The entropy-based thermodynamic basic model
First, we consider f(t) function to be the number of confirmed daily cases, as the cumulative cases number. This function is subtracted from the daily deaths and recovery cases, since death and recovery cases are also parts of players in the thermodynamic system. In this research, we have considered the epidemic as a thermodynamic system and followed the approach for the rate of vibration excitation and chemical reaction [42, 43]. Thus, we have supposed that the rate of increase (decrease) would be proportional to the cumulative cases at the previous day:
$$ \frac{{{\text{d}}f\left( t \right)}}{{{\text{d}}t}} = \alpha \left( t \right)f\left( t \right), $$
(1)
where α(t) denotes a time-dependent parameter. The determination of this parameter depends on the spreading and controlling of the epidemic. Here, we have not considered the dependence of the parameter on other quantities e.g., population.
To proceeding along, here, we impose four constraints on this parameter:
-
(i)
The parameter α(t) must have the dimension of t−1. We know that Eq. (1) is the master equation, and α(t) shows the transition probabilities for various pairs of states. The parameter is proportional to inverse time. Therefore, we have selected the simplest choice as \( \alpha \left( t \right) \sim t^{ - 1} \). With this selection, the following constraint is satisfied (\( t \to 0, \alpha \to \infty \)).
-
(ii)
There is an exponential enhancement of smooth spread to start at the initial stage, \( \alpha \left( 0 \right) \to \infty \).
-
(iii)
On a given day, \( t = L \), the rate must decrease. Mathematically, it means that \( \frac{{{\text{d}}^{2} f\left( t \right)}}{{{\text{d}}t^{2} }} \) vanishes at t = L; this is called inflexion date. Substituting Eq. (1) into \( \frac{{{\text{d}}^{2} f\left( t \right)}}{{{\text{d}}t^{2} }} = 0 \) yields
$$ \frac{{{\text{d}}\alpha }}{{{\text{d}}t}} + \alpha^{2} = 0,\quad t = L $$
(2)
-
(iv)
f(t) has a maximum; this means that for t = D, α(D) = 0.
According to above imposed constraints, it is supposed that α(t) to be an analytical function of time. The following function would satisfy the four constraints, and it is sufficiently efficient
$$ \alpha \left( t \right) = - \frac{b}{t}\ln \left( {\frac{t}{D}} \right) $$
(3)
where \( b = \frac{1 - \ln (L/D)}{{\ln^{2} (L/D)}} \) . It should note that above function is not unique, and we can select other functions which satisfy the constraints. To find the function of f(t) analytically, we have chosen α(t) as Eq. (3) that is sufficiently simple. Inserting Eq. (3) into Eq. (1), we obtain the following relation:
$$ f\left( t \right) = \frac{k}{{\sqrt {2\pi } \sigma t}}\exp \left( { - \frac{{\left( {\ln \left( t \right) - \mu } \right)^{2} }}{{2\sigma^{2} }}} \right) $$
(4)
Here σ and k are proportion constants and μ = ln D + σ2. It is noteworthy that σ can be computed by of the principle of the extreme rate of entropy production.
Extreme rate of entropy production
The principle of the extreme rate of entropy production can be employed as a reliable tools to calculate model parameters [44, 45]. Santos et al. [46] have shown that the entropy production of a quantum system undergoing open-system dynamics can be formally split up into a term that only depends on population unbalances. Tsuruyama [47] has considered the locality of the second law of thermodynamics and showed that entropy can be divided into entropy derived from a chemical reaction and entropy produced by the diffusion of signaling molecule. It is to be noted that the width of the curve f(t) ∝ t can be characterized by σ. The wider the curve is, the larger is the (Shannon) entropy. The intrinsic spread mechanism of virus and the large mixing activity of the population tend to make the curve wider.
As the maximum dissipation rate is obtained, the width would cease to enhance. Therefore, maximum dissipation rate corresponds to the extreme rate of entropy production. It is related to
$$ \frac{{{\text{d}}^{2} S\left( {\sigma ,\eta } \right)}}{{{\text{d}}\sigma^{2} }} = 0, $$
(5)
where S(σ, η) is the Shannon entropy and it can be written as
$$ S\left( {\sigma ,\eta } \right) = - \mathop \int \limits_{0}^{\infty } F\left( t \right)\ln F\left( t \right){\text{d}}t^{\eta } $$
(6)
Here, \( F\left( t \right) = t^{1 - \eta } f\left( t \right) \) with η = 3. This value is in the usual entropy definition [44].
The solution of the integral is
$$ S\left( {\sigma ,\eta } \right) = \eta \left( {\ln \left( {\sqrt {2\pi } \sigma } \right) + \eta \left( {\ln D + \sigma^{2} } \right) + \frac{1}{2}} \right) $$
(7)
Equation (5) holds if and only if
$$ \sigma = \frac{1}{{\sqrt {2\eta } }} \approx 0.0408, \quad {\text{for}}\quad \eta = 3 $$
(8)
With the value of σ, the intrinsic spread mechanism is balanced by the dissipation mechanism. To obtain the relationship between two typical dates D (maximum number of confirmed daily cases) and L (inflexion date), we should employ the definion of inflexion point. The inflexion point is the date at which the multiple controlling measure takes effect.
We inserted Eq. (4) into \( \left. {\frac{{{\text{d}}^{2} f\left( t \right)}}{{{\text{d}}t^{2} }}} \right|_{t = L} = 0 \); then we have
$$ D = L \exp \left( {\frac{1}{2}\sigma^{2} + \frac{1}{2}\sqrt {4\sigma^{2} + \sigma^{4} } } \right) $$
(9)
Applying Eq. (4), one can determine a relation between f(D) and f(L) as
$$ f\left( D \right) = F\left( L \right) \exp \left( { - \sigma^{2} - \frac{1}{2}\sqrt {4\sigma^{2} + \sigma^{4} } + \frac{1}{2}\left( {\frac{3}{2}\sigma + \frac{1}{2}\sqrt {4 + \sigma^{2} } } \right)^{2} } \right) $$
(10)
According to Eq. (8), the following important relations can be grasped
$$ \left. {\frac{D}{L}} \right|_{\eta = 3} = 1.649,\quad \left. {\frac{f\left( D \right)}{f\left( L \right)}} \right|_{\eta = 3} = 2.120 $$
(11)
By applying the above relations, we can predict the maximal number of confirmed daily cases and the day in which this maximum appears. The maximum number of infected individuals and the time at which this maximum occurs can be related to the number and time corresponding to the critical date.
First, we compute the following relation
$$ \left. {\frac{{{\text{d}}f\left( t \right)}}{{{\text{d}}t}}} \right|_{t = L} = - \frac{1}{{\sigma^{2} L}}f\left( L \right) \ln \frac{L}{D} $$
(12)
which yields
$$ L = \left( {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{4}{{\sigma^{2} }} + 1} } \right)\frac{f\left( L \right)}{{\left. {\frac{{{\text{d}}f\left( t \right)}}{{{\text{d}}t}}} \right|_{t = L} }} = \frac{3 f\left( L \right)}{{\left. {\frac{{{\text{d}}f\left( t \right)}}{{{\text{d}}t}}} \right|_{t = L} }} $$
(13)
We want to know the initial date for which the spreading of the epidemic triggers. For this purpose, we must determine the value L. This value has been obtained in Eq. (13).