New Compartmental Models for COVID-19

Three years after the initial outbreak in China in 2019, COVID-19 continues to spread worldwide, except for Africa.^1,2A number of case studies have been reported over the last three years. According to reported data, COVID-19 exhibits the following unusual characteristics:

1. 1.

   Presymptomatic patients are infectious³.

2. 2.

   The majority of patients are asymptomatic and infectious^{Four,Five,6,7,8,9,Ten,11}.

3. 3.

   Patients with infectious diseases can be detected with a polymerase chain reaction (PCR) test¹².

4. Four.

   Patients with symptoms are to be isolated in hospital or at home¹³.

5. Five.

   Significant proportion of patients lost to follow-up¹³.

Therefore, it is important to distinguish between symptomatic and asymptomatic patients.

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

In the standard SIR (Susceptible-Infected-Removed) model^17,18,19presymptomatic patients are assumed to be non-infectious, and isolated patients are included in the infectious compartment, so they can become infected without coming into contact with susceptible people.

The modified compartmental model is the SEIR (E stands for ‘exposure’) model, which treats presymptomatically exposed patients as compartments and incorporates the incubation period for infected patients.^{20,twenty one,twenty two}There were models that treated recovered and deceased patients as different compartments^{twenty three}.

SIR models cannot provide information about the relationship between the number of daily new infections and the number of infected patients. To overcome this shortcoming of his SIR type model, the SIQR model was introduced.^{twenty four,twenty five,26,27} An isolated patient is considered a compartment.The peak position of the number of isolated patients appeared later than the peak position of the number of infected patients, indicating that the total number of infected people can be estimated from the number of new infections per day.²⁷.

However, because asymptomatic patients are not treated as a compartment, the SIQR model cannot provide information on the number of lost-to-trace patients presumed to have been infected from asymptomatic patients. Based on analysis of the transmission process of COVID-19, it was shown that the incidence can be estimated from the proportion of patients lost to follow-up²⁸.

A new compartment consisting of susceptible individuals (S), presymptomatic patients (P) to obtain the relationship between daily confirmed new cases, incidence rate, percentage of patients lost to follow-up, and total number of infected cases. propose a model. Asymptomatic patients (A), isolated patients (Q), and excluded individuals (R) including recovered and deceased patients. More precisely, the P and A compartments should be considered pre-quarantine and commonly infected patients, respectively. This model is a modification of the SIQR model in which the I compartment of the latter is separated into the P and A compartments of the former. I present the time evolution of the epidemic and argue that the proportion of untraceable cases and the number of infected people may be related to the number of new confirmed cases each day.

First, basic ordinary differential equations are introduced to relate various quantities to observables. Numerical solutions to a set of ordinary differential equations are then presented, showing the time dependence of the proportion of patients lost to follow-up and the ratio of infected patients to new confirmed cases daily. We will discuss the results at the end.