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A General Approach To Modeling Covid-19

Isea Raul ¹

1 Fundacion IDEA, Hoyo de la Puerta, Baruta, Venezuela.

Abstract

The present work shows that it is possible to analytically solve a general model to explain the transmission dynamics of SARS-CoV-2. First, the within-host model is described, and later a between-host model, where the coupling between them is the viral load of SARS-CoV-2. The within-host model describes the equations involved in the life cycle of SARS-CoV-2, and also the immune response; while that the between-Host model analyzes the dynamics of virus spread from the original source of contagion associated with bats, subsequently transmitted to a host, and then reaching the reservoir (Huanan Seafood Wholesale Market in Wuhan ), until finally infecting the human population.

Author Contributions

Received Apr 10, 2023 Accepted May 10, 2023 Published Jul 07, 2023

Copyright© 2023 Isea Raul.
License
This work is licensed under a Creative Commons Attribution 4.0 International License. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Competing interests

The authors have declared that no competing interests exist.

Funding Interests:

Citation:

Isea Raul (2023) A General Approach To Modeling Covid-19 Journal of Model Based Research. - 2(2):1-19

DOI 10.14302/issn.2643-2811.jmbr-23-4556

Introduction

The World Health Organization (WHO) reported 27 cases with a new severe respiratory syndrome of unknown etiology from Wuhan (Hubei province) in China 1. Later identifies that it is a β-Coronavirus after sequencing the first days of January 2020 2. They originally called it 2019-nCOV, and after a month, it was renamed Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), where the disease it produces is Covid-19. The WHO declared it an international public health emergency on January 30, 2020, and it was later declared a pandemic on March 11, 2020 3.

So far the world has faced three coronavirus associated outbreaks. .The first was called SARS-CoV-1 4, and it originated in the province of Guangdong Province in China: eight thousand cases were confirmed with a little less than eight hundred deaths in 2002 (affecting 29 countries until January 2004).The second incident was traced in Saudi Arabia, and for this reason, it was named MERS-CoV (Middle East Respiratory Syndrome Coronavirus), its first case was reported in June 2012 until November 2018, registered 2,494 cases after affecting 27 countries 5. SARS-CoV-2 has infected more than 757 million people and 6 million deaths approximately until February 23, 2023, according to Johns Hopkins University (data available at coronavirus.jhu.edu).

It is currently a public health problem. Multiple vaccines have been developed to control the spread of the disease and to reduce the number of infections recorded daily in various parts of the world. However, it is necessary to do more studies, for example, on how to reduce the application of multiple doses regimens for a person, and so on. It can probably do identification of genomic patterns capable of generating an immune response in people derived from epitopes, as has been tested in other diseases 678, and later can be automated with the help of specialized workflows for automatic data handling 9.

Until this is achieved, it is necessary to develop mathematical models to design the public policies necessary to contain said infections based mainly on the implementation of social distancing measures, the use of face masks, quarantine programs and vaccination campaigns, etc.

In the scientific literature, there is a wide range of mathematical models that explain, for example, the contagion dynamics in various countries without a consensus on the prediction methodology (some examples in 1011). Other works explain the transmission of viral particles in the environment 12, and also studies have also been carried out where they analyze the response of the immune system 1314. However, it is necessary to integrate all these approaches on the same time scale. This process is known as immuno-epidemiological models of infectious disease systems 15 where they are described from a cellular level to the spread in the population. This type of model has been used in the dynamics of contagion in HIV-1 with a system of six differential equations 1617, as well as the paper of Murillo et al 18 by studying a multiscale model to explain the Influenza infection in 2013.

The present work presents a general model to explain the transmission dynamics of SARS-CoV-2, dividing the model in two sections. We first studied the within-host scale which we will divide into two different levels. The first consists of twelve compartments to describe the life cycle of SARS-CoV-2, and later the immune response. Finally, the between-host model consists of twelve compartments to describe the transmission of the virus from the source that gave rise to the virus (bats), the host (pangolins probably), the reservoir (Huanan Seafood Wholesale Market in Wuhan), until reaching the population human where we have added a compartment to explain the environmental viral load, a key element to integrate both scales.

Mathematical model

The mathematical model describes the contagion dynamics at two different sections scales, i.e., within-host model and between-host models. Each of them is described below.

Within-host model

The within-host model include the transmission dynamics from the life cycle of SARS-CoV-2, and subsequently the model that reproduces the immune response will be explained.

SARS-CoV-2 life cycle

The virus is a single-stranded RNA virus belonging to the Coronaviridae family, Coronavirinae subfamily 19. The first reading frame, known as ORF1ab (the largest gene), occupies two-thirds of the virus sequence, approximately 28-32 kb. It is a 5’-capped and 3’-polyadenylated positive-sense single-strand RNA (+ssRNA), non-segmented and similar to the structure found in messenger RNA of eukaryotic cells.The replication of the virus begins when the S protein of SARSCoV-2 binds directly to the Angiotensin-Converting Enzyme 2 (ACE2) receptor. Once the virus has entered the host cell, it is released into the cytoplasm, starting the replication process that will give rise to non-structural proteins and also accessory proteins, with four structural proteins. The formation of RNA(-) and also replication and transcription of RNA subgenomics. These subgenomic RNAs(-) are transcribed into mRNAs(+) which encode the structural proteins S, M, E, N, and accessory proteins. During the replication process, the N protein of the virus binds to the genome, while the M protein associates with the membranes of the endoplasmic reticulum (ER). Finally, the virions are secreted by exocytosis 19.

