The Biologist

(Lima)

The Biologist (Lima), 2021, vol. 19 (1), 29-39.

ORIGINAL ARTICLE / ARTÍCULO ORIGINAL

1Angewandte Nanophysik, Institut fur Physik, Technische Universität Ilmenau, Ilmenau 98693, Germany

2Pontiﬁcia Universidad Católica del Perú, Mechatronic Master Program and Energy Laboratory, Lima 32, Peru

3Department of Protocol and Investigation National Hospital Guillermo Almenara, Perú.

4 Northern (Artic) Federal University named after MV Lomonosov, Arkhangelsk Russia.

*Corresponding author: alan.calderon@pucp.edu.pe

1,2,* 2,3 2,4

J. Alan Calderón Ch ; Julio Guevara Guevara ; John Lozano Jauregui ;

2 2

Juan Lengua Arteaga & Gonzalo Solano

ABSTRACT

Keywords: COVID-19 – Modulating Functions – thermodynamic

In this research COVID-19 transmission is analyzed in relation to thermodynamic and energy balance

between geographic areas and its correlation with possible COVID-19 transmission between two persons

at least. In order to achieve parameters for medical doctors, as for example the minimal distance among

two infected people, who have this virus, there were designed mathematical models that were based in

statistical data to get information of COVID-19 propagation as the dependence on temperature of

geographic areas, moreover the thermal effect of the minimal distance between two people avoiding

COVID-19 infection. With this work, answers are sought to the questions: if it could be possible to find a

relation between temperature and virus transmission? Or if it could be possible to get a correlation among

thermal variables with minimal distance separation (it was described above) for two people? the goal was

to find answers to these questions to support medical doctors, who are trying to find solution for COVID-

19 propagation. It is worth mentioning that this research can be extended to more complex areas such as

street markets, street fairs or enclosed marketplaces, where products and services are sold, moreover, not

every area has an air conditioning system in Peru., This research sheds light on issues such as appropriate

ventilation parameters, the minimal distance that people need to be separated to minimise virus

transmission. Furthermore, some geometrical/material characteristics for air filters and ultraviolet (UV)

disinfection at the entrance of the main air duct are proposed.

The Biologist (Lima)

ISSN Versión Impresa 1816-0719

ISSN Versión en linea 1994-9073 ISSN Versión CD ROM 1994-9081

doi:10.24039/rtb2021191879

OPTIMAL THERMAL ANALYSIS OF CORONAVIRUS (COVID-19) TRANSMISSION DURING

QUARANTINE DAYS IN PERU

ANÁLISIS TÉRMICO ÓPTIMO DE LA TRANSMISIÓN DEL CORONAVIRUS (COVID-19)

DURANTE DÍAS DE CUARENTENA EN PERÚ

https://orcid.org/0000-0002-6486-5105

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

RESUMEN

Palabras clave: COVID-19 – funciones moduladoras – termodinámica

En esta investigación se analiza la transmisión de COVID-19 por equilibrio termodinámico y energético

entre áreas geográficas y su correlación con la posible transmisión de COVID-19 entre al menos dos

personas. Con el fin de lograr parámetros para los médicos, como por ejemplo la distancia mínima entre

dos personas infectadas que tienen este virus, se diseñaron modelos matemáticos basados en datos

estadísticos para obtener información sobre la propagación de COVID-19 como la dependencia de la

temperatura de áreas geográficas, además, el efecto térmico de la distancia mínima entre dos personas

evitando la infección por COVID-19. Con este trabajo, se buscan respuestas a las preguntas: ¿Si fuera

posible encontrar una relación entre la temperatura y la transmisión del virus? ¿O si fuera posible obtener

una variable de correlación entre variables térmicas con una separación mínima de distancia (se describió

anteriormente) para dos personas? Por lo tanto, se esperan respuestas a estas preguntas debido al apoyo de

los médicos, que están tratando de encontrar una solución contra la propagación de COVID-19. Vale la

pena mencionar que esta investigación puede extenderse a áreas más complejas como mercados y ferias

abiertas al aire libre o mercados cerrados, donde se venden productos y servicios, además, no todas las

áreas tienen un sistema de aire acondicionado en Perú. Sin embargo, en esta investigación se logra la

técnica, cómo resolver esta tarea: obtener parámetros de ventilación apropiados como la dependencia de

la distancia mínima que las personas necesitan para separarse, para evitar la transmisión del virus entre sí.

