New compartment model for COVID-19

COVID-19 is still rampant across the world, except for Africa, three years after the first outbreak in China in 2019^1,2. Many case studies have been reported over these three years. According to reported data, COVID-19 has the following unusual characteristics:

1. 1.

   Pre-symptomatic patients are infectious³.

2. 2.

   A significant portion of patients are asymptomatic and contagious^{4,5,6,7,8,9,ten,11}.

3. 3.

   Infectious patients can be detected by the polymerase chain reaction (PCR) test¹².

4. 4.

   Symptomatic patients are supposed to be quarantined in a hospital or at home¹³.

5. 5.

   A significant fraction of patients cannot be found¹³.

It is therefore important to distinguish symptomatic patients from non-symptomatic patients.

Many theoretical models have been proposed to understand the spread of COVID-19 infection^14,15,16.

In the standard SIR (Susceptible-Infected-Removed) model^17,18,19pre-symptomatic patients are assumed to be non-infectious, and quarantined patients are included in the infected and therefore infectious compartment although they do not come into contact with susceptible people.

A modified compartment model is the SEIR model (E stands for “exposed”.) in which patients exposed before showing symptoms are treated as a compartment to incorporate the incubation period of infected patients^20,21,22. There is a model that treats recovered and deceased patients as different compartments²³.

The SIR-like model cannot provide any information about the relationship between daily confirmed new cases and the number of infectious patients. In order to overcome this deficiency of the SIR type model, the SIQR model was introduced.^24,25,26,27 in which quarantined patients are considered a compartment. It is shown that the peak position of the number of quarantined patients appears later than the peak position of infected patients, and the total number of infected individuals can be estimated from the number of daily confirmed new cases²⁷.

The SIQR model, however, cannot provide any information about the number of untraceable patients who are assumed to be infected by asymptomatic patients, since asymptomatic patients are not treated as a compartment. Based on an analysis of the infection process of COVID-19, it was shown that the occurrence rate can be inferred from the fraction of untraceable patients²⁸.

In order to obtain relationships between daily confirmed new cases, occurrence rate, fraction of untraceable patients and total number of infectious people, I propose a new compartment model composed of susceptible individuals (S), patients pre-symptomatic (P), asymptomatic patients (A), quarantined patients (Q) and withdrawn persons (R) which includes recovered and deceased patients. To be more precise, the P and A compartments should be considered as pre-quarantine patients and general infected patients, respectively. This model is a modified version of the SIQR model where compartment I in the latter is separated into compartments P and A in the former. I present the temporal evolution of the epidemic and argue that the fraction of untraceable patients and the number of infectious individuals can be related to the number of new cases confirmed daily.

First, basic ordinary differential equations are introduced and various quantities are related to observables. Next, the numerical solution to the set of ordinary differential equations is presented and the time dependence of the fraction of untraceable patients and the ratio of the number of infectious patients to daily new confirmed cases are given. The results are discussed at the end.