key: cord-0842426-n21hhr71
authors: Ghanbari, Ahmad; Khordad, Reza; Ghaderi-Zefrehei, Mostafa
title: Non-extensive thermodynamic entropy to predict the dynamics behavior of COVID-19
date: 2021-09-30
journal: Physica B Condens Matter
DOI: 10.1016/j.physb.2021.413448
sha: 927af3d1f2852c37f2bc2352fb750236a46d5ae3
doc_id: 842426
cord_uid: n21hhr71

The current world observations in COVID-19 are hardly tractable as a whole, making situations of information to be incompleteness. In pandemic era, mathematical modeling helps epidemiological scientists to take informing decisions about pandemic planning and predict the disease behavior in the future. In this work, we proposed a non-extensive entropy-based model on the thermodynamic approach for predicting the dynamics of COVID-19 disease. To do so, the epidemic details were considered into a single and time-dependent coefficients model. Their four constraints, including the existence of a maximum point were determined analytically. The model was worked out to give a log-normal distribution for the spread rate using the Tsallis entropy. The width of the distribution function was characterized by maximizing the rate of entropy production. The model predicted the number of daily cases and daily deaths with a fairly good agreement with the World Health Organization (WHO) reported data for world-wide, Iran and China over 2019-2020-time span. The proposed model in this work can be further calibrated to fit on different complex distribution COVID-19 data over different range of times.

In medicine, infectious disease is a process caused by various biological agents including bacteria, viruses, fungi, protozoans and worms impairing individual health [1] . In many cases, infectious diseases like HIV [2] and malaria [3] [4] [5] [6] can be directly or indirectly spread from person to person yielding a significant burden to international health and global economics [7] leading to current cause of increasing premature rate in humans and other species [8] .

Coronaviruses were first described in 1966 [14] . They are enveloped, positive single-stranded large RNA viruses that infect humans, but also a wide range of animals. Severe acute respiratory syndrome (SARS) that began in southern China and resulted in 774 deaths out of 8098 infected patients in 29 countries and the Middle East respiratory syndrome (MERS) originated in Saudi Arabia and responsible for 848 deaths among 2458 patients in 27 countries [11] [12] [13] . In the 2019, an outbreak of epidemic was triggered in December 2019 by a novel coronavirus in Wuhan, China [10, 14] and called COVID-19 [9] . This is one kind of seventh known coronavirus that spread quickly between people through unknown mechanisms [10] and it is sought that COVID-19 likely has an animal origin [15] [16] [17] [18] [19] . Important achievements have been made for deciphering the COVID-19 transmission dynamic using various mathematical models [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] . Amongst them are growth models, DE equations [31] [32] [33] [34] [35] [36] [37] . However, statistical mechanics paradigm has scarcely, if any thoroughly, used in this area. Statistical mechanics is a branch of theoretical physics and chemistry that applies thought through tools to study systems containing large number of particles [38] . In the past few years, many works have adopted statistical mechanics in various branches of science like computational neuroscience, biomechanics, engineering and physical and chemical approach [39] [40] [41] [42] [43] [44] . Entropy is related to the missing information on the concrete state of a system and it shows a measure of the disorder of a system. Herein, we use an entropic-based statistical mechanics model for predicting COVID-19 daily cases and deaths of people in world-wide, Iran and China. Also we attempt to develop an entropy-based approach to predict dynamic of the COVID-19 epidemic worldwide. The proposed models are based on a modified susceptible-exposed-infectiousrecovered (SEIR) computational framework. It is now evident that behavior of COVID-19 epidemic depends on many factors such as medical interventions, public-service announcements, isolation of people, restriction of individual and social activities. Therefore, it would be very difficult to find all those details of the epidemic by a mathematical model.

However, in this study we try to lay out a way to put these factors in mathematical model using non-extensive entropy method. Unlike the most general physical systems in which the non-J o u r n a l P r e -p r o o f extensivity entropy is very Naive, we believe that COVID-19 epidemic systems have visible and very strong non-extensivity entropy.

In our approach, we first defined density function ( ) as the number of daily COVID-19

cases. Daily deaths due to COVID-19 and recovery ones were subtracted from this parameter since death and recovery are two active agents in thermodynamic systems. The epidemic is remarkably vague and cannot be expressed by an explicit functional form. Therefore, we supposed the epidemic to be as a thermodynamic system and according to the approach for the rate of vibration excitation and chemical reaction [45] , we considered the rate of change of ( ) as

Where ( ) is a time-dependent parameter. The balance between the epidemic spread and the control mechanisms during the public intervention are two important rationale tools that could make up ( ) parameter. Accordingly, a coupled system of differential equations should be applied to model this scenario. To this end, we use a statistic background by supposing that all the spreading and controlling mechanisms effects could be incorporated into the ( ) parameter.

To proceeding along, here, we impose four constraints on this ( ). parameter: (ii) There is an exponential enhancement of smooth spread at the initial stage, (0) → ∞.

(iii) On a given day, = , the rate must decrease. Mathematically, it means that 2 ( ) 2 vanishes at = , this is so called inflexion date. Substituting Eq. (1) into 2 ( ) 2 = 0 yields

J o u r n a l P r e -p r o o f (iv) ( ) has a maximum, this means that for = , ( ) = 0.

As could be seen in the above, it is further assumed that ( ) has an analytical function. Many functions could be searched for, however, the sufficiently simple function satisfying the four constraints could be as follows

and is cumulated days to reach the inflexion point counting from the initial date. Inserting Eq. (2) into Eq. (1) and by some computation, we could deduce the following relation

Here is proportion constant, = ln + 2 and is proportion constant and determined through the use of the principle of the extreme rate of entropy production.

The principle of entropy production has attracted the wide attention of many researchers.

Ptaszynski et al. [46] have studied entropy production in open systems. They showed that the entropy production in small-open systems coupled to environments is predominantly caused by the displacement from equilibrium. Santos et al. [47] showed that the entropy production of a quantum system undergoing open-system dynamics can be formally split into a term that only depends on population unbalances, and one that is underpinned by quantum coherences.

Shiraishi et al. [48] investigated the fundamental relation between entropy production and heat current. They establish the fact that quick energy exchange inevitably induces large entropy production in a quantitative form. Tsuruyama [49] considering the locality of the second law of thermodynamics, addressed that entropy can be divided into entropy derived from a chemical reaction and entropy produced by the diffusion of signaling molecule. He showed that the conserved production rate can be defined as the channel capacity of the given signal J o u r n a l P r e -p r o o f transduction cascade. We used the principle of maximum/minimum entropy production as a useful method to determine model parameters. In this way, the characterizes the width of the ( ) density curve. As the curve gets wider, the entropy became larger and as the maximum dissipation rate is reached, the width would cease to enhance. The dissipation rate is proportional to the rate of entropy production. Maximum dissipation rate corresponds to the extreme rate of the entropy production which again corresponds to

Where ( ) is the tsallis entropy and express as

Where is the total number of possibilities of the system and is a positive constant.

Substituting Eq. (4) into Eq. (6), the entropy is obtained by 

Before using our thermodynamic model, we need to determine the parameters. To do this firstly, we specified the value = ( ) using the reported data at the inflexion point (the date (b) At places with always small reported data, it is difficult to observe a clear peak.

(c) Due to new outbreak, sometimes there are multiple inflexion points.

Also, as mention above, is calculated numerically as In this case, as → 1, we see a better agreement and generally, there is a good agreement between the model and reported data. Fig. (4) 

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:J o u r n a l P r e -p r o o f