Autores: Ortigoza Capetillo Gerardo Mario, Ponce de la Cruz Herrera Roberto Iñaki

**Introduction**

On December 2019, the province of Hubei in the city of Wuhan, China, was the epicentre of a new type of pneumonia that did not respond to known treatments [29], in a few days the infections rose exponentially not only in the place of origin but also in different countries. The cause of the dis-ease was a new type of coronavirus, the severe acute respiratory syndrome coronavirus type 2 or SARS-CoV-2, initially called the new coronavirus of 2019 or 2019-nCoV. On March 11, 2020, the World Health Organization declared this disease a pan-demic that in just a few months has spread rapidly to practically almost the entire world. This disease, epidemiologically linked to a wholesale market for live and unprocessed fish, shellfish, and animals in Hubei province [18], rapidly spread due to the inap-propriate management of infected patients and in-ternational air traffic [4]. Covid 19 presents symptoms such as fever, dry cough, loss of smell or taste, and tiredness, but also other less com-mon ones such as aches and pains, nasal conges-tion, headache, conjunctivitis, burning and sore throat, including in some cases diarrhea, skin rashes or color changes in the fingers or toes. The symptoms are usually mild and come on gradually, in fact, some of the infected people have only mild cold-like symptoms. Approximately between 80% and 90% of those infected recovers from the dis-ease without being hospitalized, but around 20% of people who get infected end up presenting a severe condition and experience difficulties in breathing, requiring intensive care and a mechan-ical respirator. Older adults and patients with pre-vious medical conditions such as high blood pres-sure, heart or lung problems, type 2 diabetes mellitus, chronic kidney disease, obesity (body mass index 30 or higher), COPD (chronic obstruc-tive pulmonary disease), immunosuppressed (weakened immune system) from solid organ transplantation or cancer, are more likely to have severe conditions leading to death.

The first confirmed case has been traced back to 17 November 2019 in Hubei. As of 28 October 2022, more than 626.3 million cases have been re-ported around the world, resulting in more than 6,566,000 deaths. The COVID-19 pandemic in Mexico began on February 27, 2020. The first con-firmed case was detected in Mexico City (a Mexi-can who had traveled to Italy and had mild symp-toms); a few hours later, another case was con-firmed in the state of Sinaloa, and a third case, again, in Mexico City. Thus, on March 18, 2020, the first death by Covid in Mexico was docu-mented. In the state of Veracruz, the first two cases of Covid 19 were registered in the metropol-itan area Veracruz-Boca del Río on March 17, 2020, and the first death from Covid 19 in the state of Veracruz was reported in Veracruz City on 29 March 2020. On March 23, 2020, the national cam-paign called healthy distance begins in Mexico, where recommendations are issued to society such as: teaching the citizen to identify Covid symptoms, covering coughs, frequent hand wash-ing, using disinfectant gel on the hands, avoiding handshakes and kisses, maintaining physical dis-tance from others, classes were suspended at all educational levels as well as economic-productive activities were reduced to a group of essential ac-tivities.

In fact, the effect of the Covid in the state of Ver-acruz has been evident, affecting even medical staff [9]. This makes it mandatory to develop reli-able methods for early and efficient detection of Covid.

