The authors have declared that no competing interests exist.
This paper resolves analytically a mathematical model that reproduces the transmission of Covid19 in three interactive populations,
The World Health Organization declared a new pandemic caused by a virus called as Severe Acute Respiratory Syndrome Coronavirus 2 (SARSCoV2) on March 11, 2020, after the first case detected in the Seafood Market of Wuhan province in China in December 2019. Until January 02, 2021, a little more than 84 million confirmed infections and approximately two million deaths had been reported.
Uptodate, there is a wide range of mathematical models that explain the spread of the virus, but most of all are based on contagion in the human population
For this reason, the objective of this paper is to propose a mathematical model that is capable of reproducing all the possible scenarios of virus contagion in the transmission of the virus from the bat population to the human infection.
The model is inspired in the paper published by Chen
This new model is based on the following suppositions:
The populations susceptible to contracting Covid19 are S_{B}, S_{H} and S_{P} corresponding to the bat, pangolin and human populations, respectively. They are the healthy population who have not yet infected.
Λ_{B}, Λ_{H} and Λ_{P} son the newborn bats, pangolins and human, respectively. These values should be obtained by fitting the model with the collected data.
The contagion rate in bats, pangolins and humans is given by the constants β_{B}, β_{H}, and β_{P}, respectively. These values should be obtained by fitting the model with the collected data.
m_{B}, m_{H} and m_{P} represent the death rate of bats, pangolins and human, respectively. In the case of the human population, m_{P} is equal to 0.0000456.
N_{B}, N_{H}, and N_{P} are the total populations of bats, pangolins and human, respectively.
Populations infected by the virus are represented by I_{B}, I_{H} and I_{P} corresponding to the bat, pangolin and human populations, respectively.
ω_{B}, ω_{H} and ω_{P} are the infectious period of bats, pangolins and human, respectively. These values should be obtained by fitting the model with the collected data.
The reservoir is denoted as W.
R_{B}, R_{H} and R_{P} represent the recovered bat, pangolin, and human populations, 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}_{P }is from reservoir to human. These values should be obtained by fitting the model with the collected data.
ε is the lifetime of the virus in the Reservoir. Given that each country maintains different policies to combat the virus, and for this reason, it is necessary to fit according to infected data.
µ is the transmission period from I_{P} to W.
The system of differential equations proposed in this paper is the following:
Finally, indicate that there are three major differences in the model developed in this work with respect to the one proposed by Chen
The analytical resolution of the system of equations is based on the same procedure used and validated in a wide range of scientific works
The first critical point (PC^{1}) is the trivial solution of the system, that is, when there are no infections in any of the populations:
This result indicates the population that may be susceptible to contracting the virus are equal to Λ_{B}/Λ_{B}, Λ_{H}/m_{H} and Λ_{P}/m_{P} and corresponding to the populations of bats, pangolins and humans, respectively.
The next step is to determine the equilibrium conditions, it means, the result of evaluating the Jacobian of the system in this critical point. By doing this, three different eigenvalues are obtained:
where three new variables (Δ_{H}, Δ_{B}, and Δ_{P}) have been introduced indicated in (

The second critical point (pc^{2}) is when only the bat population is infected by Covid19, and the susceptible population of pangolins can infected by the virus, without them being able to infect humans:
This result reveals that the population of pangolins that can be infected directly proportional to the rate of infection between bats and pangolins. By repeating the procedure described above, the eigenvalues are
The first eigenvalue indicates the equilibrium condition in the bat population, while the third corresponds to the human population (which should be the same as the previous scenario). Finally, the second eigenvalue indicates the condition when the parameters that relate the contagion rate between bats and pangolins are combined.
The third critical point (pc^{2}) relates to the scenario when the pangolin population (host) is infected:
It should be noted that the human population that can be infected by the virus depends directly on the rate of infection with the environment (Reservoir). So it is necessary to control and reduce the presence of the virus in the environment in order to eradicate the chain of infections in the human population.
The eigenvalues obtained are:
The second eigenvalue is the one that gives us new information as it is a combination of conditions between the pangolin and human populations.
The fourth critical point (pc^{4}) represents the solution of the human population when it is infected by Covid19:
Where the new variables indicated in this critical point are listed in Table 1. The three eigenvalues in this scenario are:
The third eigenvalue is a mixture of the different parameters corresponding to human and pangolin populations.
The fifth critical point (pc^{5}) corresponds to the case when the population of bats and pangolins are infected by Covid19, so that the virus is also present in the Reservoir, and the human susceptible population can transmit the disease without being infected, that is:
This solution reveals how the environment (W^{*}) increases the number of people susceptible to contracting the disease. In this case, only two eigenvalues were determined:
As can be seen above, the first eigenvalue depends exclusively on the parameters of the bats, while the second is a combination of the parameters between the three populations.
The sixth critical point (pc^{6}) corresponds to the case when bat populations and humans are infected, that is:
The eigenvalues are:
The first two eigenvalues depend on the parameters of the bats and the human population, respectively; while the latter is a combination between the parameters between the populations of pangolins and bats.
The seventh critical point (pc^{7}) corresponds to the case when pangolin populations and humans are infected:
The human population that is susceptible to contracting the virus increases over time. In this case, only two expressions of the eigenvalues can be obtained:
that is, the eigenvalues corresponding to the bat population.
The last critical point (pc^{8}) corresponds to the case when the three populations are infected by Covid19:
This critical point is when everyone is infected, and unfortunately it has not been possible to determine the eigenvalues for this system, in view of the complexity of the equations.
The main advantage of this mathematical model is that each scenario can be considered separately. As an example, the
The observed contagion data was obtained from John Hopkins University. The values obtained by least squares were: β^{W}_{P} = 1.708, β_{p}= 1.215, ω_{P} = 0.001, Λ_{P} = 81.212, µ = 1.100, ε = 0.001, (setting the m_{P} value to 0.0000456). According to these results, the population susceptible to contracting the virus in Australia is equal to 1,780,965 (maximum value).
This is the first mathematical model that studied the different transmission scenarios of Covid19 in three different populations, it means, bats, pangolins and humans. The model is based on a compartmental scheme of the SIRtype (SusceptibleInfectedRecovered), and despite the simplicity of the model, eight critical points are obtained representing the different infection scenarios among the three populations considered in this paper. In the next publication, the model will be analyzed considering the measures of social distancing as well as the effectiveness of vaccines in various countries that are infected by Covid19.
The author appreciates the unconditional help of Rafael MayoGarcia when reading this work immersed in a sea of differential equations. Finally, thank God for his continuous inspiration in the development of this manuscript.