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 ==2 Introduction==
-In  December 31, 2019, Wuhan Municipal Health Commission in Hubei Province, China reported a cluster  of 27 pneumonia cases with unknown etiology. The outbreak is associated with common exposures in one  wholesale seafood and living animals market in Wuhan City including seven serious cases <span id='citeF-1'></span>[[#cite-1|[1]]].  The starting point  for the first symptomatic case was  notified on December 2019. By the 7th of  January 2020, the Chinese authorities identified as the causative agent of this outbreak a new type of coronavirus from the family coronaviridae  that  was called later  SARS-CoV-2 which  genetic sequence  was shared by the Chinese authorities on January the 12th <span id='citeF-2'></span>[[#cite-2|[2]]]. The via of transmission between human beings, it is similar to the one described by other coronaviruses, by  secretions of infected people, mainly by having direct contact with breathing droplets up  to 5 microns (allowing them to be transmitted over  distances closer than 2 meters) and the contaminated hands or  fomites with such secretions, followed  by the  direct contact of  mucosa emissions from the mouth, nose or eyes <span id='citeF-3'></span>[[#cite-3|[3]]]. The virus SARS-CoV-2 has been founded  into the nasopharynx excretions too,   including the saliva <span id='citeF-4'></span>[[#cite-4|[4]]]. On March 11 2020 OMS  declared COVID-19 as a world  pandemic, since then it is considered the main health problem not only by the mortality rate but   also by the secondary effects presented on people such as: Psychological illness, economic loss and the negative impact on their daily activities <span id='citeF-5'></span>[[#cite-5|[5]]].   From March 20th  to May 31st The Secretariat of Health Mexico implemented the social distancing, best known as Healthy Distance (quarantine) because of the COVID-19 pandemic, then to avoid  a catastrophic impact on the economy's country, on June 1st, a “The New Normality” started and the goverment  reported weekly health alert per region symilar to the traffic light sytem, so with  colors there was indicated the type of authorized activities in all economic, labor,  educational and social spheres.  In this context, the mathematical models of the dynamic transmission on infected diseases have  taken an unprecedented relevance. They were applied to help the government`s actions against COVID-19. These models play  an important role so as  to quantify some possible strategies for mitigation and control of  infected diseases <span id='citeF-6'></span><span id='citeF-7'></span>[[#cite-6|[6,7]]]. There   are several models for the infected diseases, since  the Clasical SIR Model to more complex proposals <span id='citeF-8'></span>[[#cite-8|[8]]].  The applied mathematical models, such as  Gompertz and Logistic <span id='citeF-9'></span><span id='citeF-10'></span>[[#cite-9|[9,10]]], have been used with some success to predict the number of infected people with COVID-19. Unfortunately these ones predict on right  acurately in short time periods and   a huge quantity of data is requested for a good  sickness estimation, as it  is shown on a recent work done by Torrealba Rodriguez et al. <span id='citeF-11'></span>[[#cite-11|[11]]], for a COVID-19 estimation in México. On his contributions,   Denaiuru et al. <span id='citeF-12'></span>[[#cite-12|[12]]], makes the proposal for a compartmental mathematical model to the spread of COVID-19 disease, by giving special emphasis on the super-spreaders individuals transmissibility. The problem by working with extensive  compartmental models, it is the lack of information for most of the parameters. In a recent  work, Neves Armando G.M et. al <span id='citeF-13'></span>[[#cite-13|[13]]]  propose a slightly generalized version of the A-SIR model and propose a scheme for fitting the parameter for the model to real data using the time series only of the deceased individuals. The scheme is applied to the concrete cases of Lombardy, Italy and So Paulo state, Brazil, showing different aspects of the epidemic. In both cases they see strong evidence that the adoption of social distancing measures contributed to a lower increase in the number of deceased individuals when compared to the baseline of no reduction in the infection rate.