Además, se sugieren algunas características geométricas / materiales para los filtros de aire y la

desinfección mediante ultravioleta (UV) en la entrada del conducto de aire principal.

INTRODUCTION

For the time, while this research is prepared, in

China it was decreased the quantity of infected

people. In Italy this situation leaves slowly from

the maximal value of infected and deceased people.

Furthermore, the situation in Peru is on the road to

get its maximal value of infected people. For

countries, where the health system was not enough

prepared for COVID-19 and simultaneously no

prepared by technical equipment to research for a

vaccine, in these countries it needs to decrease

infected people curve through statistical analysis of

the virus treatment (MINSA, 2020).

Nevertheless, this task is so probabilistic and it

depends very much from coordination between

government and people. This is the reason, why in

this research it is looking for a scientific calculation

to warrant a parameter of distance separation

between two people at least, owing to care from

virus transmission (Anekal, 2009; Crammer, 2018;

Bouchnita & Jebrane, 2020; Bourouiba, 2020;

Fuk-Woo, 2020; Harcourt, 2020; Dai & Zhao,

2020; Kapur, 2020; Schoen, 2020). From other

side, there are many questions such as, whether the

temperature can be a positive factor to reduce the

transmission? If really the virus can be transmitted

through the air? These questions are analyzed in

this research, because to find an appropriate answer

that depends from parameters of every

geographical area of the world, which means

dynamic curve of infected people could be similar

for some countries, however, can be very different

with others.

As the main target of this research, it is looking for

to achieve a relation of parameters that could be a

support for people's interaction during COVID-19

quarantine days in Peru and after the pandemic

could start to decrease the quantity of infected and

deceased people. It is problematic to know, what

exact distance separation must to be between two

people according to avoid virus transmission.

There are proposals for 1 meter, sometimes 2

meters and so on, even though, the ideal context is

to be separated as maximal as possible, however, to

keep isolation even also when the infected curve

tends to decrease quantity. This is complicated

owing to how not to not damage the economical

behavior of a country. Therefore, according to

continue economical tasks, it is necessary to know,

what could be the ideal distance between people in

markets, schools, banks and especially in Intense

Care Unit (ICU) between medical doctors and

Calderón-Chavarri et al.

with the same situation in countries with the cold

climate. Therefore, if warm areas of Peru could be

appropriated places, where people could recover

from COVID-19? Is it true and why? These

questions are analyzed in the mathematical

analysis.

Figure 1 shows the behavior of the infected

quantity of people as the dependence of days under

infection and as the dependence of temperature for

three different climate behavior in Peru. Therefore,

the red color curve corresponds to Piura (this is the

city in Peru with the hot climate, where during

almost all the year is up 30°C and this was

represented in figure 1 as points R1, R2 and R3 of

this curve), the blue color curve corresponds to

Lima (this is the city in Peru with the warm climate,

where during almost all the year is around 17°C-

22°C and this was represented in figure 1 as points

B1, B2 and B3 of this curve), and the green color

curve corresponds to Puno (this is the city in Peru

with the cold climate, where during almost all the

year is down 10°C and this was represented in the

figure 1 as points G1, G2 and G3 of this curve).

patients, due to it is necessary to correlate thermal

variables too.

This research is made to keep support for medical

doctors, who are trying to find solutions against

COVID-19, even though, in the beginning of

epidemic, Peruvian hospitals were not ready to

give enough attention to people in country under

extreme conditions. Nevertheless, in this research

is analyzed thermal characteristics for the

environmental conditions of ICU room, which are

correlated with thermodynamic variables of the

patient, equipment around and the medical doctor.

