2.2.1 Free circulation, quarantine, and detection as components of contagion.

We treat quarantine (q) and free circulation (f) as a flow that affects the total population: the individuals in free circulation could enter quarantine and vice versa. We define q (quarantined) as the sub-index to indicate isolation and f (free) as the sub-index to indicate free circulation. Now, S_q will denote the isolated susceptible populations, S_f will denote those that remain in free circulation, and so on for other compartments as I. We refer to the flow between f and q as f ⇌ q. Isolation is relevant for the susceptible and infected since the lack of interactions protects them against contagion and isolated carriers can not spread the disease. Every compartment that belongs to the epidemic flow (e.g., SIR, SEIR, SIS) also allows for a partition into individuals of q and f kind (see Fig 1).

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Fig 1. Compartmental diagram for a SEIRD model with f ⇌ q|j and infection process from environmental reservoir T.

This include exposed (E) and death (D) compartments.

https://doi.org/10.1371/journal.pone.0275546.g001

We define the flow state compartments as probabilities: λ_qf is the probability of change from state q to f for the next time step, and λ_fq the probability of the other flow direction. The population staying in q (e.g. S_q or I_q) at time t + 1 only depends on λ_qf, since infectious interactions are possible only for individuals in f (e.g. S_f and I_f). Finally, we propose an extra flow, the infected detection j, which is the process of carrier identification (e.g. I_j), which can also be treated as a variation of the q (f ⇌ q|j flow). Thus, j behaves as an absorbant state as long as the individual recovers (R).

For free circulating populations, it is essential to formulate a function that describes the probability that a susceptible contracts the disease. Thus, we implemented novel contagion functions proposed in [28], as the reader can consult in section 3.1.1.

2.2.2 Diffusion systems in transmission models.

Some authors consider that the time elapsed from the moment an individual enters a compartment is a determinant for dynamics such as infectious capacity (pathogen load) or recovery from disease [29]. Some mathematical expressions that include such information during modeling are integrodifferential terms or large chains of coupled compartments with different output rates (Erlang models). However, both approaches are equivalent in certain epidemiological circumstances, as pointed out in [30]. Thus, we propose a discrete analog to the Erlang model approach, considering the f ⇌ q|j flow.

In Fig 1, we show how a SEIRD model turns into a directed graph where each node has an associated population at every time step. Individuals in node can move to another node on the next time-step, provided an edge connecting both nodes. Such displacements obey probabilities, but we treat them as proportions, the discrete equivalent of rates. We can model some of the flows between compartments as random walks, i.e., through a matrix approach. We called the set of compartments that admit random walk as a diffusion system. Thus, it is possible to represent the entire state as a vector where each position corresponds to a node. To estimate these proportions we introduced to the model a parameter related to scaling (γ) and two parameters related to the shape of a beta distribution function (a, b), as described in detail in section 3.1.2.

2.2.3 Mathematical and biological considerations during modeling.

To formulate the model, we considered the social and biological behavior of the spread of COVID-19 over a population. However, we limited the scope by stating some assumptions which we summarize in the 15 items in Box 1.

3.1.1 Contagion probabilities overcoming homogeneous assumption.

From assumptions 1, 3, and 4, it follows that infectious interactions that involve P_f are less likely to result in new infections than the interactions involving L_f. We define the probability of infection after infectious interaction with presymptomatic, β_P, and after interaction with symptomatic, β_L, treating each type of interaction as a different process. Considering assumption 2, a natural expression for the number of interactions that the infected individuals has with the susceptible population is zI_fS_f/N, being z the number of average interactions that a regular individual has by time-unit. In this way, the presymptomatic contact function would be given by where the expression ℵ(⋅) represents the heterogeneity of the population (ℵ(⋅) ≈ 0 is a homogeneous population), and it is introduced to decrease the probability of contact whenever I_f increases or S_f decreases. ν is the parameter of intrinsic isolation. We define the symptomatic contact function as Finally, to define the probability of infection from an environmental source, let T(t) represents the size of the reservoir at time t and set k_L as the mean contribution to the reservoir of a typical symptomatic carrier L(t) per unit of time. It would be impossible to avoid contact with the reservoir if all the individuals become infected. Then we assume that a reservoir size of k_LN(t), where N(t) is the size of the whole population, represents an extreme scenario, i.e., that T(t) ≤ k_LN(t). Now, assuming that k_LN(t) ≈ k_LN(0), we consider that the probability of having contact with the reservoir Φ_T must be equal to the numerical closeness between T(t) and k_LN(0). So, we defined the contact function with the environmental source as To achieve the related contagion functions, it only remains to multiply each contact function by its respective probability of infection (β_P, β_L, β_T). From assumption 14, the probability of not being infected would be the multiplication of the complements of infection probabilities given by the contagion functions.

