# Successive Pandemic Waves with Different Virulent Strains and the Effects of Vaccination for SARS-CoV-2

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## Abstract

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## 1. Introduction

- Social distancing: the obvious way to reduce susceptible-infected interaction and subsequent contagion;
- Mask wearing and hygiene: this was implemented once it became known that transmission is mainly through respiratory droplets of infected patients and contact with surfaces infected by aerosol;
- Vaccines: a correct vaccination program can decrease overall transmission and the intensity of the disease symptoms among those infected and vaccinated, reducing the public health collapse risk and the mortality rates, as susceptible but vaccinated people become asymptomatic. Still, the virus will circulate, and the lack of a proper vaccination will create outbreaks due to contact between an increasing number of “asymptomatic” people with susceptible people. As [5] demonstrated, the existence of transient collective immunity may prolong an epidemic, and a bad vaccine scheme may exacerbate this pattern.

## 2. Materials and Methods

#### The Model

_{i}variable defines the health state of each individual: susceptible (S

_{i}= 0), infected (S

_{i}= 1), and recovered or immunized (S

_{i}= 2). In this work, we have included a fourth state: dead (S

_{i}= 3). Factors such as age, sex, or race are not considered.

_{mov}, and this probability defines the social isolation mobility.

_{i}= N

_{0}, that is, the city allows all of its residents to leave at the same time. There are several public areas with different K

_{i}= {10, 000, 1, 000, 100, 50}, so that the sum of the capacity of each area of the same type is equal to 0.25 N

_{0}. The number of places of a specific kind i is the total number of individuals they can support divided by its K

_{i}. Thus, the number of sites that support fewer people is higher than those which support more people: in other words, there are more smaller stores than big stadiums. When individuals can move to other people’s houses, we define a carrying capacity of 12 individuals.

_{mov}. Each individual may visit other places a maximum of three times during the day. If a person is going to move, the places to which they will go are chosen at random: houses, small shops, or large stores. Its maximum capacity gives the likelihood of going to a location: they are more likely to go to a large store than to someone else’s home. If a selected place reaches its capacity, a new site is drawn until the person moves, ensuring that whoever was chosen to move will make a move. COVID-19 has a high transmissibility during the pre-illness period, and the model assumed seven days of transmission before causing any disease, which was estimated by the pandemic for previous variants and the Delta variant. The mobility defined in the model emphasizes the transmissibility by pre-symptomatic and asymptomatic individuals, as no constraint on the movement of an infected person is imposed until someone dies [34,35,36].

_{infected}is the total number of infected persons at that place (house or mall) and N

_{max}is the maximum capacity of that place.

_{i}× β

_{i}, the higher the probability of being infected, where n

_{i}is the total number of individuals infected by the strain β

_{i}at that location. The infected individual then becomes a potential transmitter of the disease.

_{i}= 1 or 2), nothing happens to those who are already at that place. The arrival of an infected person will create the conditions for those who arrive later to be infected. Hence, there is an asymmetry in the model. If an infected person arrives at a place, we do not test if they will infect the susceptible ones already there. This kind of procedure corresponds to a sequential order in the contact/contagion process. It is understood that for huge populations and many days of simulation, the results will not be different from those here reported.

_{i}that can be in four states: susceptible (0); infected (1); recovered or immunized (2); and dead (3). The entire population starts the simulation as susceptible: S

_{i}= 0. A single person among the residents is infected at t = 0 with the less lethal variant β = 0.2. The dynamics is given by the process shown in Figure 1.

_{i}= 0 can become infected S

_{i}= 1 with the probability given by P

_{contact}× β

_{i}. β

_{i}is the transmission rate of an individual which contaminates a susceptible individual, as described above. An infected individual S

_{i}= 1 can die S

_{i}= 3 if their time of infection is greater than 7 days and with probability of death given by P

_{death}. In case of contamination with the most lethal variant β + δβ, the value of P

_{death}is increased by δ

_{death}. This antagonism creates a tension between the strain’s lethality and the individual’s probability of death. That is, hosts with a more lethal variant are more likely to die.