Until it is unknown a vaccine against COVID-19,

the patient for this research is considered as a black

box (not without a solution, due to there are many

situations, when people were recovered from deep

infection). Therefore, whether the main task is to

keep in steady-state ICU variables of the patient, it

is necessary to keep in steady-state environment

condition variables too. Gray (2020) and Wang et

al. (2020) analyzed by statistical data that people in

countries with the warm climate get better

recovering from the virus in contrast if to compare

Figure 1. The quantity of infected people curve in Lima, Puno and Piura.

depicted that in Lima it keeps 20°C approximately

during all 60 quarantine days (it kept similar

tendency until the last quarantine day too) and the

total quantity of infected people did not cross

number of 100000 by the official declaration of

MINSA. And finally it is depicted that in Puno it

keeps in average 2°C approximately during all 60

In figure 1 is depicted that in Piura it keeps 32°C

approximately during all 60 quarantine days (it

kept similar tendency until the last quarantine day

too) and the total quantity of infected people did not

cross number of 3000 by the official declaration of

the Health Ministry of Peru (MINSA or Ministerio

de Salud) (MINSA, 2020). Furthermore, it is

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Thermal analysis of COVID-19 in Peru

following chapter. Notwithstanding, this research

can not be useful, if authorities do not keep order to

apply movement rules in quarantine days.

Mathematical advance that was supported in

optimal solutions for thermodynamic calculations

between populations and bodies of people was

optimized with statistical data in order to get an

optimal algorithm to achieve parameters that were

described above. Moreover, this research can be

used to study solutions of the virus treatment

through physical parameters that were required by

medical doctors and results that were achieved

from the condition of the patient. That variables can

be integrated to achieve a mathematical model that

could be useful to control the virus.

The figure 2 represents two people as cylinders A

and B. The heat transmission between each other is

given because they are both heat sources as it was

represented by spheres with radius R1 and R2. The

spheres centers and cylinders centers are not

necessary to be for the same point (this is the ideal

situation). Furthermore,  represents the airflow

from A1 to A2 that can not be necessary to flow

from that direction. The task is to obtain the

minimum separation distance d between both

people according to avoid virus transmission.

quarantine days (it kept similar tendency until the

last quarantine day too) and the total quantity of

infected people did not cross the number of 1000 by

the official declaration of MINSA. Therefore, for

cities with the hot and warm climate (Piura and

Lima) as a comparison with the cities with the cold

climate (Puno) it was not verified the possibility

that high levels of temperature could decrease the

number of infected people. In Lima were more

infected people, maybe because they could not

keep the distance between each other, while they

were getting interactions. However, in Puno with

the very cold climate, it was less quantity of

infected people as the comparison with the same

situation in Lima and Piura.

Furthermore, after to analyze big areas as it was

described above, became some questions: if this

virus can be transmitted between two people at

least, even there is no contact among them? If an

infected person has a high body temperature, what

could be the minimal distance that other persons

could be around? (ECDPCCD, 2020; Li et al.,

2020). If the minimal distance could have the

dependence of airflow? These questions will

correlate the paragraph above according to achieve

a scientific result for medical doctors in the

Figure 2. Scheme of the thermal interaction between two people.

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Calderón-Chavarri et al.

As it was proposed in paragraphs above, thermal

variables such as flow, temperature and humidity

can be important parameters according to analysis

virus propagation. Therefore, the minimum

distance between people could not be the same

value in different geographical areas, owing to the

dependence of thermal variables. That is quite

important due to interaction needs specific

limitations according to the maximal

complications of virus propagation that can be

necessary to care about people. Nevertheless,

economical activities can not stop as also

interaction among patients and medical doctors.

Therefore, it comes the question: what could be the

minimal distance? To propose a model regarding

minimal distance between people, there are many

authors, such as Bouchnita & Jebrane (2020), that

started their analysis with the Second Newton Law

as it was described by the equation (1) in which for

every index i, the mass of people is m, the velocity

is v and forces are f.

Therefore, by some consequences of Chaos

Theory, it was adapted the solution for the last

patient. Therefore, it is possible to design a

mathematical model according to find an

explanation of the correlation of all these variables

and as a consequence to prevent the virus

transmission between people.

