Submit your papersSubmit Now
For Enquiries: [email protected]
IIARD LogoIIARD

Mathematical Modelling of Epidemiological Infectious Disease Transmission Dynamics: Public Health Intervention Strategies

Christiana Nkuturum

Abstract

This study explores the epidemiological infectious disease transmission dynamics with public health intervention strategies using nonlinear ordinary differential equations with four compartments in a mathematical modeling approach. The study identified two major public intervention strategies as preventive and treatment measures. The findings of this study are that the region D of the subset is a positively-invariant set mathematically and epidemiologically from the model equation (1) as a result of the two intervention measures as t→ . The negative eigenvalues depict that the DFE of the system is locally asymptotically stable for Re< 1. Re, is the effective reproduction number of the STIs/STDs where public health preventive intervention strategies such as outreach campaign programmes and treatment for the control already exist leads the disease to annihilation with respect to time. This study also investigated varying conditions for the existence of endemic equilibrium: no endemic equilibria when ψ= ρ= 0 and Re< 1; for Re> 1 and ψ= ρ= 0 there is endemic equilibrium if Re< 1. ρ= ψ= 0 leads to the backward bifurcation of the system at Re= 1; there is a Hopf bifurcation about the positive equilibriumEe implies that prevention and treatment are not equal when r= r∗; with ρ= 0 and ψ> 0 the system is globally asymptotically stable in the region for Re> 1; ρ> 0 and ψ= 0 shows no delay in preventive measures but there is delay in the treatment strategies; that preventive intervention strategies is greater than treatment for the infectious disease and gives birth to a backward bifurcation in the model system at Re= 1; R0is the basic reproduction number of the model replicated itself in terms of Rρ as the effective reproduction number in the presence of preventive approach and Rψis the effective reproduction number in the presence of treatment control this means that the DFE is GAS in the region if Re< 1.

Keywords

Epidemiological disease; Transmission dynamics; Mathematical modelling; Public health intervention strategies.