_{i}= 1, either recovers or is immunized, S

_{i}= 2. After T days, the immunized individual, S

_{i}= 2, returns to the susceptible condition, S

_{i}= 0. In the case of a vaccination program in that community, a susceptible or infected individual, either S

_{i}= 0 or S

_{i}= 1, can be immunized with probability P

_{vac}. This probability is related to the vaccination rate, which is the percentage of the total population that is vaccinated every day after the campaign started. The status of a vaccinated individual becomes S

_{i}= 2. Similar to the recovered one, the immunization protects an individual for T days. During this period, they cannot be infected with any variant of the new coronavirus [38,39].

_{vac}= 1/200, representing the typical value of vaccination campaigns in Brazil, when close to 1 million people are vaccinated per day. Brazil has a public health system formed by about 40 thousand health centers, belonging to three levels of administration, but forming a cooperative network. This system has vast experience on vaccination campaigns and can easily reach a rate of 1 million vaccinations per day, roughly 1/200. The second value we have tested is a rate of 1/1000, which represents the values that we have observed for the vaccination against COVID-19 in the first two months of vaccination, close to 200 to 300 thousand people per day. From 23 January 2021 to 29 March 2021, 14 million people received the first dose of one of the two available vaccines. In Brazil, the vaccines that were used were those that required two doses. The vaccination rate increased in April but then oscillated, since Brazil had no plan of vaccine acquisition. In the USA, the vaccination rate reached 0.006 of the whole population in March. By the beginning of March, Israel had already vaccinated practically its entire population of around 9 million people.

## 3. Results

#### 3.1. Model with Two Strains

_{mov}= 0.6, 0.5, or 0.3 were adopted, meaning an average isolation of 40%, 50%, and 70%. In the case of Brazil, the average isolation, measured while individuals stay in their homes for a day, has fluctuated between 30% to 40%, and on weekends, it can reach 50%. It increased a little in February and March 2021 due to the imposition of new measures of restriction and operation of the establishments. At the beginning of the pandemic, in March 2020, there were higher values in Brazil, in the range of 60%.

_{mov}= 0.1, representing the people who were most often at home. As described above, visitors would come and go. We assumed a proportion of visitors of 0.1% of the total initial population per day. The people who arrived and left remained in the city for only one day. A ratio of 1/1000 may be contaminated. The contamination strain had β = 0.2. In total, 1/90 of the contaminated individuals had a more lethal variant, with β = 0.25. Visitors had a probability of movement equal to 1.0 and always visited three locations (drawn at random, as described above).

_{contact}is given by Equation (1), β = 0.2, δβ = 0.05, P

_{death}= 0.01, δ

_{death}= 0.004, 1/γ = 14, and T = 120 days.

_{mov}= 0.6, without vaccination. The values represent the densities relative to the initial number of residents, in this case, 874,975 people. There is an oscillating evolution of the numbers caused by the ongoing entry of infected individuals: the visitors. The curve called β

_{av}represents the average value of β taken among all individuals with S = 1. It is observed that in the peaks of infection, the most lethal variant tends to dominate. The decrease in the infected population is typical of the SIR model, but the permanence of the most lethal variant is an important characteristic. The number of individuals who died is in the order of 10% of the original population. Certainly, this is a very high number when compared to real data. This model and the simulations do not intend to project expected values, but they represent the dynamics of disseminating the most lethal variant qualitatively.

_{mov}= 0.3. In this case, the susceptible population remains basically between 0.95 and 1, the density to the original number of residents. Infected people cannot contaminate significant fractions of the population, and the infection stops.

_{av}is closer to 0.2. It is important to remember that the rate of contaminated visitors is the same in the three cases discussed so far. The number of deaths also stabilizes with the blockade caused by vaccination.

_{mov}= 0.6; β = 0.2 for residents; number of visitors, 1/1000; contaminated visitors, 1/1000; contaminated visitors with the highest transmissibility rate, 1/90. Vaccination, when it occurs, starts on the 300th day.

