**1. Introduction**

Mathematical modeling of infectious diseases attracts much attention due to successive epidemics, such as HIV, emerging in the 1980s and still continuing [1,2], SARS epidemic in 2002–2003 [3,4], H5N1 influenza in 2005 [5,6] and H1N1 in 2009 [7,8], Ebola in 2014 [9,10]. The ongoing COVID-19 pandemic has stimulated unprecedented efforts of mathematical modeling in epidemiology. A wide variety of mathematical approaches are developed to study epidemiological problems. However, sufficiently simple and validated models still remain in the focus of mathematical modeling in epidemiology.

Modern studies in mathematical epidemiology begin with the SIR model developed in the works by W. O. Kermack and A. G. McKendrick [11–13], stimulated by the Spanish flu epidemic in 1918–1919. Among many developments of such models, we can cite multi-compartment models [14–16], models with a time-varying or nonlinear disease transmission rate [17,18], multi-patch models [19–21], multi-group models incorporating the effect of the heterogeneity of the population [22], and epidemic models with vaccination and other control measures [23,24]. Random movement of individuals in the population is considered in spatiotemporal models in order to describe spatial distributions of susceptible and infected individuals [25,26]. A more detailed literature review can be found in the monographs [27,28] and review articles [29,30].

The conventional SIR model, which includes susceptible (*S*), infected (*I*), and recovered (*R*) compartments, and similar models assume that recovery and death rates at time *t* are proportional to the number of actively infected individuals *I*(*t*) at the same moment of

**Citation:** Ghosh, S.; Volpert, V.; Banerjee, M. An Epidemic Model with Time Delay Determined by the Disease Duration. *Mathematics* **2022**, *10*, 2561. https://doi.org/10.3390/ math10152561

Academic Editor: Alexandra Kashchenko

Received: 9 June 2022 Accepted: 15 July 2022 Published: 22 July 2022

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time. This assumption does not take into account disease duration, and it can lead to a large error. In our previous work [31], we showed that this assumption leads to an overestimation of actual recoveries and deaths. Instead, if we use distributed recovery and death rates, properly chosen from real data, the description of the epidemic progression becomes more precise. However, since distributed recovery and death rates are not easily available, we develop a simpler delay model in this work. It gives close results, but it does not require precise immunological data. The model considered in [31] involves distributed recovery and death rates. The model considered in [32] is an extension of the model by incorporating the vaccinated compartment, and the resulting model is an immuno-epidemic model. The delay model, with disease duration delay, considered here is derived from the model proposed in [31] with an appropriate assumption on the recovery and death rate functions. The present model is quite different from the delay model considered in [33] as the delay parameter involved with the earlier model was the measure of the incubation period and the departure of infected individuals from the infected compartment was due to the imposition of quarantine measure. The model proposed and analyzed here is solely dependent on the disease duration period and without imposed quarantine.

Most of the existing delay epidemic models consider time delay either in the disease incidence function or in the susceptible recruitment function (Appendix A). The delay in the recovery and death rates has not been studied yet thoroughly. In this work, we introduce time delay in recovery and death rates with the average disease duration considered as the delay parameter.

In Section 2, we discuss the distributed model and derive the delay model where the discrete time delay estimates average disease duration. We obtain epidemic characterization of the delay model in Section 3. Then, in Section 4, we perform a numerical comparison among the distributed model, delay model, and conventional SIR model with the equivalent parameter values. Next, we discuss a method to estimate the value of disease duration using the real data of disease incidence in Section 5. In Section 6, we validate our delay model with epidemiological data collected during the COVID-19 epidemic. The main outcomes of the proposed model and its epidemiological implications are discussed in the concluding section.