3.1.2 Diffusion systems in transmission models with quarantine compartments.

According to assumptions 7, 8, and 9, we divided the classical compartment L and H into time-since-event compartments, because we can represent them as flows, as in [15, 34]. First, we assumed to have a sampling time unit of one day and set a maximum number of days-since-event to model within the selected compartment. Then we represented by sub-compartments (node with level) L^τ and H^τ where , the population of individuals that remain infected after τ-days since they became infected. We refer to each sub-compartment as a node, each compartment as a state, where the super-indices (τ) as the level of the node (related to disease time development). On the other hand, sub-indices will indicate the type of the node (f, q, or j).

The advantage of introducing multiple levels for symptomatic states, resides in the fact that we can define the probability of recovery from disease, the probability of dying because of the disease, and the capacity of disease transmission, as functions of the number of days since symptoms development (similar to the case of integrodifferential terms for continuous-time case).

However, in this paper, we focus on recovery and death functions. To estimate those functions we define three parameters: a and b as the shape parameters of a beta-distribution function, and γ ∈ [0, 1] as a parameter of scale. Then, we split the domain of the beta-distribution function into equally spaced points to achieve a discrete function and normalize the values of such discrete functions to fit in the interval [0, γ], being 0 the lowest value of the discrete function and γ the highest value. Such beta-shaped function evaluated at each point will give the probability of recovery or death for the corresponding level (see Fig 2).

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Fig 2. Example of two cases for diffusion systems.

We propose discretized beta-shape functions to assign a probability for recovery or death in each level. Assigning an enumeration for the nodes and a priority for flows, it is possible to define correspondent random walk matrices.

https://doi.org/10.1371/journal.pone.0275546.g002

We found in the literature that the virus could remain for 41 days or less for some reported cases [35]. Hence, we set and define each recovery function (for L and H) and death function (for H) as explained previously, i.e, through a parameterized and discrete beta-shaped function. In this way, we get three types and 41 levels for state L, for a total of 123 nodes, and a single type with 41 levels for H (41 nodes). Once the nodes have been established and numerated, it only remains to define random-walk matrices to easily model the flow inside L and H states. We enumerated the nodes as can be seen in Fig 3 and chose the following priority of events: recovery > dead > f ⇌ q|j. Priority means that an individual who recovers cannot die, and so on. Thus, we defined a matrix for each flow and multiply them to achieve the random walk matrix. For instance, the matrix in Fig 3 represents the f ⇌ q|j transversal flow; the matrix represents the process of recovery from disease for L; represents the process of dying from disease for H; and the process of recovery from the disease for H.

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Fig 3. Example of diffusion matrices for L and H deduced from an order for the nodes (enumeration).

Note that individuals in L have f ⇌ q|j flow (matrix ) and recover flow (matrix ). λ_qf and λ_fq are the probabilities of incoming or leaving quarantine, respectively; ϑ is the probability of being identified as infected. Individuals in H can recover (matrix ) or die (matrix ). Values for γ_i and μ_i are obtained as shown in Fig 2. For the mathematical description of the matrix, see sections 3.2.4 and 3.2.5. (a) Diffusion process for L, (b) Diffusion process for H.

https://doi.org/10.1371/journal.pone.0275546.g003

Finally, the random walk matrix for L is given by (1) and the random walk matrix for H is given by (2) which is strictly triangular superior. In sections 3.2.4 and 3.2.5, we define mathematically the diffusion systems, which include a beta-distribution function to describe the probabilities to flow from one compartment to another one.