+On  December 31, 2019, Wuhan Municipal Health Commission in Hubei Province, China reported a cluster  of 27 pneumonia cases with unknown etiology. The outbreak is associated with common exposures in one  wholesale seafood and living animals market in Wuhan City including seven serious cases <span id='citeF-1'></span>[[#cite-1|[1]]].  The starting point  for the first symptomatic case was  notified on December 2019. By the 7th of  January 2020, the Chinese authorities identified as the causative agent of this outbreak a new type of coronavirus from the family coronaviridae  that  was called later  SARS-CoV-2 which  genetic sequence  was shared by the Chinese authorities on January the 12th <span id='citeF-2'></span>[[#cite-2|[2]]]. The via of transmission between human beings, it is similar to the one described by other coronaviruses, by  secretions of infected people, mainly by having direct contact with breathing droplets up  to 5 microns (allowing them to be transmitted over  distances closer than 2 meters) and the contaminated hands or  fomites with such secretions, followed  by the  direct contact of  mucosa emissions from the mouth, nose or eyes <span id='citeF-3'></span>[[#cite-3|[3]]]. The virus SARS-CoV-2 has been founded  into the nasopharynx excretions too,   including the saliva <span id='citeF-4'></span>[[#cite-4|[4]]]. On March 11 2020 OMS  declared COVID-19 as a world  pandemic, since then it is considered the main health problem not only by the mortality rate but   also by the secondary effects presented on people such as: Psychological illness, economic loss and the negative impact on their daily activities <span id='citeF-5'></span>[[#cite-5|[5]]].   From March 20th  to May 31st The Secretariat of Health Mexico implemented the social distancing, best known as Healthy Distance (quarantine) because of the COVID-19 pandemic, then to avoid  a catastrophic impact on the economy's country, on June 1st, a “The New Normality” started and the goverment  reported weekly health alert per region symilar to the traffic light sytem, so with  colors there was indicated the type of authorized activities in all economic, labor,  educational and social spheres.  In this context, the mathematical models of the dynamic transmission on infected diseases have  taken an unprecedented relevance. They were applied to help the government`s actions against COVID-19. These models play  an important role so as  to quantify some possible strategies for mitigation and control of  infected diseases <span id='citeF-6'></span><span id='citeF-7'></span>[[#cite-6|[6,7]]]. There   are several models for the infected diseases, since  the Clasical SIR Model to more complex proposals <span id='citeF-8'></span>[[#cite-8|[8]]].  The applied mathematical models, such as  Gompertz and Logistic <span id='citeF-9'></span><span id='citeF-10'></span>[[#cite-9|[9,10]]], have been used with some success to predict the number of infected people with COVID-19. Unfortunately these ones predict on right  acurately in short time periods and   a huge quantity of data is requested for a good  sickness estimation, as it  is shown on a recent work done by Torrealba Rodriguez et al. <span id='citeF-11'></span>[[#cite-11|[11]]], for a COVID-19 estimation in México. On his contributions,   Denaiuru et al. <span id='citeF-12'></span>[[#cite-12|[12]]], makes the proposal for a compartmental mathematical model to the spread of COVID-19 disease, by giving special emphasis on the super-spreaders individuals transmissibility. The problem by working with extensive  compartmental models, it is the lack of information for most of the parameters. In a recent  work, Neves Armando G.M et. al <span id='citeF-13'></span>[[#cite-13|[13]]]  propose a slightly generalized version of the A-SIR model and propose a scheme for fitting the parameter for the model to real data using the time series only of the deceased individuals. The scheme is applied to the concrete cases of Lombardy, Italy and So Paulo state, Brazil, showing different aspects of the epidemic. In both cases they see strong evidence that the adoption of social distancing measures contributed to a lower increase in the number of deceased individuals when compared to the baseline of no reduction in the infection rate.
 In  this work, we consider a adaptation of the model in <span id='citeF-12'></span>[[#cite-12|[12]]] by discarding the super-infectious class but considering the asymtomatic class as an infectious vector. Also we have adjusted the parameters of the resulting model to the Mexican country situations. So we present a consistent compartmental model involving to susceptible  class <math display="inline">S</math>, exposed class <math display="inline">E</math>, infectious with symptoms class <math display="inline">I</math>, asymptomatic infectious class <math display="inline">A</math>, Hospitalized class <math display="inline">H</math>, recovery class <math display="inline">R</math>, and death class <math display="inline">F</math>. Then a qualitative analysis of the model is done. We show the basic reproduction number and the local stability of the free disease equilibrium. The model is solved numerically to predict the dynamic's pandemic in two federal entities which were considered “red spotlights” for the COVID-19 : the CDMX and Tabasco State.  As in the whole Country, in both   the National Health Distance and  the “New Normality” were implemented. Finally we present the estimations with the Classic SIR Model in order to be compared with the proposed model predictions.