The figure 3 shows thermodynamic variables

correlations for ICU room, in which R represents

the ICU room,  represents the airflow inside the

room (from entrance A and exit B), V1 and V2

represent windows, E depicts ICU equipment, M

represents a medical doctor and P represents the

Figure 3. The scheme of thermodynamic variables correlations for icu room.

MATERIALS AND METHODS equation (from Chaos Theory analysis) to

particular cases of COVID-19 and effects in Peru:

In which Γ is a function of internal parameters γ and

. In this context, it was possible to get a proposal

calculation to correlate dij as the minimal distance

separation between two people (infected or not)

and with d0 as the proposed minimal distance,

because of COVID-19 propagation analysis from

laboratories calculations, α and β are positive

constants, γ is a parameter between 0 and 1,  is the

angle between the desired velocity and force fi is in

the movement direction. Therefore, the equation

(3) shows the relation between γ and :

The last two equation solutions were analyzed and

optimized as it was described in the following

paragraphs. Nevertheless, it was possible to get a

numerical result according to suggest minimal

distance separations to avoid virus transmission in

a special situation that Peruvian government is

trying to cross nowadays to return social working

activities, but avoiding virus transmission.

However, it is necessary to analyze the mechanic of

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Thermal analysis of COVID-19 in Peru

Da-Yan, 2015): therefore, it is proposing by ( 7 )

author (Aldoghaither & Da-Yan, 2015):

( 8 )in which,

therefore, it is proposing by author (Aldoghaither &

Da-Yan, 2015):

in which, is the Modulating Function, also:

also

in which:

Finally, according to solve the diffusion equation ,

there were obtained the coefficients of the general

diffusion equation:

Moreover,

therefore:

flows (air) that gets ventilation inside rooms, where

people solve tasks by interaction. It means, it is

necessary to achieve a numerical understanding of

airspeed, airflow, temperature, humidity. All of

them are correlated with the minimal separation

distance between people.

Notwithstanding, as it was proposed by Anekal

(2009), the dynamic of virus transmission can be

described by the diffusion equation (4):

In which, Vf is the virus concentration in the fluid, v

is the velocity fluid, Dvf is the virus diffusivity in

the fluid layer. Moreover, the solution is given by

the equation (5):

for which n, D and l are constants to fix space x and t

is the connotation of time.

Therefore, by correlating the last both equations

and by Modulating Functions, the general solution

can be achieved as equation (6):

However, as it was proposed by Modulating

Function solution analysis for diffusion equation

(Anekal, 2009), it was solved diffusion equation

through this methodology due to get adapted

models to specific applications, such as

coefficients solutions can get information of

geometrical, thermal and mechanical parameters,

which can be adjusted to necessities and

applications.

That is the reason, why Modulating Function in this

analysis gives plenty strategies.

The equation (7) describes space fractional

advection dispersion equation (Aldoghaither &

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Calderón-Chavarri et al.

concentration,

in which, M is the indoor pollutant generation, v is

the room volume, n is the air purifier collection

efficiency, t is the time, q is the air purifier flow

volume.

According to optimize the solutions of the

proposed models, it was analyzed the error among

a desired value (theoretical solution) and obtained

the equation solution (23), where solution error

analysis e(t) is the discrete error, and V keeps the

Fourier series coefficients.

furthermore, α is the frequency parameter function:

for which, the nonlinear model for error analysis is

given by equation (25):

therefore, the costing function is given by equation

(26):

also, according to get parameters of the main model

it was achieved the derivation:

where parameters are showed in equation (28) as

the dependence of the adaptive coefficients W.

in which by author (Aldoghaither & Da-Yan,

2015), obtained:

Also,

also

Furthermore by author (Aldoghaither & Da-Yan,

2015),

moreover,

and

Therefore,

Furthermore, it is known the equation (22) (IUVA,

2020), in which C is indoor pollutant

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Thermal analysis of COVID-19 in Peru

And the response estimation Y is given by equation

(29):

Finally, the adaptive response estimation is given

by equation (30):

RESULTS AND DISCUSSION

After to achieve the mathematical model to

estimate the airflow and its optimal values for

rooms of 100 meters square approximately, there

were prepared a summary table 1 to make

calculation of the physical variables (from the

mathematical models, which were analyzed in

section 3), moreover curves that are shown in the

following figures.