3.2.1 Susceptible state (S).

We divide the susceptible population into two sub-compartments: those in free circulation, S_f, and those in voluntary quarantine, S_q. Event probabilities and their complements define the flow between compartments. For example, the number S_f is given by the probability of not becoming infected from any infectious resource ((1 − Φ_L(⋅)β_L), (1 − Φ_P(⋅)β_P), (1 − Φ_T(⋅)β_T)), and do not entered into quarantine (1 − λ_fq) plus those who left quarantine (λ_qf). Similarly, S_q depends on the probability of entering quarantine due to not becoming infected and leaving quarantine. But, if there is an infectious interaction in S_f, they become exposed and pass to the E compartment. The successive compartments follow the same idea of probabilities and their complement, even with more events, as we will see below. (3)

3.2.2 Exposed state (E).

Exposed individuals could circulate freely (E_f) or stay in quarantine (E_q) with probabilities λ_fq and λ_qf; they have not developed enough viral load to be infectious. Some individuals can stay over two days in this subsystem because the flow from E² to P is given by . Those individuals who have already completed the latency period become P^L or P^H.

3.2.3 Low and high presymptomatic states (P).

E passes to P depending on the severity of the symptoms they develop. Note that individuals in presymptomatic compartments have enough viral load to infect a susceptible individual through direct contact, and they also contribute to the viral load in the environmental reservoir. The flow from P to L and H are defined by and , respectively which represent probabilities of transitions between the compartments.

• Low presymptomatic (P^L): we defined them as individuals that would develop mild symptoms or no symptoms. We assumed that the disease effect of this population does not cause death. Also, this compartment has sub-compartments that represent free-circulation, with probability λ_qf, quarantine or self-isolation, with probability λ_fq, and identified, . The last one is a new sub-compartment, which causes obligatory isolation because the identification by the health system. We described it with probabilities ϑ_E (for those new presymptomatic) and ϑ_P (for those circulating presymptomatic). Finally, δ is the proportion of individuals that will develop severe symptoms and 1 − δ is its complement.
• High presymptomatic (P^H): As in the P_L compartment, we divided this compartment into three sub-compartments as (i) free-circulation, , (ii) quarantine, , and (iii) identified, ; with similar probabilities as exposed in the P_L compartment case.

3.2.4 Low symptomatic state (L).

Individuals in this compartment have sufficient viral load to transmit the disease and manifest mild symptoms or be asymptomatic. They will not die but will recover with some time-since-infection probability. Individuals in L can move between the free circulation and quarantine states or compartments. Also, they could be identified and isolated in L_j. Low symptomatic carriers have a greater viral load than presymptomatic ones; thus, they contribute more to the environmental concentration of the virus.

We took into account some parameters related to COVID-19 control. First, we modified flows from f to q (λ_fq) and from q to f (λ_qf) for L, by defining a parameter η_L and implemented it as and . Note that the higher the value for η_L the more similar the behavior of L and P. In contrast, when η_L → 0, the entire population in L_{f + q} ≔ L_f + L_q moves and remains in L_q unless they are detected. Also, we assumed the identification of presymptomatic individuals to be lesser than the identification of symptomatic ones. Hence, we proposed that percent (where η_ϑ ∈ [0, 1]) of the population in L_f+q is detected and moves into L_j for the next time step (whether they do not recover). Note that all the people in P that do develop severe symptoms are immediately detected as they flow into H_j. Further, from assumption 7, the L has f, q, and j compartments, and the only compartment available for H is j.

As mentioned in section 3.1.2, we consider three types and 41 levels for the L state for a total of 123 nodes (or sub-states). Instead of writing down those 123 state-equations we defined a vector with 123 positions (one for each node) and a random walk matrix. However, it is quite difficult and unintuitive to present our results based on the position of that vector. So, we proceed as follow: let ℓ₁(⋅) be a function that receives a node and returns its level, and ℓ₂(⋅) a function that receives a node and returns its type. Also, let N_s denote the starting node for the flow, N_a the arrival node for that flow, and let Q the condition ℓ₁(N_a) = ℓ₁(N_s) + 1, for all N_a and N_s in the diffusion system. Flows within the diffusion system can be modeled through the function while inside/out flows of L can be modeled by being the the discrete beta-shaped function we mentioned in section 3.1.2 and g_γ(x) a beta-distribution function parameterized by a_L and b_L.

It is worth to mention that the incoming flow of a node becomes the total content of the node for the next time step. That is because all individuals in time t change their level for time t + 1. For instance individuals in level k ≥ 1 and type f for time-step t + 1 would be given by where the sum does not consider any other levels than k − 1 because of the condition Q in function . The expression in the multiplication above gives us the proportion of individuals in the starting node that will move into the arrival node for the next time-step following the transversal flow. Multiplication by allows us to only consider those individuals that do not recover. On the other hand, it is clear that It only remains to note that the first level for L would run empty for the next time-step unless we consider the income of new low-symptomatic individuals. Thus L¹(t + 1) would be given by

Note that we include three main strategies for the identification of the carriers of COVID-19: (i) tracing of contagion networks, i.e., the isolation of individuals who had suspected contact with an identified carrier; (ii) the identification of those presymptomatic carriers who eluded the first strategy. Finally, (iii) the isolation of those carriers that do exhibit symptoms. We model such strategies by including three parameters for the identification of infectious individuals. To comply with such assumptions, we proposed a parameter ϑ_E that represents people in E moving into P_j because the public system traces the contagion networks. Now, ϑ_P represents people in P_f+q = P_f + P_q, that do not develop symptoms but move into P_j, which models the second strategy. For the third strategy, the identification of symptomatic carriers to be greater than the identification of presymptomatic ones, we proposed that (where η_ϑ ∈ [0, 1]) of the population in L_f+q (analog to P_f+q) move into L_j.

3.2.5 High symptomatic state (H).

Individuals in H have enough viral load to transmit the disease and they develop severe symptoms; thus, the health system rapidly identifies and isolates them, so there is a single compartment H_j in which these individuals stay in mandatory isolation. We assume these individuals can die or recover with time-since-infection probability. It follows that the diffusion system for H does not have flows of the form f ⇌ q/j, and therefore random walk matrices will be made up of probabilities of death and recovery. Functions that consider the characteristics of H, analogous to those given in the subsection 2.2.4, are defined as follows: where the functions g_μ(x) and g_H(x) are parameterized by beta-distribution functions defined as Function will be used in the following sections to define inside/out flows. In the same way, it must be clear that the income of new high-symptomatic individuals is given by while the total of high-symptomatic individuals is given by

Finally, as an example, the individuals in level k ≥ 1 for time-step t + 1 would be given by which denotes the proportion of individuals in H^k−1(t) that do not die and do not recover.

3.2.6 Recovery and death states.

The disease dynamics finish with recovered individuals from both L and H, and dead individuals from H. These are two separate compartments, and are similar in structure because they acts as sinks.

3.2.7 Environmental reservoir.

Assume we have an environmental reservoir of the pathogen constantly renewed by emissions from the infected individuals I(t). Susceptible individuals S(t) do have contact with contaminated surfaces at some rate, and then they might get infected. The contact rate is a non-decreasing function of the reservoir size, and without emissions from I(t), the reservoir would run out. That is the basic idea introduced in [27] to formulate a continuous-time model. We proposed a discrete-time equivalent for that idea defining a threshold from an extreme scenario and setting the behavior of the equation according to that threshold.

In this way, we defined T(t) as a compartment for the virus concentration in the environment according to the contributions of infectious micro-droplets expelled by P and L individuals in free circulation. The environmental reservoir can remove the virus from the system by a flow . The susceptible individuals could become infected during direct interaction with this reservoir, as exposed in section 3.1.1. We set T(t) ≤ k_LN(t) because our threshold scenario is the one where all the population becomes infected and symptomatic. Hence, a natural expression for this compartment is given by where k_P is the contribution to the reservoir from P and k_L is the contribution of symptomatic carriers to the reservoir. Further, the relation between k_P and k_L turns into how much symptomatic individuals contribute to the environmental reservoir more than presymptomatic. Then, we assumed that the increment is related to the viral load within the carrier. The viral load for a member of L could be from 1 to 10 times greater than that of a member of P according to [40].

3.2.8 Additional equations for model fitting.

When we perform parameter estimation, it is ideal to have the greatest amount of information available to make the model fit. For example, reference intervals for the parameters to be estimated and real data to which the model will fit: time series of the active cases, recovered, and dead (equivalent to node D) identified. Note that no explicitly available node is equivalent to said time series in the model; however, it is possible to propose equations that allow the fit without increasing the complexity. For instance, we propose in Eq (4) the following equations for model fitting: J as the number of active detected carriers at time t, and R_J as the number of detected carriers that have recovered at an specific time. (4)

3.2.9 Simulation of complex phenomena.

We identified migration, vaccination, and immunity loss, as complex phenomena with a high impact on the design of public policies. However, we gave up estimating their impact from available data because of the lack of information we would need to validate those estimations and the complexity shift they would cause in the model structure. Then, for the first two, we decided to include them as parameterized model inputs. We modeled such inputs as time series that indicate the impact of the input over the model at each time step. On the other hand, we treated immunity loss as a pulse that allocates a fraction of the recovered individuals into the susceptible ones at the start of the simulation. We based this last decision in the concept of extension, which we mentioned in detail in section 2.3. Next, we briefly introduce the parameterizations we proposed to simulate those phenomena.

