Modeling the role of information and limited optimal treatment on disease prevalence. (English) Zbl 1369.93567
Summary: Disease outbreaks induce behavioural changes in healthy individuals to avoid contracting infection. We first propose a compartmental model which accounts for the effect of individual’s behavioural response due to information of the disease prevalence. It is assumed that the information is growing as a function of infective population density that saturates at higher density of infective population and depends on active educational and social programmes. Model analysis has been performed and the global stability of equilibrium points is established. Further, choosing the treatment (a pharmaceutical intervention) and the effect of information (a non-pharmaceutical intervention) as controls, an optimal control problem is formulated to minimize the cost and disease fatality. In the cost functional, the nonlinear effect of controls is accounted. Analytical characterization of optimal control paths is done with the help of Pontryagin’s maximum principle. Numerical findings suggest that if only control via information is used, it is effective and economical for early phase of disease spread whereas treatment works well for long term control except for initial phase. Furthermore, we observe that the effect of information induced behavioural response plays a crucial role in the absence of pharmaceutical control. Moreover, comprehensive use of both the control interventions is more effective than any single applied control policy and it reduces the number of infective individuals and minimizes the economic cost generated from disease burden and applied controls. Thus, the combined effect of both the control policies is found more economical during the entire epidemic period whereas the implementation of a single policy is not found economically viable.
MSC:
| 93D30 | Lyapunov and storage functions |
| 91D99 | Mathematical sociology (including anthropology) |
| 92D50 | Animal behavior |
References:
| [1] | Ahituv, A.; Hotz, V.; Philipson, T., The responsiveness of the demand for condoms to the local prevalence of AIDS, J. Hum. Resour., 31, 869-897 (1996) |
| [2] | Bauch, C. T.; Galvani, A. P.; Earn, D. J., Group interest versus self-interest in smallpox vaccination policy, Proc. Natl. Acad. Sci., 100, 18, 10564-10567 (2003) · Zbl 1065.92038 |
| [3] | Behncke, H., Optimal control of deterministic epidemics, Opt. Control Appl. Methods, 21, 6, 269-285 (2000) · Zbl 1069.92518 |
| [4] | Brauer, F.; Castillo-Chávez, C., Mathematical Models in Population Biology and Epidemiology (2012), Springer: Springer New York · Zbl 1302.92001 |
| [5] | Buonomo, B.; d’Onofrio, A.; Lacitignola, D., Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci., 216, 1, 9-16 (2008) · Zbl 1152.92019 |
| [6] | Buonomo, B.; d’Onofrio, A.; Lacitignola, D., Rational exemption to vaccination for non-fatal SIS diseasesglobally stable and oscillatory endemicity, Math. Biosci. Eng., 7, 3, 561-578 (2010) · Zbl 1260.92048 |
| [7] | Buonomo, B.; d’Onofrio, A.; Lacitignola, D., Globally stable endemicity for infectious diseases with information-related changes in contact patterns, Appl. Math. Lett., 25, 7, 1056-1060 (2012) · Zbl 1243.92044 |
| [8] | Buonomo, B.; d’Onofrio, A.; Lacitignola, D., Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl., 404, 2, 385-398 (2013) · Zbl 1304.92118 |
| [9] | Castilho, C., Optimal control of an epidemic through educational campaigns, Electron. J. Differ. Equ., 2006, 125, 1-11 (2006) · Zbl 1108.92035 |
| [10] | Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1, 2, 361-404 (2004) · Zbl 1060.92041 |
| [13] | Coddington, E.; Levinson, N., Theory of Ordinary Differential Equations (1955), Tata McGraw-Hill Education: Tata McGraw-Hill Education New Delhi · Zbl 0064.33002 |
| [14] | Collinson, S.; Heffernan, J., Modelling the effects of media during an influenza epidemic, BMC Public Health, 14, 376, 10 (2014) |
| [15] | Cui, J.; Sun, Y.; Zhu, H., The impact of media on the control of infectious diseases, J. Dyn. Differ. Equ., 20, 1, 31-53 (2008) · Zbl 1160.34045 |
| [16] | Cui, J.; Tao, X.; Zhu, H., An SIS infection model incorporating media coverage, Rocky Mount. J. Math., 38, 5, 1323-1334 (2008) · Zbl 1170.92024 |
| [17] | d’Onofrio, A.; Manfredi, P., Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, J. Theor. Biol., 256, 3, 473-478 (2009) · Zbl 1400.92570 |