@@ Line 408: / Line 408: @@
 |}
-The figures [[#img-4a|4a]] and [[#img-4|4]]<nowiki/>b show the cumulative case estimates for the CDMX, using the SIR model and the model ([[#eq-1|1]]), respectively. We can note that, both models adjust appropriately the official data. However, they differ in the total number of people who were infected during the pandemic. The SIR model predicted around 48,000 total cases and the model ([[#eq-1|1]]) predicted 105,000 total cases. On the other hand, the Healthy Secretariat reported a total of 99,564 infections at September 1st, 2020.
+Figures [[#img-4a|4a]] and [[#img-4|4]]<nowiki/>b show the cumulative case estimates for the CDMX, using the SIR model and the model ([[#eq-1|1]]), respectively. We can note that, both models adjust appropriately the official data. However, they differ in the total number of people who were infected during the pandemic. The SIR model predicted around 48,000 total cases and the model ([[#eq-1|1]]) predicted 105,000 total cases. On the other hand, the Healthy Secretariat reported a total of 99,564 infections at September 1st, 2020.
 <div id='img-5a'></div>
@@ Line 428: / Line 428: @@
 ==6 Numerical results (New Normality Data)==
-On June 1st, the Mexican government activated the so-called new normality strategy in order to open up activities which are considered essential to reactivate the country's economy. This situation caused a  minor increase in almost the entire national territory, furthermore, it caused two plateaus to be observed in the epidemiological curve. If we consider the data on the state of Tabasco after confinement until August 21st, 2020, the parameters were adjusted to this normality and we can see that the model predicts the maximum number of daily cases by mid-July as seen in figure [[#img-6|6]]. The official reports indicated that the maximum number of daily cases in Tabasco ocurrred in July 12th.
+On June 1st, the Mexican government activated the so-called new normality strategy in order to open up activities which are considered essential to reactivate the country's economy. This situation caused a  minor increase in almost the entire national territory, furthermore, it caused two plateaus to be observed in the epidemiological curve. If we consider the data on the state of Tabasco after confinement until August 21st, 2020, the parameters were adjusted to this normality and we can see that the model predicts the maximum number of daily cases by mid-July as seen in figure [[#img-6|6]]. The official reports indicated that the maximum number of daily cases in Tabasco ocurrred on July 12th.
 <div id='img-6'></div>
@@ Line 531: / Line 531: @@
 ==8 Conclusion==
-We have introduced a compartmental mathematical model to understand the dynamic of the COVID 19 pandemic. This model considers a constant population  splitted into  susceptible class <math display="inline">S</math>, exposed class <math display="inline">E</math>, infectious with symptoms class <math display="inline">I</math>, asymptomatic infectious class <math display="inline">I_a</math>, hospitalized class <math display="inline">H</math>, recovery class <math display="inline">R</math>, and death class <math display="inline">F</math>. The used data correspond to two entities: Mexico city and the State of Tabasco, both considered during the pandemic as “red spotlights”. First of all we showed the existence and stability of the free illnes equilibrium and then we considedered two stages for the numerical study of model: The National Healthy Distance period and the "New normality" period. For comparison reasons, our simulations were done using our model an the classical SIR model. It must be mentioned that for all the  numerical simulations, most of the parameters were taken from official government data such as the CONACYT and the updated reference literature linked to the presented models up to 2020.
+We have introduced a compartmental mathematical model to understand the dynamic of the COVID 19 pandemic. This model considers a constant population  splitted into  susceptible class <math display="inline">S</math>, exposed class <math display="inline">E</math>, infectious with symptoms class <math display="inline">I</math>, asymptomatic infectious class <math display="inline">I_a</math>, hospitalized class <math display="inline">H</math>, recovery class <math display="inline">R</math>, and death class <math display="inline">F</math>. The used data correspond to two entities: Mexico city and the State of Tabasco, both considered during the pandemic as “red spotlights”. First of all we showed the existence and stability of the free illnes equilibrium and then we considered two stages for the numerical study of model: The National Healthy Distance period and the "New normality" period. For comparison reasons, our simulations were done using our model an the classical SIR model. It must be mentioned that for all the  numerical simulations, most of the parameters were taken from official government data such as the CONACYT and the updated reference literature linked to the presented models up to 2020.