1. Migration: We considered the migratory process as a flow of individuals towards the system. We assumed that symptomatic individuals refrain from migrating, which is a consequence of socially correct behavior. Then, by definition, the migratory flow must consist of susceptible, exposed, and recovered individuals in free circulation. To parameterize time-series generation for migration, we defined seven parameters distributed as follows: two parameters for shape (a_m, b_m), one parameter for scale (γ_m), two parameters for proportion of infectious and recovered migrants , and two parameters for initial and final time of migration .
   By using the shape parameters we compute a normal distribution function f(x) with domain restricted to the interval defined by , for instance, g(x) = f ↾_A (x). The range of g(x) is then scaled into the interval [0, γ_m] defining a new function using the following transformation The function h(x) allows us to achieve a time series when evaluating it at every time step of the simulation. This time series does have the shape of a discrete normal distribution function and a maximum value given by γ_m. Finally, we were able to take into account migration slightly modifying the equations for S_q, E², and R as follows. The relation holds.
2. Vaccination: Given the nature of the proposed model, we modeled the vaccination as a flow of people from compartment S to compartment R. In this way, we were able to treat vaccination as a time series that represents the number of susceptible individuals becoming immune to the disease at each time step, i.e., they acquire the status of recovered. To parameterize this time series, we followed a similar strategy that the one we designed for the recovery/mortality probabilities in diffusion systems (see section 3.1.2). To model vaccination we defined seven parameters as follows: two shape parameters for a beta distribution (a_v and b_v), one parameter for vaccine effectiveness (γ_v), one parameter for strategy of vaccination (δ_v), two parameters for the initial () and the final time () of vaccination program, and a last parameter for the total amount of vaccines (v). The total number of effective vaccines would be given by vγ_v. We used time parameters to set the domain of the beta distribution function. Finally, we chose a parameter of scale k_v such that being β(a_v, b_v, t) the beta distribution function with parameters a_v, b_v and domain , and N_j(t) the total amount of detected individuals at time t (including the recovered ones that were previously detected). In this way, the vaccination time series would be given by the function that we included in the system modifying the equations for S_f, S_q, and R in the following way The meaning of the strategy of vaccination parameter δ_v is the proportion of vaccines to be dosed that requires a displacement of the individuals to some place, which makes them behave as individuals in free circulation, regarding the vaccines to be dosed without exposure of the individuals to infectious sources. Note that proportion S(t)(N(t) − N_j(t))⁻¹ models the uncertainty involved when applying vaccines without knowing if the receptor is a real susceptible individual. This expression might be removed according to the particular context of each locality.
3. Immunity loss: Reinfection or cross-immunity, is a phenomenon we expected to happen because of the biological properties of the disease alongside its pandemic status. We aimed to model immunity loss by modifying the recovered compartment to flow again into the susceptible one. Hence, we proposed a function Υ(t) that allocates a small proportion of R into S (given by immunity-loss parameter k_I) at the beginning of the simulation for each extension (see section S1 Appendix for model extensions) (5) and we modified equations for recovered and susceptible individuals to include Υ(t) as follows One could expect parameter k_I to stay close to 0 for each extension. Therefore, we recommend keeping it as 0 until most of the population recovers () or the estimation algorithm starts giving poorly feasible parameters (suggesting a high effect of immunity-loss in the dynamics of the disease). We included λ_qf/(λ_qf + λ_fq) and λ_fq/(λ_qf + λ_fq) in equations above because we assumed that there is an implicit f ⇌ q flow within the recovered compartment. Being R absorbant, it follows that those terms represent the proportion of individuals in the f and q sub-states, respectively.

3.3.1 Web platform: An evaluation of public policies.

We step forward from the design and fitting of the mathematical model to its implementation in a web platform called MathCOVID. It is a platform open to the public that was used as a reference by the epidemiological units in Colombia for all the locations mentioned in this study. Authorities associated with the control implemented the model in the platform to evaluate the effects of public policies on epidemiological indicators (see S2 Appendix). The link to the platform is available in GitHub repository 2.

