Эпидемическая модель малярии без вакцинации и при ее наличии. Ч. 2. Модель малярии с вакцинацией

Serigne M. Ndiaye; Elena M. Parilina

doi:10.21638/11701/spbu10.2022.410

Authors

Serigne M. Ndiaye St Petersburg State University, 7-9, Universitetskaya nab., St Petersburg, 199034, Russian Federation https://orcid.org/0000-0001-5642-4564
Elena M. Parilina St Petersburg State University, 7-9, Universitetskaya nab., St Petersburg, 199034, Russian Federation https://orcid.org/0000-0003-3976-7180

DOI:

https://doi.org/10.21638/11701/spbu10.2022.410

Abstract

The article proposes a mathematical model of a malaria epidemic with vaccination in a population of people (hosts), where the disease is transmitted by a mosquito (carrier). The malaria transmission model is defined by a system of ordinary differential equations, which takes into account the level of vaccination in the population. The host population at any given time is divided into four subgroups: susceptible, vector-bitten, infected, and recovered. Sufficient conditions for the stability of a disease-free equilibrium and endemic equilibrium are obtained using the theory of Lyapunov functions. Numerical modeling represents the influence of parameters (including the vaccination level of the population) on the disease spread.

Keywords:

epidemic model, malaria, vaccination, SEIR model, endemic equilibrium

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References

Литература

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Aldila D., Seno H. A. Population dynamics model of mosquito-borne disease transmission, focusing on mosquitoes’ biased distribution and mosquito repellent use // Bull. Math. Biol. 2019. Vol. 81. P. 4977-5008.

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References

Sokolov S. V., Sokolova A. L. HIV incidence in Russia: SIR epidemic model-based analysis. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2019, vol. 15, iss. 4, pp. 616-623. https://doi.org/10.21638/11702/spbu10.2019.416

Smith D. L, Battle K. E., Hay S. I., Barker C. M., Scott T. W., McKenzie F. E. Ross, Macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens. PLoS Pathog., 2012, vol. 8, no. 4, Art. no. e1002588. https://doi.org/10.1371/journal.ppat.1002588

MacDonald G. Epidemiological basis of malaria control. Bull. World Health Organ., 1956, vol. 15, no. 3-5, pp. 613-626.

Zakharov V. V., Balykina Yu. E. Prognozirovanie dinamiki epidemii koronavirusa (COVID-19) na osnove metoda precedentov [Predicting the dynamics of the coronavirus (COVID-19) epidemic based on the case-based reasoning approach]. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2020, vol. 16, iss. 3, pp. 249-259. https://doi.org/10.21638/11701/spbu10.2020.303 (In Russian)

Feng Z., Hernandez V. Competitive exclusion in a vector-host model for the dengue fever. Math. Biol., 1997, pp. 523-544.

Qiu Z. Dynamical behavior of a vector-host epidemic model with demographic structure. Computers & Mathematics with Applications, 2008, vol. 56, no. 12, pp. 3118-3129.

Sirbu A., Lorento V., Servedio V., Tria F. Opinion dynamics: models, extensions and external effects. Physics and Society, 2016, vol. 5, pp. 363-401.

Bushman M., Antia R., Udhayakumar V., de Roode J. C. Within-host competition can delay evolution of drug resistance in malaria. PLoS Biol., 2018, vol. 16, no. 8, Art. no. e2005712. https://doi.org/10.1371/journal.pbio.2005712

Turner A., Jung C., Tan P., Gotika S., Mago V. A comprehensive model of spread of malaria in humans and mosquitos. SoutheastCon, 2015, pp. 1-6. https://doi.org/10.1109/SECON.2015.7132968

Hong H., Wang N., Yang J. Implications of stochastic transmission rates for managing pandemic risks. Rev. Financ. Stud., 2021, Feb. 9, no. hhaa132. https://doi.org/10.1093/rfs/hhaa132

Gomez-Hernandez E. A., Ibarguen-Mondragon E. A two patch model for the population dynamics of mosquito-borne diseases. J. Phys.: Conference Series, 2019, vol. 1408, no. 1, Art. no. 012002.

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Cai L., Tuncer N., Martcheva M. How does within-host dynamics affect population-level dynamics? Insights from an immuno-epidemiological model of malaria. Mathematical Methods in the Applied Sciences, 2017, vol. 40, no. 18, pp. 6424-6450.

Maliki O., Romanus N., Onyemegbulem B. A mathematical modelling of the effect of treatment in the control of malaria in a population with infected immigrants. Applied Mathematics, 2018, vol. 9, pp. 1238-1257.

Baygents G., Bani-Yaghoub M. A mathematical model to analyze spread of hemorrhagic disease in white-tailed deer population. Journal of Applied Mathematics and Physics, 2017, vol. 5, no. 11, pp. 2262-2282.

Wiwanitkit V. Unusual mode of transmission of dengue. Journal of Infect. Dev. Ctries, 2009, vol. 4, pp. 051-054.

Ndiaye S. M., Parilina E. M. Epidemicheskaja model' maljarii bez vakcinacii i pri ee nalichii. Ch. 1 Model' maliarii bez vaktsinatsii [An epidemic model of malaria without and with vaccination. Pt 1. A model of malaria without vaccination]. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2022, vol. 18, iss. 2, pp. 263-277. https://doi.org/10.21638/11701/spbu10.2022.207 (In Russian)

Ndiaye S. M. Modelisation dun systeme de pecherie avec maladie [Modeling a fishery system with disease]. Bachelor Thesis, supervise par M. Lam, F. Mansal. Dakar, Universite Cheikh Anta Diop de Dakar Press, 2017, pp. 3-10.

Aldila D., Seno H. A. Population dynamics model of mosquito-borne disease transmission, focusing on mosquitoesТ biased distribution and mosquito repellent use. Bull. Math. Biol., 2019, vol. 81, pp. 4977-5008.

Lipsitch M., Cohen T., Cooper B., Robins J. M., Ma S., James L., Gopalakrishna G., Chew S. K., Tan C. C., Samore M. H., Fisman D., Murray M. Transmission dynamics and control of severe acute respiratory syndrome. Science, 2003, vol. 300, pp. 1966-1970.

Britton T. Stochastic epidemic models: A survey. Mathematical Biosciences, 2010, vol. 225, iss. 1, pp. 24-35.

Diekmann O., Heesterbeek A. P., Roberts M. G. The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface, 2010, vol. 7, pp. 873-885.

Van den Driessche P. Reproduction numbers of infectious disease models. Infect. Dis. Model., 2017, vol. 2, pp. 288-303.

Jones J. H. Notes on R0. Department of Anthropological Sciences. Stanford, CA, USA, 2007, vol. 323, pp. 1-19.