Several mathematical models have been defined in order to get a better understanding of the Covid transmission process, recent approaches combine fractal theory and fuzzy logic in order to forecast COVID-19 time series [6] or neural network and fuzzy models. A vast amount of epidemical models is deterministic, based on compartments [5], and defined as nonlinear systems of ordinary differen-tial equations [2,15, 17, 23,16]. These models are variants of the famous model due to McKendrick and Kermack [1]. Epidemiological quantities such as the basic reproduction number [10]. transmis-sion, latency, and recovery rates are estimated by fitting the models to historical data. Deterministic models assume an initial state at t = 0 and allow the computation of the state of the variables at time t + 1 by using the state of the variables at the previous time t, they also assume that the pop-ulations are perfectly mixed, with no spatial or so-cial aspect to the epidemic. Most existing models cannot deal with heterogeneities (population het-erogeneity and distribution heterogeneity) and complex contagion patterns (mainly caused by the human interaction induced by mobility) in the real world. Cellular automata (CA) formulations allow us to consider local interactions of infected due to neighboring cells and global interactions due to human mobility probability. At each cell, control strategies can be implemented by assuming that parameters such as transmission and recovery rates depend on time and space. Moreover, space heterogeneities are locally accounted by using population densities and seasonality can be easily implemented by considering incubation rates and transmission probabilities as functions of weather variables (time-space dependent). All these fea-tures make cellular automata an attractive method to perform Covid spread simulations. Ghosh and Bhattacharya [11] defined on a rectangular grid a probabilistic SEIQR CA with a genetic algorithm to estimate parameters of the dynamics of Covid spread but they do not include human mobility. Schimit [26] proposes a probabilistic SEIR CA on rectangular grids to analyze the impact of the so-cial isolation features on the population dynamics with no human mobility; Zhang et al. [31] conduct a dynamic evolutionary analysis of COVID spread based on probabilistic SEIQRD CA. They used a random walking on a rectangular grid to simulate population mobility. Sree and Nedunuri [24] pro-posed a hybrid nonlinear CA classifier, no spatial spread is considered only time series of data are trained to predict the number of infected and the number of Covid deaths. Zhou et.al [32] defined an extended susceptible, antibody infected re-moved CA to estimate levels of risk between coun-tries, no grid is used only geodesic distances and parameters for intercountry traveling routes. Ghosh and Bhattacharya [12] defined a SEIQR prob-abilistic CA on a rectangular grid with no human mobility but using a sphere of influence to model degrees of contact. Pokkulurian and Usha [25] con-sidered a nonlinear CA classifier in time to predict the Covid recovery rate in India. Dascalu et al. [8] present numerical simulations obtained with an enhanced cellular automaton with autonomous agents, they considered healthy, sick, and immune people inside rooms, on a rectangular grid (with walls) the defined probability of movement (right, left, forward, backward) to estimate the number of infected. Wang et al. [28] implemented an SEIR CA on a rectangular grid with no population den-sity and no human mobility to forecast the spread of Covid in Wuhan. Lugo and Alatriste [20] pro-posed a SI CA-based defined on a rectangular grid with no population density and no human mobility for selecting the most desirable testing frequency and identifying the best fitting size of random trails on local urban environments to diagnose SARS-CoV-2 and isolate infected people. Hidawi [13] makes a literature review of CA applied as classifi-ers for Covid spread predictions. Lu [19] performs simulations with a SIRD CA defined on rectangular grids to estimate the effects of lockdowns in Covid spread. Our approach implements a CA over an unstructured triangulated grid (no anisotropy is in-duced by rectangular grids or neighborhoods), it has the flexibility to be defined on complex domain geometries and includes geographic information such as population density distributions and local contact interactions. This provides us with the pos-sibility to estimate the time evolution of the trans-mission rate at different locations; by monitoring this information we can take decisions about im-posing or relaxing control measures in different lo-cations at different times. The work is organized as follows: Section methods describes the main as-sumptions of the unstructured cellular automata for Covid spread on an unstructured triangular grid. These include compartments, states, neigh-borhoods, the stochastic transition function, epi-demic parameters, boundary, and initial condi-tions. Section Results shows some numerical sim-ulations. Finally, we include some conclusions from this work.

**Material and Methods**

A cellular automaton A is a tuple (d, S, N, f ) where d is the dimension of space, S is a finite set of states, N a finite subset of Zd is the neighborhood and f: SN → S is the local rule, or transition func-tion, of the automaton. A configuration of a cellu-lar automaton is a coloring of the space by S, an element of ????????????. The global rule G: SZd → SZd of a cellular automaton maps a configuration c ∈ SZd to the configuration G(c) obtained by applying f uniformly in each cell: for all position z ∈????????????, G(c)(z) = f (c(z + v1), ..., c(z + vk)) where N = {v1, ..., vk} [30].

A cellular automata approach can be suitable to simulate local evaluations of diseases models where diseases parameters (transmission rates, death rates, incubation rates among others) can be assumed to be time and space-dependent, al-lowing in this way an easy implementation of sea-sonality due to ambient variables. Moreover, sto-chastic effects may be significant at the early stages of an outbreak and also when patch popu-lations are small, thus it seems to be reasonable to consider stochastic cellular automata in order to model and simulate Covid spread. Our starting point is the SEIRD (Susceptible, Exposed, In-fected, Recovered, Death) ODE model.

Here ???? is the recovery rate, ???? is the recip-rocal of the latency period, ???? is the transmission rate constant, ???? is the case fatality ratio (CFR), where a mass action incidence is assumed. No reinfection

is included (recovered people acquire some immunity that lasts the entire epidemic), finally, no births and natural deaths are considered. For this model, the basic reproduction number is given by R0=????/y

**Parameters**

The values of parameters β, η, γ were obtained by fitting an SEIR model to weekly data (weeks 12 to 18) reported by the public health ministry of the state of Veracruz, while φ was estimated by the daily historical data [22], table 1 reports the parameters and their fixed values. All the data employed in this study were obtained at the official website of the public health ministry of the state of Veracruz [7]. Our area of study is the state of Veracruz, it is situated on the coast of the Mexican Gulf, with latitude North between 17o 09′ and 22o28′, longitude West between 93o36′ and 98o39′. It has a total area of 71,826 km2, a total population of 8,112,505 (census 2015) [14] and consists of 212 municipalities. Using a geographical information system (GIS) the polygonal defining the boundary of the state was obtained. This boundary polygonal was input to a grid generator, Figure 1 shows the polygonal and the generated unstructured triangular grid. Table 2 presents some information about the triangular mesh. Classical CA implementations on rectangular grids have the disadvantage that the geometry of the underlying grid and of the chosen neighborhood strongly influence the time-dependent and stationary patterns, in such a way that anisotropy effects appear on arbitrarily large scales. Unstructured triangular grids provide us with the ability to handle complex computational domains (domains whose boundary is a planar straight-line graph also with the possibility of listing domains with holes) and implement boundary conditions and visualizations [21].

**Numerical Experiments**

In this section, all the numerical experiments assumed the initial condition: S(0) = 8112503, E(0) = 0, I(0) = 2 located at Veracruz City, R(0) = 0, D(0) = 0, figure 1.People are assumed to be distributed over the whole domain (212 municipalities) using the population density calculated with demographic data provided by the census INEGI (Instituto Nacional de Estadística y Geografía) [14], figure 2 shows this initial population distribution.Setting appropriate initial conditions is rel-evant because spatially heterogeneous transmis-sion may arise due to spatial variation in human-populated areas and human population density. In spite of all these features, only a few cellular au-tomata models have been defined to simulate the spread of Covid. Mathematical models of human mobility have shown a great potential for infec-tious disease epidemiology in contexts of data scarcity. Two of the most commonly used models are the gravity and radiation model. The gravity model requires parameter tuning and is thus diffi-cult to implement when no data are provided. On the other hand, the radiation model based on pop-ulation densities is parameter-free, but biased [27]. Some other transportation models from the fields of economics, geography, and network sciences include Stouffer’s rank model, Fotheringham’s competing destinations model, and the radiation model of Simini [3]. In this work, we consider a very basic scheme where a randomly chosen portion λ of the population is allowed to move with a mobil-ity probability pmob to a randomly chosen destina-tion. Moreover, we include the scenario where a visitor’s attractor is identified. People are classified according to their disease status: susceptible, ex-posed, infected, recovered, or dead. In the first set of numerical experiments, we compare no movement with the movement of people in one of the three compartments: Susceptible, Exposed, and Infected.