 As a first stage, the study is done during the lockdown (National Healthy Distance) strategy implemented from March 20th to May 31st. For both models, ours and SIR, the obtained numerical estimations fit in an excellent way the reported data, as  the accumulated number of infectious disease and  the daily infectious cases as well as for the daily deaths (see Figures [[#img-4a|4a]]-[[#img-5|5]]). However, it can be noticed that  our model predicts better the infected accumulated cases, the accumulated deaths and the duration of the pandemic (see Figures [[#img-5|5]] and [[#img-8|8]]). With respect to the reproductive number, for the model ([[#eq-1|1]]) and using data and parameters for the CDMX we obtain <math display="inline">R_0=1.79</math>, while for the SIR model the value is <math display="inline">R_0=1.21</math>. This means that our model is more realistic with respect to pandemic transmision speed, which can be corroborated with the monthly data until May 31, 2020.
@@ Line 537: / Line 537: @@
 In a second stage, simulations were done using the updated data until August 22th 2020,  corresponding to the “New Normality” period when the  Mexican  government open the activities in sectors which they considered  important  for the economy, but when the peak's pandemic was not reached yet.  The obtained numerical results with the model ([[#eq-1|1]]) fit in a satisfactory way and the predictions improve with respect to the obtained with the classic SIR Model (see Figure [[#img-6|6]]-[[#img-8|8]]).
-Finally it is important to mention that the inclusion of classes in the population, such as hospitaized and deaths cases, the models provide valuable information to the goverments during the management of pandemics.
+Finally it is important to mention that the inclusion of classes in the population, such as hospitalized and deaths cases, the models provide valuable information to the goverments during the management of pandemics.
 ===BIBLIOGRAPHY===

@@ Line 24: / Line 24: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_LOPEZ_323363510-diagrama.png|480px|Transference diagram for the proposed COVID-19 model.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 1:''' Transference diagram for the proposed COVID-19 model.

@@ Line 78: / Line 78: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>x'=\Psi(x,y)-\Omega(x,y),</math>
+| style="text-align: center;" | <math>x'=\Psi(x,y)-\Theta(x,y),</math>
 |}
 |}
@@ Line 102: / Line 102: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\Omega= \left( \begin{array}{c}  kE\\  -k(1-\sigma )E+(\phi _h+\gamma _s+\delta _s)I\\  -k\sigma E+ \mu A\\  -\phi _h I+(\gamma _h+\delta _h)H \end{array}  \right) </math>
+| style="text-align: center;" | <math>\Theta= \left( \begin{array}{c}  kE\\  -k(1-\sigma )E+(\phi _h+\gamma _s+\delta _s)I\\  -k\sigma E+ \mu A\\  -\phi _h I+(\gamma _h+\delta _h)H \end{array}  \right) </math>
 |}
 |}
@@ Line 122: / Line 122: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>V=\left[\frac{\partial \Omega_i}{\partial x_j}(E_0)\right]=\left[ \begin{array}{cccc}  k & 0 & 0 &0\\  -k(1-\sigma ) &h_1& 0 & 0\\  -k\sigma & 0 & \mu & 0\\  0 & -\phi _h & 0 & h_2 \end{array} \right] </math>
+| style="text-align: center;" | <math>V=\left[\frac{\partial \Theta_i}{\partial x_j}(E_0)\right]=\left[ \begin{array}{cccc}  k & 0 & 0 &0\\  -k(1-\sigma ) &h_1& 0 & 0\\  -k\sigma & 0 & \mu & 0\\  0 & -\phi _h & 0 & h_2 \end{array} \right] </math>
 |}
 |}
@@ Line 423: / Line 423: @@
 |}
-Fgures [[#img-5a|5a]] and [[#img-5|5b]] show the estimated cumulative case for Tabasco using the SIR model and the model ([[#eq-1|1]]), respectively. The SIR model estimates around 12,000 total infected cases while the model ([[#eq-1|1]]) estimates around 29000 cumulative infected cases. In September 1st, 2020, the Healthy Secretariat reported 28,471 confirmed infected cases.
+Figures [[#img-5a|5a]] and [[#img-5|5b]] show the estimated cumulative case for Tabasco using the SIR model and the model ([[#eq-1|1]]), respectively. The SIR model estimates around 12,000 total infected cases while the model ([[#eq-1|1]]) estimates around 29000 cumulative infected cases. In September 1st, 2020, the Healthy Secretariat reported 28,471 confirmed infected cases.
