×

A dynamic model for infectious diseases: the role of vaccination and treatment. (English) Zbl 1352.92166

Summary: Understanding dynamics of an infectious disease helps in designing appropriate strategies for containing its spread in a population. Recent mathematical models are aimed at studying dynamics of some specific types of infectious diseases. In this paper we propose a new model for infectious diseases spread having susceptible, infected, and recovered populations and study its dynamics in presence of incubation delays and relapse of the disease. The influence of treatment and vaccination efforts on the spread of infection in presence of time delays are studied. Sufficient conditions for local stability of the equilibria and change of stability are derived in various cases. The problem of global stability is studied for an important special case of the model. Simulations carried out in this study brought out the importance of treatment rate in controlling the disease spread. It is observed that incubation delays have influence on the system even under enhanced vaccination. The present study has clearly brought out the fact that treatment rate even in presence of time delays would contain the disease as compared to popular belief that eradication can only be done through vaccination.

MSC:

92D30 Epidemiology

Software:

dde23
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Naheed, Afia; Singh, Manmohan; Lucy, David, Numerical study of SARS epidemic model with the inclusion of diffusion in the system, Appl Math Comput, 2209, 480-498 (2014) · Zbl 1365.92128
[2] Scherer, Almut; Mclean, Angela, Mathematical models of vaccination, Br Med Bull, 62, 187-199 (2002)
[3] Beretta, E.; Kuang, Y., Modeling and analysis of a marine bacteriophage infection, Math Biosci, 149, 57-76 (1998) · Zbl 0946.92012
[4] Billarda, L.; Dayananda, P. W.A., A multi-stage compartmental model for HIV-infected individuals: I waiting time approach, Math Biosci, 249, 92-101 (2014) · Zbl 1308.92095
[5] Bhunu, C. P., Mathematical analysis of a three-strain tuberculosis transmission model, Appl Math Model, 35, 9, 4647-4660 (2011) · Zbl 1225.37101
[6] Chichigina, O.; Valenti, D.; Spagnalo, B., A simple noise model with memory for biological systems, Fluct Noise Lett, 5, 2, L243-L250 (2005)
[7] Diehmann, O.; Heesterbeek, J., Mathematical Epidemiology of Infectious Disease: Model building, analysis and interpretation (2000), Wiley: Wiley New York
[8] Erbe, L. H.; Freedman HI, H. I.; Sree Hari Rao, V., Three species food-chain models with mutual interference and time delays, Math Biosci, 80, 57-80 (1986) · Zbl 0592.92024
[9] Fiasconaro, A.; Valenti, D.; Spagnolo, B., Nonmonotonic behavior of spatiotemporal pattern formation in a noisy Lotka-Volterra system, Acta Phys Pol B, 35, 491-1500 (2004)
[10] Freedman, H. I.; Sree Hari Rao, V., Stability criteria for a system involving two time delays, SIAM J Appl Math, 46, 4, 552-560 (1986) · Zbl 0624.34066
[11] Freedman, H. I.; Sree Hari Rao, V.; Jaya Lakshmi, K., Stability, persistence and extinction in predator - prey system with discrete and continuous time delays, WSSIAA, 1, 221-238 (1992) · Zbl 0832.34078
[12] Forouzannia, Farinaz; Gumel, Abba B., Mathematical analysis of an age-structured model for malaria transmission dynamics, Math Biosci, 247, 80-94 (2014) · Zbl 1282.92018
[13] Huanga, Gang; Takeuchi, Yasuhiro; Ma, Wanbiao; Wei, Deijun, Global stability for SIR and SEIR epidemic models with nonlinear incidence rate, Bull Math Biol, 72, 5, 1192-1207 (2010) · Zbl 1197.92040