i) No movement assumption: Figure 3 shows a 240 days simulation of the diseases spread with no human mobility. On the right side, we observe an approximately circular wave front-outward spreading from the initial infected point (Veracruz City); Susceptible areas are green col-ored while infected are red. On the left, we note that a low number of infected is obtained when no movement of people is assumed.Before we start the numerical experiments to discuss human mobility, let us make some re-marks about the human population distribution. Ode’s models make the assumption that the hu-man population is well mixed (homogeneous dis-tribution over the space domain). However, hu-man settlements are not homogeneous, they can range from: excessively populated, moderately populated, sparsely populated, or even depopu-lated places. Figures 4 and 5 compare the Covid spread over a one-year period using a homogene-ous human population distribution (each cell has the same population total human population/total number of cells), versus covid spread with the density population displayed at figure 2, no human mobility is assumed. The peak size of the hetero-geneous distribution doubles the peak size of the homogeneous distribution, on the other hand, the final size (percentage of people who escape the diseases) reduces by approximately 30% in heter-ogeneous distribution when compared with the homogeneous distribution for a one-year simulation. These observations suggest that the use of cellular automata models with the assump-tion of homogeneous population distributions could result in an underestimation of the severity of the disease.

ii) Human mobility Eulerian approach

a) Moving Susceptible. A portion of cells, λ = 0.5, is randomly cho-sen and a fraction of susceptible people are as-sumed to move to a different cell (where they set-tled) with a probability pmob = 0.5. No significant differences are observed between the Eulerian movement of the susceptible people and the case of no movement of people as depicted in figure 3.

b) Moving Infected. A portion of cells, λ = 0.5, is randomly cho-sen and a fraction of infected people are assumed to move to a different cell (where they settled) with a probability pmob = 0.5. In figure 6, we ob-serve an increase in the peak size for both, in-fected and exposed.

c) Moving Exposed. A subset of cells, λ = 0.5, is picked at ran-dom, and a subset of exposed people is presumed to relocate to a different cell (where they settled), with a probability pmob = 0.5. We see a similar pat-tern as when infected people are transferred. Ex-posed individuals who are relocated to a different location eventually get infected. Figure 7 shows the ultimate state after 60 days, with a wavefront spreading outward from the initial infected spot. Simultaneously, the mobility of sick (or exposed) people creates new geographical foci of infection and new wave fronts, speeding up the disease’s spread. In this case, numerical simulations.

iii) Human mobility Lagrangian ap-proach

Veracruz City is the most inhabited munic-ipality in the state of Veracruz, as well as an inter-national harbor with an international airport and a huge variety of commercial opportunities. In fact, the National Ministry of Public Health bases its es-timates for Covid spread over the whole the state of Veracruz by monitoring confirmed cases at Ver-acruz City.

a) Moving Susceptibles: We consider Veracruz City a visitor attraction. A portion λ = 0.5 of cells corresponding to municipalities other than Veracruz City are randomly chosen and a fraction of susceptible people are assumed to visit a ran-domly chosen cell belonging to Veracruz City with a probability pmob = 0.5. Using the number of in-fected people in the neighborhood of the visited cell and the probability of being infected, this fraction of Susceptible gets exposed. This is similar to the fact that healthy people can acquire the dis-ease when visiting regions with high Covid trans-mission, and they can bring the sickness back with them when they return home. Figure 8 shows a 60-days simulation in which susceptible individuals visited Veracruz City, were exposed, and then re-turned the same day to their point of origin, re-sulting in the formation of new foci of infection.

b) Moving Infected: This time we take Veracruz City as a starting point, and a portion λ = 0.5 of infected is chosen at random, with a prob-ability pmob = 0.5, this portion visits a randomly chosen cell in another municipality, the portion of visiting infected is added to the vicinity of the land-ing cell, and then the event of a fraction of the susceptible at the landing cell being exposed is cal-culated. In this case, we are considering one-day visits that replace work trips or visits to relatives/ friends. Figure 9 depicts the growth in peak, the mobility of infected (even for brief visits), and the creation of new wavefronts, all of which contribute to the disease’s spread.

