Local Asymptotic Stability and Sensitivity Analysis of a New Mathematical Epidemic Model Without Immunity

Year 2022, , 50 - 62, 01.03.2022

Sümeyye Çakan

https://doi.org/10.36753/mathenot.935016

Abstract

With this study it is aimed to introduce and analyze a new SIS epidemic model including vaccination
effect. Vaccination considered in the model provides a temporary protection effect and is administered
to both susceptible and new members of the population. The study provides a different aspect to the
SIS models used to express, mathematically, some infectious diseases which are not eradicated by the
immune system. The model given this study is designed by considering varying processes from person
to person in the disease transmission, the recovery from disease (recovery without immunity) and in the
loss of protective effect provided by the vaccine. The processes that change according to individuals are
explained by distributed delays used in the relevant differential equations that provide the transition
between compartments. The differences in the model are especially evident in these parts. In analyzing
the model, firstly, the disease-free and endemic equilibrium points related to the model are determined.
Then, the basic reproduction number R₀ is calculated with the next generation matrix method. Next, the
dynamics about locally asymptotically stable of the model at the disease-free and endemic equilibriums are examined according to the basic reproduction number R₀. Attempts intended to reduce the spread of the disease are, of course, in the direction supporting the lowering the value R0. In this context, the reducing and enhancing effects of the parameters used in the model on the value R₀ have been interpreted mathematically and suggestions were made to implement control measures in this direction. Also, in order to evaluate the support provided by the vaccine during the spread of the disease, the model has been examined as vaccinated and unvaccinated, and by some mathematical process, it has been seen that the vaccination has a crucial effect on disease control by decreasing the basic reproduction number. In other respects, by explored that the effect of parameters related to vaccination on the change of R₀, a result about the minimum vaccination ratio of new members required for the elimination of the disease in the population within the scope of the target of R₀<1 has been obtained.

Keywords

Local Asymptotic Stability, Sensitivity Analysis, SIS model, Vaccine Effect, Disease-free equilibrium point, Endemic equilibrium point, Basic Reproduction Number

References

• [1] Şekerci Fırat, Y.: Climate change forces plankton species to move to get rid of extinction: mathematical modeling approach. Eur. Phys. J. Plus. 135:794, 1-20 (2020).
• [2] Özarslan, R., Şekerci Fırat, Y.: Fractional order oxygen plankton system under climate change, Chaos. 30(3), 33131 (2020).
• [3] Şekerci Fırat, Y., Özarslan, R.: Dynamic analysis of time fractional order oxygen in a plankton system. Eur. Phys. J. Plus. 135(1), 1-13 (2020).
• [4] Daşbaşı, B., Boztosun, D.: Stability analysis of the palomba model in economy by fractional-order differential equations with multi-orders. The Journal of International Social Research. 11 (59), 1-7 (2018).
• [5] Yavuz, M., Sene, N.: Stability analysis and numerical computation of the fractional predator-prey model with the harvesting rate. Fractal Fract. 4(3), 35 1-22 (2020).
• [6] Yavuz, M., Özdemir N.: Analysis of an epidemic spreading model with exponential decay law. Mathematical Sciences and Applications E-Notes. 8(1), 142-154 (2020).
• [7] Yavuz, M., Bonyah E.: New approaches to the fractional dynamics of schistosomiasis disease model. Physica A: Statistical Mechanics and its Applications. 525, 373-393 (2019).
• [8] Kermack, W. O., McKendrick, A. G.: A contributions to the mathematical theory of epidemics. Proc. Roy. Soc. A. 115, 700-721 (1927).
• [9] Çakan, S.: Dynamic analysis of a mathematical model with health care capacity for COVID-19 pandemic. Chaos, Solitons and Fractals. 139, 110033, (2020).
• [10] Aghdaoui, H., Alaoui A. L., Nisar K. S., Tilioua, M.: On analysis and optimal control of a SEIRI epidemic model with general incidence rate, Results in Physics, 20, 103681 1-9 (2021).
• [11] Gölgeli, M., Atay F. M.: Analysis of an epidemic model for transmitted diseases in a group of adults and an extension to two age classes. Hacet. J. Math. Stat. 49 (3), 921-934 (2020).
• [12] Naik, P. A.: Global dynamics of a fractional-order SIR epidemic model with memory. Int J Biomath. 13(8), 2050071 (2020).
• [13] Naik, P. A., Zu, J., Owolabi M.K.: Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order. Physica A. 545, 123816 (2020).
• [14] Naik, P. A., Zu, J., Ghoreishi, M.: Stability analysis and approximate solution of SIR epidemic model with Crowley-Martin type functional response and holling type-II treatment rate by using homotopy analysis method. J Appl Anal Comput. 10(4), 1482-1515 (2020).
• [15] Naik, P. A., Yavuz, M., Qureshi, S., Zu, J., Townley, S.: Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. Eur. Phys. J. Plus. 135(10), 795 (2020).
• [16] Naik, P. A., Yavuz, M., Zu, J.: The role of prostitution on HIV transmission with memory: A modeling approach. Alexandria Eng J. 59(4), 2513-2531 (2020).
• [17] Naik, P. A., Owolabi M. K., Yavuz, M., Zu, J.: Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos Solitons & Fractals. 140, 110272 (2020).
• [18] Yavuz, M., Co¸sar, F. Ö., Günay, F., Özdemir, F. N.: A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign. Open Journal of Modelling and Simulation. 9(3), 299-321 (2021).
• [19] Li, J., Ma, Z.: Qualitative analyses of SIS epidemic model with vaccination and varying total population size. Math. Comput. Model. 35, 1235-1243 (2002).
• [20] Cai, L. M., Li, X. Z.: Analysis of a SEIV epidemic model with a nonlinear incidence rate. Applied Mathematical Modelling. 33 (5), 2919-2926 (2009).
• [21] McCluskey C. C.: Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Anal. RealWorld Appl., 11 (1), 55-59 (2010).
• [22] Elazzouzi, A., Alaoui, A. L., Tilioua, M., Torres, D. F. M.: Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse. Statistics, Optimization and Information Computing, 7(3), 545-557 (2019).
• [23] Elazzouzi, A., Alaoui A. L., Tilioua, M., Tridane, A.: Global stability analysis for a generalized delayed SIR model with vaccination and treatment, Adv. Difference Equ., 532, 1-19 (2019).
• [24] Li, J., Ma, Z.: Global analysis of SIS epidemic models with variable total population size. Math. Comput. Model. 39, 1231-1242 (2004).
• [25] Lakshmikantham, S., Leela, S., Martynyuk, A. A.: Stability analysis of nonlinear systems. Marcel Dekker, Inc., New York (1989).
• [26] Driessche, P. V. D., Watmough J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math.Biosci. 180, 29-48 (2002).