 ==6 Numerical results (New Normality Data)==

@@ Line 13: / Line 13: @@
 ==2 Introduction==
-In  December 31 2019, Wuhan Municipal Health Commission in Hubei Province, China reported a cluster  of 27 pneumonia cases with unknown etiology. The outbreak is associated with common exposures in one  wholesale seafood and living animals market in Wuhan City including seven serious cases <span id='citeF-1'></span>[[#cite-1|[1]]].  The starting point  for the first symptomatic case was  notified on December 2019. By the 7th of  January 2020, the Chinese authorities identified as the causative agent of this outbreak a new type of coronavirus from the family coronaviridae  that  was called later  SARS-CoV-2 which  genetic sequence  was shared by the Chinese authorities on January the 12th <span id='citeF-2'></span>[[#cite-2|[2]]]. The via of transmission between human beings, it is similar to the one described by other coronaviruses, by  secretions of infected people, mainly by having direct contact with breathing droplets up  to 5 microns (allowing them to be transmitted over  distances closer than 2 meters) and the contaminated hands or  fomites with such secretions, followed  by the  direct contact of  mucosa emissions from the mouth, nose or eyes <span id='citeF-3'></span>[[#cite-3|[3]]]. The virus SARS-CoV-2 has been founded  into the nasopharynx excretions too,   including the saliva <span id='citeF-4'></span>[[#cite-4|[4]]]. On March 11 2020 OMS  declared COVID-19 as a world  pandemic, since then it is considered the main health problem not only by the mortality rate but   also by the secondary effects presented on people such as: Psychological illness, economic loss and the negative impact on their daily activities <span id='citeF-5'></span>[[#cite-5|[5]]].   From March 20th  to May 31st The Secretariat of Health Mexico implemented the social distancing, best known as Healthy Distance (quarantine) because of the COVID-19 pandemic, then to avoid  a catastrophic impact on the economy's country, on June 1st, a “The New Normality” started and the goverment  reported weekly health alert per region symilar to the traffic light sytem, so with  colors there was indicated the type of authorized activities in all economic, labor,  educational and social spheres.  In this context, the mathematical models of the dynamic transmission on infected diseases have  taken an unprecedented relevance. They were applied to help the government`s actions against COVID-19. These models play  an important role so as  to quantify some possible strategies for mitigation and control of  infected diseases <span id='citeF-6'></span><span id='citeF-7'></span>[[#cite-6|[6,7]]]. There   are several models for the infected diseases, since  the Clasical SIR Model to more complex proposals <span id='citeF-8'></span>[[#cite-8|[8]]].  The applied mathematical models, such as  Gompertz and Logistic <span id='citeF-9'></span><span id='citeF-10'></span>[[#cite-9|[9,10]]], have been used with some success to predict the number of infected people with COVID-19. Unfortunately these ones predict on right  acurately in short time periods and   a huge quantity of data is requested for a good  sickness estimation, as it  is shown on a recent work done by Torrealba Rodriguez et al. <span id='citeF-11'></span>[[#cite-11|[11]]], for a COVID-19 estimation in México. On his contributions,   Denaiuru et al. <span id='citeF-12'></span>[[#cite-12|[12]]], makes the proposal for a compartmental mathematical model to the spread of COVID-19 disease, by giving special emphasis on the super-spreaders individuals transmissibility. The problem by working with extensive  compartmental models, it is the lack of information for most of the parameters. In a recent  work, Neves Armando G.M et. al <span id='citeF-13'></span>[[#cite-13|[13]]]  propose a slightly generalized version of the A-SIR model and propose a scheme for fitting the parameter for the model to real data using the time series only of the deceased individuals. The scheme is applied to the concrete cases of Lombardy, Italy and So Paulo state, Brazil, showing different aspects of the epidemic. In both cases they see strong evidence that the adoption of social distancing measures contributed to a lower increase in the number of deceased individuals when compared to the baseline of no reduction in the infection rate.