c)Moving Exposed: Finally, we return to Veracruz City as a starting place, and this time a portion λ = 0.5 of exposed are randomly picked, with a probability pmob = 0.5, this portion visits a randomly chosen cell in another town, and with an η probability, this fraction becomes infected. If this occurs, the percentage of recently infected visitors is added to the vicinity of the landing cell, and then the event of a fraction of the susceptibles at the landing cell being exposed is computed. Simula-tions respond similarly to the scenario of infected Lagrangian motion shown in figure 9. In summary, Eulerian mobility is similar to human migratory be-havior, as individuals attempt to flee sickness. For instance, students returning to their hometowns from the major metropolis, or affluent people re-treating to their country estates until the plague is eradicated. On the other hand, the Lagrangian ap-proach replicates commuting behavior by simulat-ing one-day journeys, such as work travels, shop-ping day in a large city, or visiting relatives/friends out of town, all of which may be completed in a single day. We are primarily concerned with the spread throughout the region, therefore migra-tions from town to town are taken into account. Based on the numerical experiments with respect to the Eulerian approach, we suggest that persons attempting to leave from illnesses can do so suc-cessfully if they are healthy (susceptible) and travel to a healthy location with no reports of ex-posed or infected cases (National Ministry of Public Health name them municipalities of the hope). In contrast, exposed or diseased persons traveling into a healthy area must be isolated to prevent the emergence of a new outbreak.

According to the Lagrangian approach, a one-day trip of susceptible individuals into high transmission regions carry a risk of contagion, it also raises the possibility that they may bring the illness with them when they return to their place of origin, potentially producing new foci of infec-tion. In reality, on May 24, 2020, health Secretary Roberto Ramos Alor issued a warning not to visit the towns of Coatzacoalcos, Poza Rica, and Veracruz City owing to a high risk of COVID-19 in-fection. It is apparent that affected persons must not relocate, even a single-day visit might infect others in nearby communities, causing new cen-ters of illness. Exposed persons constitute a risk; they may have no symptoms, but in only one visit, they might become infective and act as the source of new loci of infection in other locations. Tables 3 and 4 show the disease transmission in early and late phases using Eulerian and Lagrangian ap-proaches, respectively. The first column for each approach refers to the proportion of infected caused by the mobility of either susceptiles, ex-posed, or infected cells, while the second column displays the percentage of cells that at least one infected (infected cells). The percentage of infection generated by the three distinct compartments differs only slightly. The percentage of infected produced by their mobility under the Eulerian approach (marked with gray) is higher than the percentages produced by the movement of susceptible and ex-posed, while the percent of infected produced by the movement of susceptible (marked with gray) is higher than the percentage produced by and ex-posed and infected under the Lagrangian ap-proach. Indeed, under the Lagrangian approach, the movement of susceptible produces the maxi-mum percentage of infected at the early stages.

In the late stages depicted in Table 4, the percentage of infected produced by the movement of exposed (marked with gray) is higher than the percentages produced by the movement of susceptible and infected under both Eulerian and La-grangian approaches. In fact, the movement of ex-posure under the Lagrangian approach produces the maximum percentage of infected in the late phases. In terms of the percentage of cells that re-port at least one case, this percentage is near to the human mobility employed in the early stages and increases as the illness progress over time. In the early stages of Table 3, the percentage of in-fected cells produced by the movement of infected (marked with blue) is greater than the percent-ages produced by the movement of susceptible and exposed; whereas the percentage of infected cells produced by the movement of exposed (marked with blue) is greater than the percentage produced by the movement of susceptible and in-fected. In addition, under the Eulerian approach, the migration of infected cells produces the maxi-mum percentage of infected cells in the early stages. In the late stages, as shown in Table 4, un-der the Eulerian approach, the percentage of in-fected cells produced by the movement of infected (marked with blue) is higher than the percentages produced by the movement of susceptible and ex-posed cells; meanwhile, under the Lagrangian ap-proach, the percentage of infected cells produced by the movement of susceptible cells (marked with blue) is higher than the percentages produced by the movement of exposed and infected. In fact, under the Lagrangian approach, the migration of sensitive cells produces the maximum proportion of infected cells in the late phases.

