth rate and death rate, which are important from the point of view of real life. The speed of spread of COVID-19 is alarming and the devastation it is causing to humankind is increasing every day. The most important source of spread of COVID-19 is close interaction between the infected and the susceptible. The only way to control such a spread is by vaccination. A high level of protection can be provided by vaccinating the maximum percentage of population so that persons with weak immunity also get protected. In treating diseases caused by viruses, antibiotics are mostly used. Overusage of antibiotics leads to drug resistance of the virus causing the infection. Vaccines help to control the development of drug resistance; hence it is necessary to study the SIR model with the impact of vaccination, which is discussed in Section 10.3.2. The number of re-infection cases of COVID-19 is very low but the risk of spreading the disease increases if the re-infected person has no symptoms, which is usual in the case of COVID-19. In this context, it is important to study a mathematical model that addresses the impact of vaccines along with re-infection rate, as offered in Section 10.3.3. An imperative fact which one must focus on is that the exposure of a person to an infected person determines the intensity of COVID-19; therefore it is necessary to study a model which takes into account the fraction of the population exposed to an infected person, as offered in the Susceptible-Exposed- Infected-Recovered (SEIR) model in Section 10.4. The outbreak of COVID-19 can be minimized by implementing measures like quarantine. A model is studied in Section 10.5 which takes into account the fraction of quarantined and unquarantined population. A deeper and more useful study is covered in Section 10.6 in the form of a modified SEIR model, by dividing the infected population into two subgroups of isolated (in hospital) and non-isolated cases, along with the recovery rate and death rate for both groups. The most dangerous threat in the spread of COVID-19 is the asymptomatic population. The Susceptible-Exposed-Infected-Asymptomatic-Rec overed (SEIAR) model in Section 10.7 studies the impact of asymptomatic carriers in the spread of COVID-19. An extension of the SEIAR model is presented in Section 10.8, which accounts for hospitalization and death rates. It is highly important to understand the relationship between the rate of spreading of COVID-19 and the time to provide timely treatment to infected persons to control the disease. The model described in Section 10.9 derives a relationship between the rate of spreading and time. There is a need to explore the multiple ways in which the spread of COVID-19 happens, and one such factor is pathogen concentration in the environment which contributes to increase in the spread of COVID-19. The model in Section 10.10 studies the spread of COVID-19 by focusing on the indirect transmission of the disease from environment to humans. This study is crucial, keeping in mind that some studies are being conducted presently with a perspective of whether COVID-19 is airborne or not. Towards the end of the chapter, in Section 10.11, a brief account of the challenges in modeling and forecasting the spread of COVID-19 is presented, so that more accurate, effective, and realistic models can be formulated in future.