In the platform, we implemented the parameters that inherently describe control policies as sliders to allow users to evaluate different strategies to compare with the model forecast given by the current parameters. The changes in the parameters are presented in Fig. S4 Appendix as follows:

• Processes of removing the virus from the environment (ϕ_T), as cleaning surfaces or closed areas.
• The lockdown process by increasing or decreasing the probabilities of going outside or inside quarantine (λ_qf and λ_fq). We could define this process as an intermittent strategy (three days of restricted circulation).
• Connectivity level inside populations (ν), i.e., can simulate the restriction of the availability of public transport between neighborhoods or localities.
• The number of average interactions between individuals (z) as those established in public transport, supermarkets, schools, or at job places.
• The government capacity to track contagion networks (ϑ_E), identify asymptomatic and symptomatic carriers (ϑ_P and η_ϑ), and finally, the likelihood that people with symptoms will self-isolate (self-care, η_L).
• Include migration as an input given by a parameterized time-series that models the addition of new individuals to different compartments, as described in section 2.
• Include vaccination programs as an input given by a parameterized time-series that models the number of vaccines applied to individuals per unit of time. Vaccinated individuals that gained recovered status according to the effectiveness of the vaccine. We refer the reader to section 2 for further reading.

Modeling COVID-19 in this way has allowed us to identify possible bad scenarios and simulate outbreaks of COVID-19 in Colombia and some of its localities in the last two years. We present the simulation and the long-term behavior of the system according to model fitting in Figs 7 and 8 for multiple outbreaks in the country. For example:

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Fig 7. Model simulation using Colombia as example, the blue dotted line shows the limit of model fitting.

After that line we present two scenarios, one in which the simulation follows the same trend and other one in which we change some parameters related to public policies. In scenario (A) we present a change in quarantine behavior (λ_qf from 0.17 to 0.33). In the following figures, we present the original scenario but adding some public policies as (B) intermittent lock-downs as pulse-series, (C) increasing identification programs (λ_fq in 0.33, ϑ_E from 0.11 to 0.15, and ϑ_P from 0.55 to 0.60), (D) reducing the connectivity inside the country (ν from 35 to 150), and (E) a Montecarlo simulation of multiple migration processes hypothesized in November 2021.

https://doi.org/10.1371/journal.pone.0275546.g007

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Fig 8. Model simulation using Colombia as an example.

The blue line shows the limit of model fitting. After that line, we present two scenarios: one in which the simulation follows the same trend and another one in which we include vaccination policies. Here, we designed the distribution of 20000000 vaccines through an effectiveness of 0.99 and a beta distribution with parameters a = 5 and b = 20.

https://doi.org/10.1371/journal.pone.0275546.g008

Scenario (A) is reached by increasing the parameter λ_qf value (from 0.17 to 0.33) to simulate the social dynamics of the Christmas celebration in Colombia in which the people tend to free circulate triggering the fourth outbreak produced by the appearance of the Omicron strain in Latin America [41, 42]. Note that the curve obtained during this simulation meets the trend of real data in Fig 5, a dynamic behavior also validated in S4 Appendix with Montecarlo simulations. In the following figures, we simulate scenarios with a high flux of people alongside some public policies that could decrease the number of cases of the fourth outbreak. In scenario (B), we present the intermittent lock-down strategy described as pulse-series, in which λ_fq and λ_qf are as in S3 Appendix, i.e., people tend to go out every five days and then lockdown.

Scenario (C) in Fig 7, show the simulation of a high flow of people because of Christmas (λ_fq = 0.33) alongside an increase in the identification programs through parameters ϑ_E (from 0.11 to 0.15) and ϑ_P (from 0.55 to 0.60) that represents an increasing of identification programs effort. Following this strategy, we can obtain a smaller outbreak than the one obtained in scenario (A) and even in the real data presented in Fig 5. On the other hand, in scenario (D), we increased the value of ν (from 35 to 150), reducing the movement inside the country, representing a limitation of people from moving to other localities. It has greater impact among the presented scenarios. In scenario (E), we present the case of a considerable migration for a month, in which susceptible people gradually incomes to the country. We can see the effect in the long term, in which detected cases increase considerably after the migration.

We present the vaccination effect in scenario (F) (see Fig 8). We simulated a historical situation of Colombia before implementing mass vaccination (until March 2021) and after it (simulation until July 2021); note that the trend of the simulation with vaccination is similar to the real process carried out in the country during July and August 2021 (see Fig 5).