**Some Epidemic quantities affected by human mobility**

For contagion to occur, an infected individ-ual who spreads the virus must enter a group of healthy people. Thus, it is clear that the mobility of infected people, either migrating or commuting, raises the possibility of a new outbreak being gen-erated in the areas visited by the infected. In gen-eral, infected people who acquire severe symp-toms go to the hospital, but infected people who develop mild symptoms stay at home (a 14-day quarantine is recommended), significantly decreasing infected people’s mobility. According to the Lagrangian approach, susceptibles might be exposed when visiting high transmission areas and bring the disease back to their place of origin.

However, due to fear of contagion (a de-fensive instinct), most vulnerable persons usually avoid areas with high disease transmission. Ex-posed persons provide a significant challenge since they do not exhibit symptoms, it is unknown how infectious they are, and they can ultimately get sick. As a result, it is necessary to pay close attention to the mobility of exposed persons and how it influences epidemic quantities such as the peak number of infected, the time it takes to reach the peak, and the final size of the epidemic. The peak refers to the maximum number of infected persons in a day; it is critical to have an estimate of the maximum number of hospital beds neces-sary in a day as well as the date on which this can occur. Figure 10 depicts the magnitude of the peak and the time required to attain it as a function of the mobility probability. The higher the mobility, the sooner the maximum is achieved and the larger the peak. Figure 11 compares epidemic curves for the Lagrangian movement of exposed with probmob = 0.2 and probmob = 0.8. In regards to exposed curves (yellow curves), the graph on the right (greater human mobility) displays a tall, narrow yellow curve, but the graph on the left (lower human mobility) seems short and broad; thus, less movement of exposed individuals tends to flatten the exposed curve. This flattening of the exposed curve causes a delay in reaching the in-fected peak (red curves) and diminishes the size of the peak, as seen in figure 10. The ultimate size of an epidemic S∞, which indicates the number of susceptible individuals who escape the epidemic, is frequently used as a measure of the severity of the epidemic; the bigger this number, the less se-vere the sickness. Simulation studies were per-formed for a lengthy period (730 days) to estimate the eventual size; calculations for the non-move-ment case generate an average final size of 28%. For the case of human mobility under the Lagran-gian approach, values of probmob ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} were tested, but the numerical findings do not show a relationship be-tween the final size and the probability of mobility.Instead, all of these values result in a final pandemic size of less than 0.1%.

Local transmission rates: Odes models assume a constant transmission rate β throughout the time and space domain. In our model β = cpi, where pi denotes the probability of infection and c is the per capita contact rate. Another advantage of our approach is that the probability of infection is adjusted locally (at each cell) by the local population density. In fact 1 − (1 − pi)Ni , where Ni is the total number of infected individuals in the local cell’s vicinity. Disease transmission happens at various sites and at different periods; heavily populated areas have higher transmission rates than less inhabited areas. Figure 12 shows an average behavior of the time evolution of local transmission. The chosen cell is the source cell (spot where the first Covid case was confirmed). The possibility to estimate the time evolution of the transmission rate in either a cell, a group of cells (municipality), or a group of municipalities (geographic region), enables us to track the disease’s evolution in space and time. We may use this information to make decisions about applying or loosening control measures in different locations and at different times. Monitor and Control: Consider a simple monitor and control scheme c1. Taking the local calculation of β(t) and supposing a local expression of the basic reproduction number given by the ode model R_l (t)=β(t)/γ (here the recovery rate is assumed constant in time and space) we reduce the human mobility at each cell to half of its original value provided the local R_l (t)>1. This strategy is similar to the Veracruz government’s control measure; in which various towns have different mobilities based on the current color of the Mexican Covid traffic light. The traffic light color scheme was implemented in preparation for the country’s gradual reopening on June 1, 2020. It is made up of four colors (green, yellow, orange, and red) that signify the severity of the pandemic at the State level. Aside from the dynamic mobility reduction, we consider c2 the a priori control where mobility is lowered (before the beginning of the simulations) to half of the original value at cells whose density population is greater than the mean density population of the whole number of cells. Table 2.4 shows the time to peak, normalized size peak, and ultimate size of 240-days Lagrangian approach simulation with probmob = 0.8 for both,

