Constant and pulse vaccination research and comparative analysis

Cover Page

Cite item

Full Text

Abstract

In this paper, a study of various vaccination strategies is carried out for a specific model of an infectious disease, through its numerical study and qualitative analysis of the data obtained. As models of an infectious disease, one of the most common classes of models used in various fields of medicine, compartmental models, is used. In the course of the work, constant and pulse vaccination strategies are compared. These strategies are compared in terms of the proportion of the population infected, as well as the material costs of the vaccination campaign. The results showed that the best equilibrium position is observed in the case of constant vaccination. However, in the application of pulse vaccination strategy has fewer outbreaks of high amplitude disease and is more cost effective.

Full Text

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.

×

About the authors

Stanislav D. Chernyshev

Samara National Research University

Author for correspondence.
Email: chernyshev.st17@gmail.com

Student 4 course of Samara University informatic faculty

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russia

Natalia A. Slobozhanina

Samara National Research University

Email: slobogeanina@mail.ru

Candidate of Philological Sciences, Associate Professor of Samara University Department of Foreign Languages and Russian as a Foreign Language

34, Moskovskoye shosse, Samara, 443086, Russi

References

  1. Pretes D.B., Jardim C.L.T.F., Ferreira L.A.F., Silva J.M., Kritz M.V. Vaccination Strategies: a comparative study in an epidemic scenario // Journal of Physics: Conference Series. – 2016. – Vol. 738(1). – 012083. - doi: 10.1088/1742-6596/738/1/012083.
  2. Alonso-Quesada S., De la Sen M., Nistal R. On vaccination strategies for a SIR epidemic model guaranteeing the nonexistence of endemic solutions // Discrete Dynamics in Nature and Society. –2018. – Vol. 2018. – 9484121. - doi: 10.1155/2018/9484121.
  3. Owuor O.N., Johannes M., Kibet M.S. Optimal vaccination strategies in an SIR epidemic model with time scales // Applied Mathematics. – 2013. – Vol. 4(10B). – P. 1-14. - doi: 10.4236/am.2013.410A2001.
  4. Doutor P., Rodrigues P., Soares M.D., Chalub F.A. Optimal vaccination strategies and rational behaviour in seasonal epidemics // J Math Biol. - 2016. – Vol. 73(6-7). – doi: 10.1007/s00285-016-0997-1.
  5. Libotte G.B., Lobato F.S., Platt G.M., Neto A.J.S. Determination of an optimal control strategy for vaccine administration in COVID-19 pandemic treatment // Comput Methods Programs Biomed. – 2020. - Vol. 196. - 105664. doi: 10.1016/j.cmpb.2020.105664.
  6. Costantino V., Kunasekaran M., MacIntyre C.R. Modelling of optimal vaccination strategies in response to a bioterrorism associated smallpox outbreak // Hum Vaccin Immunother. - 2021. - Vol. 17(3). P. 738-746. - doi: 10.1080/21645515.2020.1800324.

Supplementary files

Supplementary Files
Action
1. JATS XML

Copyright (c) 2023 Proceedings of young scientists and specialists of the Samara University

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Proceedings of young scientists and specialists of the Samara University

ISSN 2782-2982 (Online)

Publisher and founder of the online media, journal: Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russian Federation.

The online media is registered by the Federal Service for Supervision of Communications, Information Technology and Mass Communications, registration number EL No. FS 77-86495 dated December 29, 2023

Extract from the register of registered media

Regulation of the online media

Editor-in-chief: Andrey B. Prokof'yev, Doctor of Science (Engineering), associate professor,
head of the Department of Aircraft Engine Theory

2 issues a year

0+. Free price. 

Editorial address: building 22a, room 513, Soviet of Young Scientists and Specialists, 1, Academician Pavlov Street, Samara, 443011, Russian Federation.

Address for correspondence: room 513, building 22a, 34, Moskovskoye shosse, Samara, 443086, Russian Federation.

Tel.: (846) 334-54-43

e-mail: smuissu@ssau.ru

Domain name: VMUIS.RU (Domain ownership certificate), Internet email address: https://vmuis.ru/smus.

The previous certificate is a printed media, the journal “Bulletin of Young Scientists and Specialists of Samara University”, registered by the Office of the Federal Service for Supervision of Communications, Information Technologies and Mass Communications in the Samara Region, registration number series PI No. TU63-00921 dated December 27, 2017.

© Samara University

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies