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The fields of epidemiological disease modeling and economics have tended to work independently of each other despite their common reliance on the language of mathematics and exploration of similar questions related to human behavior and infectious disease. This paper explores the benefits of incorporating simple economic principles of individual behavior and resource optimization into epidemiological models, reviews related research, and indicates how future cross-discipline collaborations can generate more accurate models of disease and its control to guide policy makers.

n this age of molecular biology, The healthcare industry, politicians and the community at large are trying to find 'magic bullet' drugs and vaccines to conquer disease. Although smallpox has been eradicated and polio may soon be a scourge of the past, many pathogens replicate rapidly and mutate prodigiously, enabling them to evolve ways to circumvent our immune systems, as well as our drugs and vaccines. To fight and win the war against new emerging infections such as HIV/AIDS, TB and now SARS (severe acute respiratory syndrome), it is important to understand the temporal and spatial dynamics of the pathogens in human and, in some cases, animal reservoirs or vector populations. It is also necessary to understand the complex web of socio-economic factors pertinent to controlling the spread of disease, so that feasible, affordable and, most importantly, effective public-health policies can be devised and implemented.

For more than a year, the COVID-19 pandemic has been a major public health issue, affecting the lives of most people around the world. With both people's health and the economy at great risks, governments rushed to control the spread of the virus. Containment measures were heavily enforced worldwide until a vaccine was developed and distributed. Although researchers today know more about the characteristics of the virus, a lot of work still needs to be done in order to completely remove the disease from the population. However, this is true for most of the infectious diseases in existence, including Influenza, Dengue fever, Ebola, Malaria, and Zika virus. Understanding the transmission process of a disease is usually acquired through biological and chemical studies. In addition, mathematical models and computational simulations offer different approaches to predict the number of infectious cases and identify the transmission patterns of a disease. Information obtained helps provide effective vaccination interventions, quarantine and isolation strategies, and treatment plans to reduce disease transmissions and prevent potential outbreaks. The focus of this paper is to investigate the spread of COVID-19 and its effect on a population through mathematical models. Specifically, we use SEIR and SEIR with vaccine models to formulate the spread of COVID-19, where S, E, I, R, and V are susceptible, exposed, infected, recovered, and vaccinated compartments, respectively. With these two models we calculate a central quantity in epidemiology called the basic reproduction number, R0. This helps examine the dynamical behavior of the models and how vaccines can help prevent the spread of the virus.

Chronic obstructive pulmonary disease (COPD), the fourth leading cause of death worldwide, has a puzzling etiology. Although it is a smoking‐associated disease, only a minority of smokers develop it. Moreover, the disease continues to progress in COPD patients, even after smoking ceases. This article proposes a mathematical model of COPD that offers one possible explanation for both observations. Building on a conceptual model of COPD causation as resulting from protease‐antiprotease imbalance in the lung, leading to ongoing proteolysis (digestion) of lung tissue by excess proteases, we formulate a system of seven ordinary differential equations (ODEs) with 18 parameters to describe the network of interacting homeostatic processes regulating the levels of key proteases (macrophage elastase (MMP‐12) and neutrophil elastase (NE)) and antiproteases (alpha‐1‐antitrypsin and tissue inhibitor of metalloproteinase‐1). We show that this system can be simplified to a single quadratic equation with only two parameters to predict the equilibrium behavior of the entire network. The model predicts two possible equilibrium behaviors: a unique stable "normal" (healthy) equilibrium or a "COPD" equilibrium with elevated levels of MMP‐12 and NE (and of lung macrophages and neutrophils) and reduced levels of antiproteases. The COPD equilibrium is induced in the model only if cigarette smoking increases the average production of MMP‐12 per alveolar macrophage above a certain threshold. Following smoking cessation, the predicted COPD equilibrium levels of MMP‐12 and other disease markers decline, but do not return to their original (presmoking) levels. These and other predictions of the model are consistent with limited available human data.

In this study, we examined various forms of mathematical models that are relevant for the containment, risk analysis and features of COVID-19. Greater emphasis was laid on the extension of the Susceptible-Infectious-Recovered (SIR) models for policy relevance in the time of COVID-19. These mathematical models play a significant role in the understanding of COVID-19 transmission mechanisms, structures and features. Considering that the disease has spread sporadically around the world, causing large scale socioeconomic disruption unwitnessed in contemporary ages since World War II, researchers, stakeholders, government and the society at large are actively engaged in finding ways to reduce the rate of infection until a cure or vaccination procedure is established. We advanced argument for the various forms of the mathematical models of epidemics and highlighted their relevance in the containment of COVID-19 at the present time. Mathematical models address the need for understanding the transmission dynamics and other significant factors of the disease that would aid policymakers to make accurate decisions and reduce the rate of transmission of the disease.

Front Cover -- A Historical Introduction to Mathematical Modeling of Infectious Diseases -- Copyright -- Dedication -- Contents -- Introduction -- Motivation and short history (of this book) -- Structure and suggested use of the book -- Target audience -- Mathematical background -- Miscellaneous remarks -- References -- Acknowledgments -- 1 D. Bernoulli: A pioneer of epidemiologic modeling (1760) -- 1.1 Bernoulli and the "speckled monster -- 1.1.1 1 through 4: Preamble -- 1.1.2 5 through 6: Mathematical foundation -- 1.1.3 7 through 9: Table 1 -- 1.1.4 11 & -- 12: Table 2 -- 1.1.5 13: Closed form solution for the counterfactual survivors -- Appendix 1.A Answers -- Appendix 1.B Supplementary material -- References -- 2 P.D. En'ko: An early transmission model (1889) -- 2.1 Introduction -- 2.2 Assumptions -- 2.3 The model -- 2.4 Simulation model -- 2.4.1 Start of the simulation -- 2.4.2 Discussion of Table 1 and Figures -- 2.4.3 An important detail: The period -- Appendix 2.A Answers -- Appendix 2.B Supplementary material -- References -- 3 W.H. Hamer (1906) and H. Soper (1929): Why diseases come and go -- 3.1 Introduction -- 3.2 Hamer: Variability and persistence -- 3.2.1 A tortuous introduction -- 3.2.2 Characteristic of periodic measles epidemics -- 3.2.3 The case of influenza -- 3.3 Soper: Periodicity in disease prevalence -- Regeneration" of the population -- Law of infection -- Mass action -- 3.3.1 Infection dynamics -- 3.3.2 The simulated epidemic -- 3.3.3 Periods -- 3.3.4 Considerations of seasonal factors and model fit to Glasgow data -- Appendix 3.A -- The discussion -- Appendix 3.B Answers -- Appendix 3.C Supplementary material -- References -- 4 W.O. Kermack and A.G. McKendrick: A seminal contribution to the mathematical theory of epidemics (1927) -- 4.1 Introduction -- 4.2 General theory: (2) through (7).