monitor and control schemes. From table 2.4 we notice that, control c1 causes a 16% delay in reaching the peak, an 8 % reduction in peak size, and a 30% increase in final size when compared to no control simulation. Control c2, on the other hand, results in a 22% delay in reaching the peak, a 12 % reduction in peak size, and a 48% increase in ultimate size when compared to no control simulation. This numerical simulation implies that it is vital not only to monitor local transmission rates and lower local human mobilities using a traffic light system but also to identify densely inhabited areas and impose more restrictive mobility controls on them. It is clear that reducing human mobility has a negative impact on the local economy, therefore, in densely populated areas where the economy is important, a slight reduction in mobility can be combined with more restrictive measures to reduce the probability of contagion, such as healthy distance and the mandatory mask use, among others.

**Discussion**

Cellular automata offer an alternative modeling approach to study spatial disease spread, allowing us to investigate local interactions at different scales. The use of population densities can be valuable to deal with heterogeneities. Population densities must be incorporated into the cellular automata model as long as they are available. Indeed, as the numerical simulations demonstrate, cellular automata models based on homogeneous population distributions may result in an underestimating of illness severity (lower size peak that heterogeneous case and higher final size that heterogeneous case). Some ways of estimating human mobility include technologies such as Google’s mobility report, Facebook’s mobility report, Twitter’s mobility report, etc. For instance, if these mobility statistics are supplied, they may be included in the cellular automata code by using a matrix of probabilities ???? with entries ????????,???? representing the probabilities that a portion from cell i visits cell j. However, in the absence of data, other approaches for simulating human motion must be considered. The numerical simulations reveal that human mobility of infected or exposed persons is a significant step in disease spread, as displayed in figure 10. When the probability of mobility increases, the size of the peak grows, and the time to reach the peak decreases. As a result, in addition to tracking daily confirmed cases and fatalities, we consider it is critical to track human mobility on a daily basis in order to limit local human mobility (traffic light system).

**Conclusions**** **

By using a SEIRD cellular automata model implementation we have obtained valuable information regarding some effects of human mobility on the Covid disease spread. Eulerian and Lagrangian approaches to human mobility using random walks have been adopted. The first is tied to human migration, whereas the latter is related to peoples’ daily commuting behavior. According to the Eulerian approach, the mobility of susceptible persons has no influence on Covid spread, unlike the movement of exposed and infected people, which give rise to new foci of infection in the destination locations. On the other hand, Lagrangian approach demonstrates that, in addition to being exposed and infected, susceptible people can also give rise to new foci of infection when they travel to high transmission areas for a short period of time, become exposed, and bring the disease back with them when they return. We focus our attention on the movement of the exposed. The simulations show that the Lagrangian movement of exposed influences peaks size and time to attain it. The higher the mobility, the sooner the maximum is reached and the larger the peak. Reducing the movement of exposed persons (reducing the mo-bility probability) tends to flatten the curve of ex-posed (who eventually get infected), it delays dis-ease transmission, and it minimizes the size of the peak. To keep health-care facilities from collaps-ing, the Mexican government has insisted on a stay-at-home campaign (Quédate en casa), which has reduced the number and mobility of individu-als in public areas. Because cellular automata models are local, it is possible to estimate the tem-poral evolution of the transmission rate in a single cell, a group of cells (municipality), or a set of mu-nicipalities (geographic region). This feature pro-vides us with a valuable tool for tracking the dis-ease’s progress in space and time. As a result, bet-ter-informed decisions regarding imposing or re-laxing control measures in various areas at distinct periods may be made. A future study might incor-porate the use of more compartments (quaran-tine, asymptomatic infected, age groups with dif-ferent transmission features), time-space variable latency, recovery and fatality rates, more accurate information relating to human mobility, and ex-plore other monitor and control schemes.

Palabras clave: Simulación autómatas celulares estocásticos movilidad humana propagación del Covid

2023-08-12 | 195 visitas | Evalua este artículo 0 valoraciones

Vol. 18 Núm.1. Enero-Junio 2023 Pags. 32-45 Rev Invest Cien Sal 2023; 18(1)