# A Mathematical Model for the COVID-19 Pandemic in Tokyo through Changing Point Calculus
## 1 Introduction  
On 11 February 2020, the International Committee on Taxonomy of Viruses (ICTV) named the virus that in December 2019 had produced an outbreak of pneumonia in Wuhan, China, as _severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)_. The same day, the World Health Organization (WHO) named the disease caused by this new coronavirus as _COVID-19_. On 11 March 2020, the WHO declared the outbreak of COVID-19 a pandemic. According to WHO data [1], since the first case of COVID-19 was detected in December 2019, until March 2023, the date on which we wrote this article, more than 750 million cases of COVID-19 have been confirmed worldwide, and more than 6.8 million people have died as a result of this disease. The COVID-19 pandemic has had a high impact on economics, social relations, and physical and mental health all over the world [2]. This has prompted researchers from different countries to develop mathematical models that allow them to determine which measures are most appropriate to deal with the pandemic. When we develop a mathematical model, the objective is to ensure that the model represents real data with high fidelity so that the parameters obtained from the model can be used to draw conclusions from the real world. The main technical issue in the construction of such mathematical model is that there is no primitive function that describes the behavior of the COVID-19 pandemic data; so, the development of a more complex mathematical model is necessary. The relative simplicity of the SIR model makes it suitable for rapidly assessing an epidemic when data on virus parameters are scarce. However, as pointed out in [3], the applicationof the classical SIR model to the COVID-19 pandemic has given poor results. Several characteristics of the SARS-CoV-2 virus can explain this fact, including a relatively high latency period, the existence of asymptomatic individuals, and the possibility of reinfection due to loss of acquired immunity. Thus, to overcome the limitations that the SIR model presents when modeling the characteristics of this new virus, several SIR-based models have been proposed, some of which we discuss below. We must keep in mind that better mathematical models provide us with tools to understand the evolution of the COVID-19 pandemic and, therefore, enable us to better confront future pandemics caused by viruses that have characteristics similar to the SARS-CoV-2 virus. One of the strategies used in the development of new models, the one we have followed, is to expand the number of compartments of the SIR model. Following this approach, authors in [4] model the COVID-19 pandemic with the SEIRD model (Susceptible, Exposed, Infective, Recovered, and Deceased), which expands the SIR model with two new compartments, Exposed and Deceased. Although with the Exposed compartment the authors take into account the latency period of the SARS-CoV-2 virus, they do not consider the asymptomatic individuals or the loss of immunity. Similarly, in [5], the researchers propose a seven compartmental model--Susceptible, Exposed, Infected, Asymptomatic, Quarantined, Fatal, and Recovered (SEIAQFR)--which takes into account both the latency period and the existence of asymptomatic individuals. However, it does not consider the loss of immunity and, therefore, reinfections. Following a different approach, the authors in [6] propose three time-dependent modified SIR models (SIR-T, SIRD-T, and SIAR-T), and in [7], researchers propose an age-structured SIR model. A study that, like ours, models the pandemic in Tokyo uses a combination of SIR models to fit the data [3]. They obtain effective contact rate values similar to those we obtain when applying our model. However, the characteristics of its procedure make it unfeasible as a tool for evaluating Non-Pharmaceutical Interventions (NPIs). To model the spread of COVID-19, other alternatives different from SIR models have also been explored, such as the use of Cellular Automaton (CA) [8; 9].