Recently Isea and Mayo-Garc´ıa 20 have proposed the first analytical solution for the life cycle of SARS-CoV-2 according to the previous description. This model can be seen in figure 1, where it has been divided into four levels in different colors, where the first explains the cell entry, the second the genome transcription and replication, then the translation of structural and accessory proteins, and finally the assembly and eventual releases of new virions from the cell. The twelve variables that are going to describe the SARS-CoV-2 life cycle model are shown in Table 1, so the equations are simply (details in 20):

Here [V_release] is the concentration of viral particles that are released in the cell, and they are precisely the ones that will trigger the immune response in people, as will be explained in the next section.

Figure 1. Diagram of the life cycle of SARS-CoV-2The level that describes the cell entry is shown in blue color and indicated with number 1. The transcription and replication of the genome is in red color (number 2), while that the translation of structural and accessory proteins, and the assembly and release of virions are show in green and orange color, respectively. Only a few constants are shown in the life cycle dynamics and all the details are described in the text.

Table 1. Description of the variables used to describe the life cycle of SARS- CoV-2 (details in 20).

Variable	Description	Variable	Description
[V_free]	Free virions outside of cell	[gRNA₋]	Negative sense genomic and subgenomic
[V_bound]	Number of virions bound to ACE2	[gRNA]	Positive sense ge- nomic and subgenomics RNAs
[V_endosome]	Number of virionsin endosomes	[N]	Concentration of Nproteins per virion
[gRNA₊]	Numberofss-Positive sensegenomic RNA	[SP]	S+M+E per virion
[NSP]	Abundance of nonstructural protein populations	[N − gRNA]	Ribonucleocapsid molecules
[V_assemble]	Assembled virions	[V_released]	Released virions

Modelling the immune response

The immune response that is triggered in people as a result of infection by SARSCoV-2 virions has been described in various models in the scientific literature 2122, but most posts solve it numerically. Furthermore, recent studies have associated a storm of cytokines generated by SARS-CoV-2 such that it leads lymphocytes to the lungs in search of infection, and thus maintains the replication and transmission of the virus 23.

Of all the models available in the scientific literature, we only consider the simplest of them, as shown in

Figure 2. Diagram illustrating the immune response (see text for more details)

Figure 2, where the target cells (cells without contagion and represented by the letter T) are going to be infected (I) as a result of SARS-CoV-2 viral particles (V), so that an immune response product of antibodies is generated (A) (details in 22). The equations that describe the above process are:

Where T, I, V, and A represent target cells, infected cells, viral particles, and antibodies, respectively. The definitions of parameters will be derived directly from the work of Danchin et al 24. So both the life cycle of the virus and the dynamics of the immune response present the viral load, which is the link between them.

Between-hosts model

Figure 3 represents a model of the transmission dynamics from the possible origin of the virus associated with bats (denoted with the subscript B), then infecting an unknown host (usually associated with pangolins, abbreviated with the letter H). Later it reaches the reservoir (i.e., Wuhan market, W), until finally infecting the human population (P)

Figure 3. The Flowchart of the model proposed in this paper. The infection source is shown in blue color and it is a bats population, represented with B subscript. The host region is green and depicted with the H subscript (some bioinformatics studies indicated that could be pangolin population). The Reservoir (violet re- gion) was the Seafood Market in Wuhan. Finally, the human people are the last section and identified with the P subscript in cyan color.

Therefore, this model that allow us to explain the transmission of the virus is given by the following differential equations:

We can observe this model is divided into twelve categories where the subscripts B, H, and P have been used to represent the bat, pangolin, and human, respectively. The Populations susceptible to contracting the virus are S_B, S_H, and S_P, and the population infected are I_B, I_H, and I_P. R_B, R_H ,and R_Prepresent the recovered bat, pangolin, and human populations, and Q_Pis the population that is in quarantine. The reservoir is denoted as W , and the last compartment is V_P, which represents the environmental viral load, respectively.

The constants Λ_B, Λ_H and Λ_P are the newborn bats, pangolins, and human, respectively. The contagion rate in bats, pangolins, and humans is given by the constants β_B, β_H, and β_P, respectively. Also, m_B, m_H, and m_Prepresent the death rate, and N_B, N_H and N_Pare the total populations of bats, pangolins, and human, respectively; ω_B, ω_H, and ω_P are the infectious period of bats, pangolins, and human, respectively. The infection rate between bats and pangolins is given by the variable β^B_H, where the superscript and subscript identify the beginning and end of the rate of infection, respectively; while the contagion rate from pangolins to the reservoir is given by β^H_w, and β^W_Pfrom reservoir to human. These values should be obtained by fitting the model with the collected data. Finally, εis the lifetime of the virus in the Reservoir. Given that each country maintains different policies to combat the virus, it is necessary to fit it according to infected data.

Journal of Model Based Research