[14] Gielen, J. L.W., A stochastic model for epidemics based on renewal equation, J Biol Syst, 8, 1-20 (2000)
[15] Gray, A.; Greenhalgh, D.; Hu, L.; Mao, X.; Pan, J., A stochastic differential equation SIS epidemic model, SIAM J Appl Math, 71, 3, 876-902 (2011) · Zbl 1263.34068
[16] Li, Guihua; Wendi, Wang; Chen, Jin, Global stability of a SIR epidemic model with constant immigration, Chaos, Solitons Fractals, 30, 1012-1019 (2006) · Zbl 1142.34352
[17] Hall, I. M., Iain Barrass, Steve Leach, Didier Pittet, Stphane Hugonnet. Transmission dynamics of methicillin-resistant Staphylococcus aureus in a medical intensive care unit, J R Soc Interface, 9, 75, 2639-2652 (2012)
[18] Hampton, T., Largest-ever outbreak of Ebola virus disease thrusts experimental therapies vaccines into spotlight., JAMA, 312, 10, 987-989 (2014)
[19] Han, L.; Pugliese, A., Epidemics in two competing species, Nonlinear Anal: Real World Appl, 10, 2, 723-744 (2009) · Zbl 1167.34358
[20] Hertz, D.; Jury, E. I.; Zeheb, E., Simplified analytical stability test for systems with commensurate time delays, IEE Proc Control Theory Appl, 131, 1, 52-56 (1984) · Zbl 0535.93054
[21] Hethcote, H. W.; Lewis, M. A.; Driessche, P. V., An epidemic model with delay and a nonlinear incidence rate, J Math Biol, 27, 49-64 (1989) · Zbl 0714.92021
[22] Hethcote, H. W.; Driessche, P. V., Some epidemic models with nonlinear incidence, J Math Biol, 29, 271-287 (1991) · Zbl 0722.92015
[23] Hethcote, H. W.; Stetch, H. W.; Driessche, P. V., Nonlinear oscillations in epidemic models, SIAM J Appl Math, 40, 1-9 (1981) · Zbl 0469.92012
[24] Adnani, Jihad; Hattaf, Khalid; Yousfi, Noura, Stability analysis of a stochastic SIR epidemic model with specific nonlinear incidence rate, Int J Stoch Anal, 2013 (2013) · Zbl 1314.92153
[25] Jiang, Z.; Wei, J., Stability and bifurcation analysis in a delayed SIR model, Chaos, Solitons Fractals, 35, 3, 609-619 (2008) · Zbl 1131.92055
[26] Nadeau, Julie; McCluskey, Connell, Global stability for an epidemic model with applications to feline infectious peritonitis and tuberculosis, Appl Math Comput, 230, 473-483 (2014) · Zbl 1410.34162
[27] Ang, Keng-Cheng, A simple stochastic model for an epidemic: numerical experiments with Matlab, Electron J Math Technol, 1, 2, 117-128 (2007) · Zbl 1166.92320
[28] Korobeinikov, A.; Maini, P. K., A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math Biosci Eng, 1, 52-60 (2004) · Zbl 1062.92061
[29] Kyrychko, Y. N.; Blyuss, K. B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal Real World Appl, 6, 3, 495-507 (2005) · Zbl 1144.34374
[30] La Cognata, A.; Valenti, D.; Dubkov, A. A.; Spagnolo, B., Dynamics of two competing species in the presence of Levy noise sources, Phys Rev E, 82, 1-9 (2010)
[31] Leung, I. K.C.; Gopalsamy, K., Dynamics of continuous and discrete time siv models of Gonorrhea transmission, Dyn Contin Discrete Impuls Syst Ser B Appl Algorithms, 19, 351-375 (2012) · Zbl 1263.92027
[32] Liu, W.; Hethcote, H. W.; Levin, S. A., Dynamical behaviour of epidemiological models with nonlinear incidence rates, J Math Biol, 25, 4, 359-380 (1987) · Zbl 0621.92014
[33] Milner, F. A.; Zhao, R., A new mathematical model of syphilis, Math Model Nat Phenom, 5, 6, 96-108 (2010) · Zbl 1204.92046
[34] Naresh, R.; Tripathi, A.; Tchuenche, J. M.; Sharma, Dileep, Stability analysis of a time delayed epidemic model with nonlinear incidence rate, Comput Math Appl, 58, 2, 348-359 (2009) · Zbl 1189.34098
