Исследование и сравнительный анализ стратегий постоянной и импульсной вакцинации

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Introduction

Mathematical models are essential tools that allow scientific and descriptive analysis, since their versatility admit the obtainment of several valuable results for a better comprehension of natural phenomena. Amongst a great number of matters studied in the mathematical modelling area, the epidemiology is one of big importance, since it may bring very useful results and is able to aid public health agencies to better deal with epidemics [1].

In this work, a relevant problem related to the spread of infectious diseases will be considered. This topic was chosen due to controversial issues about the relevance of COVID-19 vaccinations. One of the ways to stop the spread of infections is to vaccinate a healthy part of the population for creating immunity to the disease.

There are several vaccination strategies: constant vaccination, in which a part of the population is vaccinated at birth, and vaccination in pulses, in which a certain part of the susceptible population is vaccinated at certain intervals [2]. The purpose of this work is to consider both of these cases, and also to carry out their comparative analysis.

 

Compartmental models in epidemiology

This paper will consider the traditional SIR model, modified to take into account the impact of a vaccination campaign. Populations are assigned compartments with "labels" such as S, I, and R. Members of the population can move between compartments. The order of the "labels" usually shows the flow patterns between compartments. The processes between the compartments are described by some mathematical model [3]. Examples of such patterns are shown in the (fig 1).

Fig. 1. Movement between compartments in different compartment models.

 

The model has been used in this work differs from the traditional one in that a certain part of the population ρ is vaccinated immediately after their birth, which prevents the possibility of infection. Then this part of the population is introduced into the group of recovered. The equations describing the SIR model in this particular vaccination situation become:

Where

α is recovery intensity rate infected individuals ;

β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, ;

death and birth rate ;

.

Using the fact that the number of recovered people can be found by subtracting the numbers S and I from the total number of members of the population, the system (1) is simplified as follows [4]:

The purpose of this work is to study the system (2) by methods of qualitative and numerical analysis. Based on its research, it is necessary to determine which vaccination strategy is more beneficial.

 

Results and discussions

To begin with, the results obtained in the study of the strategy of constant vaccination will be reviewed. In the article [5] the average statistical values of the parameters of the system under study are given for a certain disease. They are used for numerical analysis. For given values of the system parameters, the considered fixed point will be a stable focus (fig. 2).

Fig. 2. Phase trace of system (2) near point P.

 

By approaching the stable fixed point closer to the Susceptible axis on the phase portrait, it is possible to reduce the proportion of the infected population against the total population. The position of the fixed point in the phase portrait can be changed by varying the system parameters. The study shows that with an increase in the vaccination parameter ρ, the value along the Infected axis in the phase portrait will decrease. This also can be judged from the meaning of the vaccination parameter: the more of the population is vaccinated against the disease, the less is the number of infected individuals. Further, this is verified by plotting the relation of the number of infected individuals versus time for different vaccination parameter.

(Fig. 3) shows the behavior of infected people in a population with smaller vaccination parameters ρ presents more peaks of infection and tends to an equilibrium point with more infected people compared to the behavior of the same population in which a larger proportion of people have been vaccinated.

Fig. 3. Dependence of the number of infected individuals on time for various values of the vaccination parameter .

 

To conduct a numerical study with the vaccination in pulses, the vaccination coefficient of the system (2) was replaced as follows:

Then the system will be considered in this form:

For the given values of the parameters, the phase track of the fixed point passes into the limit cycle, this can be seen in (fig. 4).

Fig. 4. Phase trace of system (2) near point P.

 

(Fig. 5) shows a graph of the dependence of the number of infected individuals on time, when the phase trajectory has just begun to pass into the limit cycle. Here one can notice multimode oscillations typical for the limit cycle: the oscillatory process has several main amplitudes, which decay exponentially from above and below [6]. Starting from the moment of transition of the phase trajectory to the limit cycle, this attenuation becomes insignificant, and the number of infected members of the population can be considered to be periodically time dependent.

Fig. 5. Dependence of the number of infected individuals on time with pulse vaccination.

 

(Fig. 6) shows a graph that shows all the cases of vaccination considered earlier: a case free from vaccination, cases of constant and pulse vaccination.

Fig. 6. Comparison of graphs of the dependence of the number of infected individuals on time for constant and pulse vaccination.

 

The most rapid reduction in the number of infected members of the population to a certain level occurs with vaccination in pulses, but with constant vaccination, the number of infected most of the time is so small that it is almost zero. In the case of vaccination in pulses, there are no significant outbreaks of infection, while the case of constant vaccination is due to a large number of peaks of the disease. The equilibrium position to which the number of diseased individuals tends at large t is the largest for the case free from vaccination. The lower value is for the case of vaccination in pulses and the smallest is for the case of constant vaccination.

From the point of view of material costs for vaccination, the strategy of vaccination in pulses is more preferable, since less vaccine is consumed compared to the case of constant vaccination.

 

Conclusion

In this paper, cases were considered in which constant and vaccination in pulses of the population is carried out in order to control the spread of a certain disease in a certain population.

A pulse vaccination strategy is more cost-effective and also results in fewer outbreaks of high amplitude disease. However, the best equilibrium position is observed in the case of constant vaccination.

Об авторах

Станислав Дмитриевич Чернышев

Самарский национальный исследовательский университет имени академика С.П. Королева

Автор, ответственный за переписку.
Email: chernyshev.st17@gmail.com

Cтудент 4 курса факультета информатики Самарского университета

Россия, ул. Московское шоссе, д. 34, г. Самара, 443086

Наталья Александровна Слобожанина

Самарский национальный исследовательский университет имени академика С.П. Королева

Email: slobogeanina@mail.ru

Кандидат филологических наук, доцент кафедры ИНЯЗ и РКИ Самарского университета

ул. Московское шоссе, д. 34, г. Самара, 443086

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