The authors have declared that no competing interests exist.
The goal of this paper is to show that it is possible to obtainan analytical solution of the life cycle of SARSCoV2 based on a deterministic model. To do this, this work solved a system of twelve differential equations where we obtained two points of equilibrium. The first critical point corresponds to the initial conditions regarding the virus entry into the cell without replication in the cell, and the second involved one is the process of the transcription and replication of the virus in the infected cell.
The new Severe Acute Respiratory Syndrome Coronavirus (SARSCoV2) is a virus belonging to the
It is a virus of approximately 2832 kb. It is a 5’capped and 3’polyadenylated positivesense single strand RNA (+ssRNA), nonsegmented, and belongs to the genus of
The replication of the virus begins when the S protein of SARSCoV2 binds directly to the AngiotensinConverting 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 nonstructural proteins and also accessory proteins, with four structural proteins
Taking into account the above description, the next step is to generate a mathematical model of the life cycle of SARSCoV2 using the law of mass actionas described in the next session. This type of study has been used in other intracellular replication processes such as HIV1
There are many scientific studies about the life cycle of SARSCoV2 (see for example
The equations that describe the process of the cell entry are the same ones used by Grebennikov et al
where the value of the constants are indicated in
The nonstructural proteins (Nsps) are responsible for the transcription and replication of genomic RNA, where Nsp12 is the central component in encoding the RNAdependent RNA polymerase (RdRp), and its function is to generate a negativesense singlestranded RNAs.
In this step, Grebennikov’s model proposes three differential equations, but we modified it (the changes are indicated in red color), where the first equation, ^{NSP}, takes into account the abundance of nonstructural protein populations, while that [gRNA_{}] takes account the negative sense genomic and subgenomic. Finally, the last equation is completely different from Grebennikov's model which describes [gRNA]
(Remember that the values of the constants are indicated in
Parameter  Value 

0,61 

12,00 

0,50 

0,50 

0,40 

1,00 

8,00 

0,20 

0,07 

0,10 

0,02 

0,04 

0,20 

0,12 

0,06 

2,10* 

2,99* 

0,20* 

0,19* 

37,32* 

4,39* 

8,01* 
We are going to consider very simple equations that describe the number of N proteins per virion (^{N}) and the total number of structural proteins (^{SP}) unlike Grebennikov, where we are not going to consider the formation of viruslike particles and the budding of new virions from the ER and Golgi compartments (ERIGC), so the equations in this model are:
The N proteins play an important role in incorporating viral RNA into particles. The virions assemble in the ERGolgi compartment by encapsulation of NRNA complexes and the newly assembled virions can leave the infected cell by exocytosis.
To explain this step, we change the equation that describesthe rates of change of the ribonucleocapsid (^{N}), while the assembled virions ([V_{assemble}]), and released ([V_{release}]) are described as:
The next step is solving these equations analytically.
This system of differential equations is solved employing the same methodology used and validated in previous works
The next step is to find the critical point. We find two critical points to be described below:
The trivial solution of the system of differential equations occurs when they are all equal to zero:
It means that there are no free virions and neither infected virus (all values are zero).
The next step is to calculate the Jacobian of the system and evaluate it at this critical point. We only showthe nonzero terms of the Jacobian (J), where the order of the subscripts correspond to the row, column of the Jacobian matrix, respectively; that is, J_{4,3} corresponds to the value of the matrix at row position 4, column 3 of the resulting Jacobian matrix. Thus, all nonzero terms are
Where k_{1},k_{2 }and k_{3 }as (k_{bind}+d_{v}),(k_{fuse}+k_{diss}+d_{v}) and (k_{uncoat}+d_{endsome}) respectively;
From these expressions, four eigenvalues of the system are obtained, which indicate the stable conditions of the system, given by:
It is interesting to note that there is no restriction for the system to be stable since the expression in the square root will always be positive. Likewise, it is surprising that the stability of the system depends precisely on the variables of the initial contagion of the virus, that is, on the entry of the virus.
This critical point reveals that the virus can be transcription and replication without being present in the environment.
Finally, we are going to determine if these equilibrium points are stable, and to do that, we calculated the eigenvalues for this critical point, but they are not indicated in the work due to the complexity of the equations, so the five eigenvalues for the second critical point are indicated, which are:
The first two eigenvalues coincide with those of the first critical point, while the last equations essentially depends on the virus release mechanism.
To demonstrate the feasibility of the model, the system of differential equations was solved using a program written in Python, where the initial values are 10,10,2,4,0,10,10000,456,2000,0,0,1 which correspond to [V_{free}], [V_{bound}], [V_{endsome}], [gRNA_{+}], [gRNA] [NSP], [gRNA_{_}] , [N],[SP], [NgRNA], [V_{assemble}] and [V_{release}] respectively.
The results are shown in
Finally, when numerically calculating the eigenvalues according to the data indicated in
The present paper shows that it is possible to analytically resolve the SARSCoV2 life cycle. In fact, we obtained two conditions of equilibrium, that is, when the virus is present in the environment, and when it is transcribed and replicated in the infected cell. Finally, more studies must be done to obtain the correct values of the constants indicated in the differential equations and from there, to be able to carry out the stability studies.