The first critical point (PC¹) 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 (Table 1) to simplify the equation in this paper. These solutions reveal that the different populations depend exclusively on each population and are not conditioned on each other.

The second critical point (pc²) is when only the bat population is infected by Covid-19, 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 (pc²) 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 (pc⁴) represents the solution of the human population when it is infected by Covid-19:

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 (pc⁵) corresponds to the case when the population of bats and pangolins are infected by Covid-19, 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 (pc⁶) 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 (pc⁷) 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 (pc⁸) corresponds to the case when the three populations are infected by Covid-19:

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 Figure 2 shows the result of the least squares fitting of the observed contagion data registered in Australia, but due to limitations of the computational equipment, only the cases that occurred from 2020/06/03 to 2021/01/02 are analyzed (corresponding to the second wave of contagions). Here, the different parameters corresponding to the populations of bats and pangolins were considered equal to zero.

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).