+In  December 31, 2019, Wuhan Municipal Health Commission in Hubei Province, China reported a cluster  of 27 pneumonia cases with unknown etiology. The outbreak is associated with common exposures in one  wholesale seafood and living animals market in Wuhan City including seven serious cases <span id='citeF-1'></span>[[#cite-1|[1]]].  The starting point  for the first symptomatic case was  notified on December 2019. By the 7th of  January 2020, the Chinese authorities identified as the causative agent of this outbreak a new type of coronavirus from the family coronaviridae  that  was called later  SARS-CoV-2 which  genetic sequence  was shared by the Chinese authorities on January the 12th <span id='citeF-2'></span>[[#cite-2|[2]]]. The via of transmission between human beings, it is similar to the one described by other coronaviruses, by  secretions of infected people, mainly by having direct contact with breathing droplets up  to 5 microns (allowing them to be transmitted over  distances closer than 2 meters) and the contaminated hands or  fomites with such secretions, followed  by the  direct contact of  mucosa emissions from the mouth, nose or eyes <span id='citeF-3'></span>[[#cite-3|[3]]]. The virus SARS-CoV-2 has been founded  into the nasopharynx excretions too,   including the saliva <span id='citeF-4'></span>[[#cite-4|[4]]]. On March 11 2020 OMS  declared COVID-19 as a world  pandemic, since then it is considered the main health problem not only by the mortality rate but   also by the secondary effects presented on people such as: Psychological illness, economic loss and the negative impact on their daily activities <span id='citeF-5'></span>[[#cite-5|[5]]].   From March 20th  to May 31st The Secretariat of Health Mexico implemented the social distancing, best known as Healthy Distance (quarantine) because of the COVID-19 pandemic, then to avoid  a catastrophic impact on the economy's country, on June 1st, a “The New Normality” started and the goverment  reported weekly health alert per region symilar to the traffic light sytem, so with  colors there was indicated the type of authorized activities in all economic, labor,  educational and social spheres.  In this context, the mathematical models of the dynamic transmission on infected diseases have  taken an unprecedented relevance. They were applied to help the government`s actions against COVID-19. These models play  an important role so as  to quantify some possible strategies for mitigation and control of  infected diseases <span id='citeF-6'></span><span id='citeF-7'></span>[[#cite-6|[6,7]]]. There   are several models for the infected diseases, since  the Clasical SIR Model to more complex proposals <span id='citeF-8'></span>[[#cite-8|[8]]].  The applied mathematical models, such as  Gompertz and Logistic <span id='citeF-9'></span><span id='citeF-10'></span>[[#cite-9|[9,10]]], have been used with some success to predict the number of infected people with COVID-19. Unfortunately these ones predict on right  acurately in short time periods and   a huge quantity of data is requested for a good  sickness estimation, as it  is shown on a recent work done by Torrealba Rodriguez et al. <span id='citeF-11'></span>[[#cite-11|[11]]], for a COVID-19 estimation in México. On his contributions,   Denaiuru et al. <span id='citeF-12'></span>[[#cite-12|[12]]], makes the proposal for a compartmental mathematical model to the spread of COVID-19 disease, by giving special emphasis on the super-spreaders individuals transmissibility. The problem by working with extensive  compartmental models, it is the lack of information for most of the parameters. In a recent  work, Neves Armando G.M et. al <span id='citeF-13'></span>[[#cite-13|[13]]]  propose a slightly generalized version of the A-SIR model and propose a scheme for fitting the parameter for the model to real data using the time series only of the deceased individuals. The scheme is applied to the concrete cases of Lombardy, Italy and So Paulo state, Brazil, showing different aspects of the epidemic. In both cases they see strong evidence that the adoption of social distancing measures contributed to a lower increase in the number of deceased individuals when compared to the baseline of no reduction in the infection rate.
 In  this work, we consider a adaptation of the model in <span id='citeF-12'></span>[[#cite-12|[12]]] by discarding the super-infectious class but considering the asymtomatic class as an infectious vector. Also we have adjusted the parameters of the resulting model to the Mexican country situations. So we present a consistent compartmental model involving to susceptible  class <math display="inline">S</math>, exposed class <math display="inline">E</math>, infectious with symptoms class <math display="inline">I</math>, asymptomatic infectious class <math display="inline">A</math>, Hospitalized class <math display="inline">H</math>, recovery class <math display="inline">R</math>, and death class <math display="inline">F</math>. Then a qualitative analysis of the model is done. We show the basic reproduction number and the local stability of the free disease equilibrium. The model is solved numerically to predict the dynamic's pandemic in two federal entities which were considered “red spotlights” for the COVID-19 : the CDMX and Tabasco State.  As in the whole Country, in both   the National Health Distance and  the “New Normality” were implemented. Finally we present the estimations with the Classic SIR Model in order to be compared with the proposed model predictions.