#### 3.2. Model with Many Strains

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

^{5}individuals. Only the proportion of new dead individuals is shown, represented by the blue line. The proportion of those who died by the most lethal variant is represented by the red line; Figure S2: ABM-SIR dynamic of COVID-19 evolution over 1000 days with 70% of social isolation, adjusted for the initial population of ∼8.75 × 10

^{5}individuals, without vaccination. Each color represents a population group, defined in the legend. The value of β

_{av}is the average for all infected people; Figure S3: ABM-SIR dynamic of COVID-19 evolution over 1000 days with 40% of social isolation, adjusted for the initial population of ∼8.75 × 10

^{5}individuals. Only the proportion of new dead individuals is shown, represented by the blue line. The proportion of those who died by the most lethal variant is represented by the red line. The vaccination rate is 1/200 of susceptible or recovered individuals per day; Figure S4: ABM-SIR dynamic of COVID-19 evolution over 1000 days, adjusted for the initial population of 1 M, with variants entering via visitors. The social distancing is 45%. The values of β and mortality for infected individuals can be seen in Table 1. In this scenario, all recovered individuals lost their immunity after 120 days; Code 1: Code-C++; Code 2: Code-Fortran95.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation of transitions between states in our model. Probabilities are discussed in the text.

**Figure 2.**ABM-SIR dynamic of COVID-19 (i.e., Susceptible-Infected-Recovered Agent-Based model) evolution over 1000 days with a social isolation of 40%, adjusted for the initial population of ∼8.75 × 10

^{5}individuals. There is no vaccination. Each color represents a population group, defined in the legend. The value of β

_{av}is the average for all infected people.

**Figure 3.**ABM-SIR dynamic of COVID-19 (i.e., Susceptible-Infected-Recovered Agent-Based model) evolution over 1000 days for social isolation of 40%, adjusted for the initial population of ∼8.75 × 10

^{5}individuals. Each color represents a population group, defined in the legend. The value of β

_{av}is the average for all infected people. The vaccination rate is 1/1000 of susceptible or recovered individuals per day.

**Figure 4.**ABM-SIR dynamic of COVID-19 (i.e., Susceptible-Infected-Recovered Agent-Based model) evolution over 1000 days for social isolation of 40%, adjusted for the initial population of ∼8.75 × 10

^{5}individuals. Each color represents a population group, defined in the legend. The value of β

_{av}is the average for all infected people. The vaccination rate is 1/200 of susceptible or recovered individuals per day.

**Figure 5.**Real data for new cases (blue line) and new vaccinations (red line) for Israel (left plot) and Portugal (right plot). All curves are standardized per million and are not cumulative.

**Figure 6.**Presence of virus in population, same scenario of Figure S4. This plot show how rapidly infection caused by variants 2 and 3 overcome the original one from the second wave.

Variant | β | Lethality (%) |
---|---|---|

1 (original) | 0.2 | 1.0 |

2 | 1.7 × 0.2 | 1.5 |

3 | 1.7 × 0.2 | 0.5 |

4 | 0.7 × 0.2 | 1.5 |

5 | 0.7 × 0.2 | 0.5 |

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**MDPI and ACS Style**

Castro e Silva, A.; Bernardes, A.T.; Barbosa, E.A.G.; Chagas, I.A.S.d.; Dáttilo, W.; Reis, A.B.; Ribeiro, S.P. Successive Pandemic Waves with Different Virulent Strains and the Effects of Vaccination for SARS-CoV-2. *Vaccines* **2022**, *10*, 343.
https://doi.org/10.3390/vaccines10030343

**AMA Style**

Castro e Silva A, Bernardes AT, Barbosa EAG, Chagas IASd, Dáttilo W, Reis AB, Ribeiro SP. Successive Pandemic Waves with Different Virulent Strains and the Effects of Vaccination for SARS-CoV-2. *Vaccines*. 2022; 10(3):343.
https://doi.org/10.3390/vaccines10030343

**Chicago/Turabian Style**

Castro e Silva, Alcides, Américo Tristão Bernardes, Eduardo Augusto Gonçalves Barbosa, Igor Aparecido Santana das Chagas, Wesley Dáttilo, Alexandre Barbosa Reis, and Sérvio Pontes Ribeiro. 2022. "Successive Pandemic Waves with Different Virulent Strains and the Effects of Vaccination for SARS-CoV-2" *Vaccines* 10, no. 3: 343.
https://doi.org/10.3390/vaccines10030343