A study of stability of SEIHR model of infectious disease transmission. (English) Zbl 1490.34053
Summary: We develop in this paper a Susceptible Exposed Infectious Hospitalized and Recovered (SEIHR), spread model. In the model studied, we introduce a recruitment constant, to take into account the fact that newborns can transmit disease. The disease-free and endemic equilibrium points are computed and analyzed. The basic reproduction number \(\mathcal{R}_0\) is acquired, when \(\mathcal{R}_0\leq 1\), the disease dies out and persists in the community whenever \(\mathcal{R}_0 > 1\). From numerical simulation, we illustrate our theoretical analysis.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 92D30 | Epidemiology |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34D23 | Global stability of solutions to ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
Keywords:
compartmental modeling; recruitment; infectious disease; reproduction number; equilibria; stability analysis; numerical simulationSoftware:
Be-CoDiSReferences:
| [1] | I. Ali and S. K. Ullah, Analysis of stochastic delayed SIRS model with exponential birth and saturated incidence rate, Chaos Solitions Fractal, (2020). · Zbl 1490.92064 |
| [2] | H. Amann, Ordinary differential equations: An introduction to nonlinear analysis, Walter de Gruyter, Berlin. New York (1990). · Zbl 0708.34002 |
| [3] | R.M. Anderson and R.M. May, Infectious Diseases of Humans. Dynamics and Control, Oxford Science Publications, (1991). |
| [4] | G. Birkhoff and G. C. Rota, Ordinary Differential Equations: Ginn, Boston, (1982). |
| [5] | F. Brauer, Mathematical epidemiology: Past, present, and future, Infectious Disease Modelling, 2, (2017) pp.113-127. |
| [6] | O. Diekmann, J.A.P. Heesterbeek and J.A.J. Metz: On the definition and the computation of the basic reproduction ratio R_0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28, (1990) pp.365-382. · Zbl 0726.92018 |
| [7] | W. H. Fleming and R. W. Rishel, Deterministic and stochastic Optimal control, Springer, New York, NY, USA, (1975). · Zbl 0323.49001 |
| [8] | A. Guiro, B. Kone and S. Ouaro, Mathematical Model of the Spread of the Coronavirus Disease 2019 (COVID-19) in Burkina Faso, Applied Mathematics, 11, (2020), pp. 1204-1218. |
| [9] | A. Guiro, D. Ouedraogo, H. Ouedraogo, Stability Analysis for a Discrete SIR Epidemic Model with Delay and General Nonlinear Incidence Function, Applied Mathematics,9 (2018), pp. 1039-1054. · Zbl 1442.49057 |
| [10] | B. Ivorra, D. Ngom and A. M. Ramos, Be-CoDis: A Mathematical Model to Predict the Risk of Human Diseases Spread Between Countries Validation and Application to the 2014-2015 Ebola Virus Disease Epidemic, Bulletin of Mathematical Biology, 17, 9 (2015) pp.1668-1704. · Zbl 1339.92085 |
| [11] | O. W. Kermack and G. A. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A. Contain Paper Math Phys Charact, (1927). · JFM 53.0517.01 |
| [12] | R. Kiran, M. Roy, S. Abbas and A. Taraphder, Effect of population migration and punctuated lockdown on the spread of infectious diseases, arXiv:2006.15010v2, (2021). · Zbl 1477.92024 |
| [13] | C. C. Lai, et al. severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and corona virus disease-2019 (COVID-19): the epidemic and the challenges, Int. J. Antimicrob. Agents, 55,3 (2020), pp.924-934. |
| [14] | V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York, (1989). · Zbl 0676.34003 |
| [15] | J. P. Lasalle, The stability of Dynamical Systems, SIAM and Philadelphia, (1976). · Zbl 0364.93002 |
| [16] | T. Liu, J. Kang, L. Lin, H. Zhong and J. Xiao, Transmission Dynamics of 2019 Novel Coronavirus (2019-nCoV), (2020). |
| [17] | D. L. Lukes, Differential Equations: Classical To Controlled, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 162 (1982). · Zbl 0509.34003 |
| [18] | D. K. Mamo, Model the transmission dynamics of COVID-19 propagation with public health intervention, Applied Mathematics, (2020). · Zbl 1468.34070 |
| [19] | R. Ross, The Prevention of Malaria, John Murray, (1911). |
| [20] | B. Tang, et al. Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9, 2, (2020), pp.462-474. |
| [21] | S. Tyagi, S. Gupta, S. Abbas, K. P. Das and B. Riadh, Analysis of infectious disease transmission and prediction through SEIQR epidemic model, Nonauto. Dyn. Sys., (2021), no.1, 75-86. · Zbl 1466.92209 |
| [22] | S. Tyagi, S. C. Martha, S. Abbas and A. Debbouche, Mathematical modeling and analysis for controlling the spread of infectious diseases, Chaos, Solitions and Fractals, 144 (2021), Paper No.110707. · Zbl 1498.92007 |
| [23] | P. Van den Driessche, J. Watmough, Reproduction numbers and subthreshold endemic equilibria for the compartmental models of disease transmission, Mathematical Biosciences,180 (2002) pp. 29-48. · Zbl 1015.92036 |
| [24] | R. Varga, matrix iterative analysis, Prentice-Hall, (1962). · Zbl 0133.08602 |
| [25] | J. T. Wu, K. Leung and G. M. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modeling study, Lancet, 395, (2020) pp.261-269. |
| [26] | Y. Xia, C. Lansun and C. Jufang, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers Mathematics with Applications, 32, 4 (1996) pp.109-116. · Zbl 0873.34061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