[35] Pongsumpun, Puntani; Tang, I-Ming, Dynamics of a new strain of the H1N1 influenza a virus incorporating the effects of repetitive contacts, Comput Math Methods Med, 2014 (2014) · Zbl 1307.92353
[36] Bisen, Prakash S.; Raghuvanshi, Ruchika, Emerg Epidemics Manag Control (2013), Wiley: Wiley New York
[37] Upadhyay, Ranjit Kumar; Kumari, Nitu; Sree Hari Rao, V., Modeling the spread of bird flu and predicting outbreak diversity, Nonlinear Anal Real World Appl, 9, 4, 1638-1648 (2008) · Zbl 1154.92324
[38] Shampine, L. F.; Thompson, S., Solving DDEs in MATLAB, Appl Numer Math, 37, 441-458 (2001) · Zbl 0983.65079
[39] Sree Hari Rao, V.; Naresh Kumar, M., Estimation of the parameters of an infectious disease model using neural networks, Nonlinear Anal Real World Appl, 11, 3, 1810-1818 (2010) · Zbl 1196.34061
[40] Sree Hari Rao, V.; Naresh Kumar, M., Predictive dynamics: modeling for virological surveillance and clinical management of dengue, (Sree Hari Rao, V.; Durvasula, Ravi, Dynamics models of infectious diseases. Dynamics models of infectious diseases, Vector borne diseases, vol. I (2013), Springer: Springer New York), 1-41
[41] Sree Hari Rao, V.; Naresh Kumar, M., A new intelligence-based approach for computer-aided diagnosis of dengue fever, Inf Technol Biomed IEEE Trans, 16, 1, 112-118 (2012)
[42] Sree Hari Rao, V.; Naresh Kumar, M., Control of infectious diseases: dynamics and informatics, (Sree Hari Rao, V.; Durvasula, Ravi, Dynamics models of infectious diseases. Dynamics models of infectious diseases, Non vector-borne diseases, vol. II (2013), Springer: Springer New York), 1-30
[43] Sree Hari Rao, V.; Phaneendra, Bh. R.M., Stability of differential systems involving time lags, Adv Math Sci Appl, 8, 2, 948-964 (1998) · Zbl 0914.34068
[44] Tchuenche, J. M.; Nwagwo, A., Local stability of a SIR model and effect of time delay, Math Models Appl Sci, 32, 16, 2160-2175 (2009) · Zbl 1173.92029
[45] Tchuenche, J. M.; Nwagwo, A.; Levins, R., Global behaviour of a SIR epidemic model with time delay, Math Models Appl Sci, 30, 6, 733-749 (2007) · Zbl 1112.92055
[46] Tian, X.; Xu, R., Stability analysis of a delayed SIR epidemic model with stage structure and nonlinear analysis, Discrete Dyn Nat Soc, 17 (2009) · Zbl 1183.34130
[47] Tuckwell, H. K.; Le Corfec, E., A stochastic model for early HIV-I population dynamics, J Theorl Biol, 195, 451-463 (1998)
[48] Valenti, D.; Tranchina, L.; Brai, M.; Caruso, A.; Cosentino, D.; Spagnolo, B., Environmental metal pollution considered as noise: effects on the spatial distribution of benthic forminifera in two coastal marine areas of Sicily(Southern Italy), Ecol Model, 213, 449-462 (2008)
[49] Van Den Driessche, P.; Wang, L.; Zou, X., Modeling diseases with latency and relapse, Math Biosci Eng, 4, 205-219 (2009) · Zbl 1123.92018
[50] Wang, J.; Zhang, J., Zhen Jin. Analysis of a SIR model with bilinear incidence rate, Nonlinear Anal Real World Appl, 11, 4, 2390-2402 (2010) · Zbl 1203.34136
[51] Wei-min, L.; Levin Simon, A.; Iwasa, Yoh, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J Math Biol, 23, 2, 187-204 (1986) · Zbl 0582.92023
[52] Yoshida, N.; Hara, T., Global stability of a delayed SIR epidemic model with density dependent birth and death rates, J Comput Appl Math, 201, 2, 339-347 (2007) · Zbl 1105.92034
[53] Zhang, T.; Teng, Z., Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence, Chaos, Solitons Fractals, 37, 5, 1456-1468 (2008) · Zbl 1142.34384
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.