@@ Line 13: / Line 13: @@
 ==2 Introduction==
-On  December 31 2019, Wuhan Municipal Health Commission in Hubei Province, China reported a cluster  of 27 pneumonia cases with unknown etiology. The outbreak is associated with common exposures in one  wholesale seafood and living animals market in Wuhan City including seven serious cases <span id='citeF-1'></span>[[#cite-1|[1]]].  The starting point  for the first symptomatic case was  notified on December 2019. By the 7th of  January 2020, the Chinese authorities identified as the causative agent of this outbreak a new type of coronavirus from the family coronaviridae  that  was called later  SARS-CoV-2 which  genetic sequence  was shared by the Chinese authorities on January the 12th <span id='citeF-2'></span>[[#cite-2|[2]]]. The via of transmission between human beings, it is similar to the one described by other coronaviruses, by  secretions of infected people, mainly by having direct contact with breathing droplets up  to 5 microns (allowing them to be transmitted over  distances closer than 2 meters) and the contaminated hands or  fomites with such secretions, followed  by the  direct contact of  mucosa emissions from the mouth, nose or eyes <span id='citeF-3'></span>[[#cite-3|[3]]]. The virus SARS-CoV-2 has been founded  into the nasopharynx excretions too,   including the saliva <span id='citeF-4'></span>[[#cite-4|[4]]]. On March 11 2020 OMS  declared COVID-19 as a world  pandemic, since then it is considered the main health problem not only by the mortality rate but   also by the secondary effects presented on people such as: Psychological illness, economic loss and the negative impact on their daily activities <span id='citeF-5'></span>[[#cite-5|[5]]].   From March 20th  to May 31st The Secretariat of Health Mexico implemented the social distancing, best known as Healthy Distance (quarantine) because of the COVID-19 pandemic, then to avoid  a catastrophic impact on the economy's country, on June 1st, a “The New Normality” started and the goverment  reported weekly health alert per region symilar to the traffic light sytem, so with  colors there was indicated the type of authorized activities in all economic, labor,  educational and social spheres.  In this context, the mathematical models of the dynamic transmission on infected diseases have  taken an unprecedented relevance. They were applied to help the government`s actions against COVID-19. These models play  an important role so as  to quantify some possible strategies for mitigation and control of  infected diseases <span id='citeF-6'></span><span id='citeF-7'></span>[[#cite-6|[6,7]]]. There   are several models for the infected diseases, since  the Clasical SIR Model to more complex proposals <span id='citeF-8'></span>[[#cite-8|[8]]].  The applied mathematical models, such as  Gompertz and Logistic <span id='citeF-9'></span><span id='citeF-10'></span>[[#cite-9|[9,10]]], have been used with some success to predict the number of infected people with COVID-19. Unfortunately these ones predict on right  acurately in short time periods and   a huge quantity of data is requested for a good  sickness estimation, as it  is shown on a recent work done by Torrealba Rodriguez et al. <span id='citeF-11'></span>[[#cite-11|[11]]], for a COVID-19 estimation in México. On his contributions,   Denaiuru et al. <span id='citeF-12'></span>[[#cite-12|[12]]], makes the proposal for a compartmental mathematical model to the spread of COVID-19 disease, by giving special emphasis on the super-spreaders individuals transmissibility. The problem by working with extensive  compartmental models, it is the lack of information for most of the parameters. In a recent  work, Neves Armando G.M et. al <span id='citeF-13'></span>[[#cite-13|[13]]]  propose a slightly generalized version of the A-SIR model and propose a scheme for fitting the parameter for the model to real data using the time series only of the deceased individuals. The scheme is applied to the concrete cases of Lombardy, Italy and So Paulo state, Brazil, showing different aspects of the epidemic. In both cases they see strong evidence that the adoption of social distancing measures contributed to a lower increase in the number of deceased individuals when compared to the baseline of no reduction in the infection rate.