References

[1] Bai, Y.& Mu, X. (2018). Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible, J. of Applied Analysis and Computation, 8, 402-412. doi: 10.11948/2018.402 [2] Brauer, F., van den Driessche, P., Wu, J. . (2008) Mathematical Epidemiology, Lecture Notes in Mathematics. Mathematical Biosciences Subseries, Vol. 1945, Springer, Berlin, Heidelberg. [3] Brauer, F., Castillo-Chavez, C., & Feng, Z. (2019). Mathematical models in epidemiology. Springer. [4] Buonomo B. and Lacitignola D. (2011) On the backward bifurcation of a vaccination model With nonlinear incidence, Nonlinear Analysis: Modelling and control, Vol. 16, No. 1, pp. 30-46. [5] Diekmann, O., Heesterbeek, J. A. P. & Roberts, M. G. (2010). The construction of next- generation matrices for compartmental epidemic models, J. R. Soc., Interface, 7, 873- 885. doi: 10.1098/rsif.2009.0386 [6] Diekmann, O., Heesterbeek, J. A. P., & Britton, T. (2013). Mathematical tools for understanding infectious disease dynamics. Princeton University Press. [7] Dubey, B., Dubey, P. and Dubey, U. S. (2016) Dynamics of an SIR Model with Nonlinear Incidence and Treatment Rate. Applications and Applied Mathematics: An International Journal, Vol. 11, iss. 1, pp. 97 – 116. [8] Elaiw, A. M. and Azoz, S. A. (2013) Global properties of a class of HIV Infection Models with Beddington-DeAngelis Functional Response. Mathematical Methods in the Applied Science, 36, 383 – 394. [9] Feng, X., Teng, Z., Wang, K. and Zhang, F. (2014) Backward Bifurcation and Global Stability in an Epidemic Model with Treatment and Vaccination. Discrete and Continuous Dynamical Systems Series B, Vol. 19, No. 4, pp. 999 – 1025. [10] Funk, S., Salathé, M., & Jansen, V. A. A. (2010). Modelling the influence of human behaviour on the spread of infectious diseases: A review. Journal of the Royal Society Interface, 7(50), 1247–1256. [11] Gao Y., Zhang W., Liu ., and Xiao Y. (2017) Bifurcation Analysis of an SIRS Epidemic Model with Saturated Incidence Rate and Saturated Treatment Function. Journal of Applied Analysis and Computation, Vol. 7( 3), pp.1070 – 1094. [12] Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599– 653. [13] Hu, Z. X., Ma, W. B. & Ruan, S. (2012). Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012), 12- 20. doi: 10.1016/j.mbs.2012.03.010 [14] Hu, Z., Ma, W. B. & Ruan, S. (2011). Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dynam. Syst. Ser. B, 15 , 93- 112. doi: 10.3934/dcdsb.2011.15.93. [15] Huang, G., Ma, W. and Takeuchi, Y. (2011) Global Analysis for Delay Virus Dynamics Model with Beddington-DeAngelis Functional Response. Applied Mathematics Letters, Vol. 24, 1199 – 1203. [16] Kaddar, A. (2010) Stability Analysis in a delayed SIR epidemic Model with a Saturated Incidence Rate”. Nonlinear Analysis Modelling and Control, Vol. 15, pp. 299 – 306. IJCSMT [17] Keeling, M. J., & Grenfell, B. T. (2002). Understanding the persistence of measles: Modelling the interplay of epidemiology and demography. Proceedings of the Royal Society B, 269(1489), 335–343. [18] LaSalle, J., Lefschez, S. (1976) The Stability of Dynamic Systems. SIAM, Philadelphia. [19] Li, G-H., Zhang, Y-X. (2017). Dynamic behaviours of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates. PLoS ONE 12(4): 30175789. [20] Li, G.H. & Zhang, Y. X. (2017). Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, Plos One, 2017, 1- 28. doi: 10.1371/journal.pone.0175789. [21] Lloyd, A. L., & May, R. M. (2001). How viruses spread among hosts. Science Journal, 292(5519), 1316–1317. [22] Lin, Y. G., Jiang, D. Q. & M. L. Jin, (2015). Stationary distribution of a stochastic SIR model with saturated incidence and its asymptotic stability, Acta Mathematica Scientia, 35 , 619- 629. doi: 10.1016/S0252-9602(15)30008-4 [23] Liu, Q., Jiang, D., Hayat, T. & Ahmad, B. (2018). Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Levy jumps, Nonlinear Analysis: Hybrid Systems, 27 , 29-43. doi: 10.1016/j.nahs.2017.08.002. [24] Lu, M., Huang,J., Ruan, S. & Yu, P. (2019). Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Diff. Eqs., 267, 1859-1898. doi: 10.1016/j.jde.2019.03.005. [25] Mehta, R. & Deshmukh, A. (2025). Advanced Mathematical Epidemiology and Dynamic Modeling Approaches for Infectious Disease Spread: Insights, Challenges, and Future Prospects. Journal of Engineering Mathematics & Statistics, 9(2), 73 -83. [26] Naji, R. K. and Abdulatee B. H. (2017). The Dynamics of SIIR Model with Nonlinear Incidence Rate and Saturated Treatment Function. Sci.Int. , 29(6), 1223 – 1236. [27] Okuonghae, D. (2018) Backward Bifurcation of an Epidemiological Model with Saturated Incidence, Isolation and Treatment Functions: Qualitative Theory of Dynamical Systems, doi:10.1007/s12346-018-0293-0 [28] Pastor-Satorras, R., & Vespignani, A. (2001). Epidemic spreading in scale-free networks. Physical Review Letters, 86(14), 3200–3203. [29] Riley, S. (2007). Large-scale spatial -transmission models of infectious disease. Science Journal, 316(5829), 1298–1301. [30] Ryu, S., Chun, J. Y., Lee, S., Yoo, D., Kim, Y., Ali, S. T., & Chun, B. C. (2022). Epidemiology and Transmission Dynamics of Infectious Diseases and Control Measures. Viruses, 14(11), 2510. https://doi.org/10.3390/v14112510 [31] Safi, M. A., Garba, S. M. (2012) Global stability analysis of SEIR model with Holling type II incidence function. Comput.,Math. Methods Med. [32] Udoo, I. J. M. (2019). A qualitative analysis of an SIIR epidemic model with saturated incidence, awareness and treatment functions. Journal of Nigerian Society for Mathematical Biology. 2(1), 145-172 [33] Zhou, L. H. and Fan, M. (2012) Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal.: Real World Appl., Vol. 13, pp. 312 – 324

More Articles from INTERNATIONAL JOURNAL OF COMPUTER SCIENCE AND MATHEMATICAL THEORY

Advances in Algorithmic Contract Scoring for Pre-Negotiation Yield Optimization and Risk Retention

Author: Ngozi Samuel Uzougbo, Michael Ominyi, Cyril Chimelie Anichukwueze, Blessing, Chika Jones

DevTest flow: Designing a Scalable Continuous Testing Pipeline for High-Velocity Software Delivery

Author: Lawal Ahmed Oladimeji, Achori Busayo, Akeju BusayoZainab, Saka Samson, Damilare, Mbah Demian Chidi, Runsewe Similoluwa Mayowa, Oladiti Luqman, Abiodun