Table 1. Calculated Physical Parameters and Optimal Estimated Airﬂow.

In figure 4 is shown the curve of airflow as the

dependence on airspeed and temperature (blue

color curve) and its optimal curve (red color curve).

It is possible to identify that airflow is increasing

while airspeed is increasing too. However, in the

other side the temperature is decreasing. Therefore,

the optimal value for airflow keeps similar

behavior of the calculated airflow (from O to O )

1 2

that can be interpreted as the suggestion to reduce

the temperature of the room progressively, the

airflow is increasing and the airspeed is increasing

too.

In figure 5 is shown the curve of airflow as the

dependence on humidity and temperature (blue

color curve) and its optimal curve (red color curve).

Figure 4. Airﬂow curve (blue color) and optimized airﬂow curve (red color) as the dependence on airspeed and temperature.

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Calderón-Chavarri et al.

It is possible to understand that airflow is

increasing while the minimal distance separation

between people is increasing too, notwithstanding,

in opposite side the relative humidity percent is

decreasing. Hence, the optimal value of the airflow

keeps linearity response behavior (from P to P),

1 2

that can be interpreted as the suggestion to increase

humidity of the room progressively, the airflow

decreasing and the minimal distance separation is

decreasing too.

In this research, there were analyzed 3 equations

models that were based in chaos theory application,

according to suggest the minimal interaction

distance between people. Diffusion equation

analysis, according to calculate airflow parameters

and thermodynamic parameters configurations and

as the consequence to keep appropriated

ventilation in rooms, where people are under high

possibilities to get virus transmission.

It is suggested to expand this study for open areas,

such as markets or rooms without air conditioning

system, but they only could use fans to achieve

appropriated correlation among the physical

variables as it was proposed in this research. This

future research can help to suggest appropriated

parameters, which authorities could try to apply

according to care people under this epidemic.

Figure 5. Airﬂow curve (blue color) and optimized airﬂow curve (red color) as the dependence on humidity and minimal

separation distance.

Finally, by UV propagation, it can be proposed a

cleaning procedure owing to enhance the

protection against COVID-19 (as a result of

equation 22) (IUVA, 2020).

It is dedicated special gratitude to Hugo Medina,

because of his teachings in Science Physics for

many different generations of engineers, he did and

he makes that physics laws could be so easy to get

understanding of nature and current life, such as for

this research. With a very good base of laws of

nature, it was possible to obtain a fundament to

correlate advanced mathematics with the

formalism that engineering applications always

need. Even though Peru was not prepared to be face

to face against a big epidemic, but it was found the

answer from many researchers (read bellow), who

supported with their points of view, suggestions

and analysis discussions to finish this research,

which is waiting to be useful for the responsible

people, who have the task to organize priority of

activities. Because humans need to return to solve

tasks with much attention in physical parameters

and with caring much distance separations, room

ACKNOWLEDGMENT

The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Thermal analysis of COVID-19 in Peru

temperature, room humidity, airflow and airspeed

between them. Therefore, it is expressed deep

warm thanks to Willy Gamboa, Christian Gozar,

Broni Huamaní, Alexánder Zutta, Leslie Vargas

and Lilian Gamarra, because of their suggestions

and opinions to analyze the consequences of this

research. It is expressed thankfulness to medical

doctors from the Health Center of PUCP, because

of their time to give suggestions in the development

of this research. It is expressed thankfulness to

researchers Hui Dai, Bin Zhao, and Lawrence J.

Schoen due to their time to share opinions and

suggestion to this research. It is expressed

thankfulness to students of the lecture

Nanotechnology MTR609, Mechatronic Master

Program, PUCP: G. Alvarado, N. Azambuja, M.

Chávez, I. Loayza, and M.J. Romero, because of

their opinions and suggestions to analyze the

consequence of this research.

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The Biologist (Lima). Vol. 19, Nº1, jan - jun 2021

Thermal analysis of COVID-19 in Peru

Received August 12, 2020.

Accepted December 31, 2020.