+In  December 31 2019, Wuhan Municipal Health Commission in Hubei Province, China reported a cluster  of 27 pneumonia cases with unknown etiology. The outbreak is associated with common exposures in one  wholesale seafood and living animals market in Wuhan City including seven serious cases <span id='citeF-1'></span>[[#cite-1|[1]]].  The starting point  for the first symptomatic case was  notified on December 2019. By the 7th of  January 2020, the Chinese authorities identified as the causative agent of this outbreak a new type of coronavirus from the family coronaviridae  that  was called later  SARS-CoV-2 which  genetic sequence  was shared by the Chinese authorities on January the 12th <span id='citeF-2'></span>[[#cite-2|[2]]]. The via of transmission between human beings, it is similar to the one described by other coronaviruses, by  secretions of infected people, mainly by having direct contact with breathing droplets up  to 5 microns (allowing them to be transmitted over  distances closer than 2 meters) and the contaminated hands or  fomites with such secretions, followed  by the  direct contact of  mucosa emissions from the mouth, nose or eyes <span id='citeF-3'></span>[[#cite-3|[3]]]. The virus SARS-CoV-2 has been founded  into the nasopharynx excretions too,   including the saliva <span id='citeF-4'></span>[[#cite-4|[4]]]. On March 11 2020 OMS  declared COVID-19 as a world  pandemic, since then it is considered the main health problem not only by the mortality rate but   also by the secondary effects presented on people such as: Psychological illness, economic loss and the negative impact on their daily activities <span id='citeF-5'></span>[[#cite-5|[5]]].   From March 20th  to May 31st The Secretariat of Health Mexico implemented the social distancing, best known as Healthy Distance (quarantine) because of the COVID-19 pandemic, then to avoid  a catastrophic impact on the economy's country, on June 1st, a “The New Normality” started and the goverment  reported weekly health alert per region symilar to the traffic light sytem, so with  colors there was indicated the type of authorized activities in all economic, labor,  educational and social spheres.  In this context, the mathematical models of the dynamic transmission on infected diseases have  taken an unprecedented relevance. They were applied to help the government`s actions against COVID-19. These models play  an important role so as  to quantify some possible strategies for mitigation and control of  infected diseases <span id='citeF-6'></span><span id='citeF-7'></span>[[#cite-6|[6,7]]]. There   are several models for the infected diseases, since  the Clasical SIR Model to more complex proposals <span id='citeF-8'></span>[[#cite-8|[8]]].  The applied mathematical models, such as  Gompertz and Logistic <span id='citeF-9'></span><span id='citeF-10'></span>[[#cite-9|[9,10]]], have been used with some success to predict the number of infected people with COVID-19. Unfortunately these ones predict on right  acurately in short time periods and   a huge quantity of data is requested for a good  sickness estimation, as it  is shown on a recent work done by Torrealba Rodriguez et al. <span id='citeF-11'></span>[[#cite-11|[11]]], for a COVID-19 estimation in México. On his contributions,   Denaiuru et al. <span id='citeF-12'></span>[[#cite-12|[12]]], makes the proposal for a compartmental mathematical model to the spread of COVID-19 disease, by giving special emphasis on the super-spreaders individuals transmissibility. The problem by working with extensive  compartmental models, it is the lack of information for most of the parameters. In a recent  work, Neves Armando G.M et. al <span id='citeF-13'></span>[[#cite-13|[13]]]  propose a slightly generalized version of the A-SIR model and propose a scheme for fitting the parameter for the model to real data using the time series only of the deceased individuals. The scheme is applied to the concrete cases of Lombardy, Italy and So Paulo state, Brazil, showing different aspects of the epidemic. In both cases they see strong evidence that the adoption of social distancing measures contributed to a lower increase in the number of deceased individuals when compared to the baseline of no reduction in the infection rate.
 In  this work, we consider a adaptation of the model in <span id='citeF-12'></span>[[#cite-12|[12]]] by discarding the super-infectious class but considering the asymtomatic class as an infectious vector. Also we have adjusted the parameters of the resulting model to the Mexican country situations. So we present a consistent compartmental model involving to susceptible  class <math display="inline">S</math>, exposed class <math display="inline">E</math>, infectious with symptoms class <math display="inline">I</math>, asymptomatic infectious class <math display="inline">A</math>, Hospitalized class <math display="inline">H</math>, recovery class <math display="inline">R</math>, and death class <math display="inline">F</math>. Then a qualitative analysis of the model is done. We show the basic reproduction number and the local stability of the free disease equilibrium. The model is solved numerically to predict the dynamic's pandemic in two federal entities which were considered “red spotlights” for the COVID-19 : the CDMX and Tabasco State.  As in the whole Country, in both   the National Health Distance and  the “New Normality” were implemented. Finally we present the estimations with the Classic SIR Model in order to be compared with the proposed model predictions.

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 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\Gamma =\{ (S,E,I,A,H,R,F)\in \mathbb{R}^7:  S\geq 0, \; E \geq 0, \; I \geq 0, \; A\geq 0,   \; H\geq 0,   \; R\geq 0, \; F\geq 0\} </math>
+| style="text-align: center;" | <math>\Gamma =\{ (S,E,I,A,H,R,F)\in \mathbb{R}^7:  S\geq 0, \; E \geq 0, \; I \geq 0, \; A\geq 0,   \; H\geq 0,   \; R\geq 0, \; F\geq 0\}. </math>
 |}
 |}
-−
 ==4 Stability analysis of disease-free equilibrium.==

← Older revision	Revision as of 05:10, 18 October 2023
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