| [18] | d’Onofrio, A.; Manfredi, P.; Salinelli, E., Bifurcation thresholds in an SIR model with information-dependent vaccination, Math. Model. Nat. Phenom., 2, 01, 26-43 (2007) · Zbl 1337.92223 |
| [19] | d’Onofrio, A.; Manfredi, P.; Salinelli, E., Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theor. Popul. Biol., 71, 3, 301-317 (2007) · Zbl 1124.92029 |
| [20] | d’Onofrio, A.; Manfredi, P.; Salinelli, E., Fatal SIR diseases and rational exemption to vaccination, Math. Med. Biol., 25, 4, 337-357 (2008) · Zbl 1154.92031 |
| [21] | Fleming, W.; Rishel, R., Deterministic and Stochastic Optimal Control, vol. 1 (1975), Springer: Springer New York · Zbl 0323.49001 |
| [22] | Funk, S.; Gilad, E.; Watkins, C.; Jansen, V., The spread of awareness and its impact on epidemic outbreaks, Proc. Natl. Acad. Sci., 106, 16, 6872-6877 (2009) · Zbl 1203.91242 |
| [23] | Funk, S.; Gilad, E.; Jansen, V., Endemic disease, awareness, and local behavioural response, J. Theor. Biol., 264, 2, 501-509 (2010) · Zbl 1406.92567 |
| [24] | Gaff, H.; Schaefer, E., Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6, 3, 469-492 (2009) · Zbl 1169.49018 |
| [25] | Gaff, H.; Schaefer, E.; Lenhart, S., Use of optimal control models to predict treatment time for managing tick-borne disease, J. Biol. Dyn., 5, 5, 517-530 (2011) · Zbl 1225.92031 |
| [27] | Goldman, S.; Lightwood, J., Cost optimization in the SIS model of infectious disease with treatment, Top. Econ. Anal. Policy, 2, 1, 1-24 (2002) |
| [29] | Grass, D., Optimal Control of Nonlinear Processes: with Applications in Drugs, Corruption and Terror (2008), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1149.49001 |
| [30] | Greenhalgh, D.; Rana, S.; Samanta, S.; Sardar, T.; Bhattacharya, S.; Chattopadhyay, J., Awareness programs control infectious disease-multiple delay induced mathematical model, Appl. Math. Comput., 251, 539-563 (2015) · Zbl 1328.92075 |
| [31] | Gupta, A.; Moyer, C.; Stern, D., The economic impact of quarantineSARS in Toronto as a case study, J. Infect., 50, 5, 386-393 (2005) |
| [32] | Hethcote, H., The mathematics of infectious diseases, SIAM Rev., 42, 4, 599-653 (2000) · Zbl 0993.92033 |
| [33] | Joshi, H.; Lenhart, S.; Li, M.; Wang, L., Optimal control methods applied to disease models, Contemp. Math., 410, 187-208 (2006) · Zbl 1148.49018 |
| [34] | Joshi, H.; Lenhart, S.; Albright, K.; Gipson, K., Modeling the effect of information campaigns on the HIV epidemic in Uganda, Math. Biosci. Eng., 5, 4, 757-770 (2008) · Zbl 1154.92037 |
| [35] | Joshi, H.; Lenhart, S.; Hota, S.; Agusto, F., Optimal control of an SIR model with changing behavior through an education campaign, Electron. J. Differ. Equ., 2015, 50, 1-14 (2015) · Zbl 1516.92109 |
| [36] | Jung, E.; Iwami, S.; Takeuchi, Y.; Jo, T., Optimal control strategy for prevention of avian influenza pandemic, J. Theor. Biol., 260, 2, 220-229 (2009) · Zbl 1402.92278 |
| [37] | Kassa, S.; Ouhinou, A., The impact of self-protective measures in the optimal interventions for controlling infectious diseases of human population, J. Math. Biol., 70, 12, 213-236 (2015) · Zbl 1312.92038 |
| [38] | Kirk, D., Optimal Control Theory: An Introduction (2012), Dover Publications: Dover Publications New York |
| [39] | Kiss, I.; Cassell, J.; Recker, M.; Simon, P., The impact of information transmission on epidemic outbreaks, Math. Biosci., 225, 1, 1-10 (2010) · Zbl 1188.92033 |
| [41] | LaSalle, J., The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics (1976), SIAM: SIAM Philadelphia · Zbl 0364.93002 |
| [42] | Lawrence, P., Differential Equations and Dynamical Systems (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0717.34001 |
| [43] | Lee, S.; Chowell, G.; Castillo-Chávez, C., Optimal control for pandemic influenzathe role of limited antiviral treatment and isolation, J. Theor. Biol., 265, 2, 136-150 (2010) · Zbl 1406.92355 |
| [44] | Lenhart, S.; Workman, J., Optimal Control Applied to Biological Models, vol. 15 (2007), CRC Press: CRC Press Boca Raton · Zbl 1291.92010 |
| [45] | Li, Y.; Ma, C.; Cui, J., The effect of constant and mixed impulsive vaccination on SIS epidemic models incorporating media coverage, Rocky Mt. J. Math., 38, 5, 1437-1455 (2008) · Zbl 1167.34015 |
| [46] | Liu, Y.; Cui, J., The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1, 1, 65-74 (2008) · Zbl 1155.92343 |
| [47] | Liu, R.; Wu, J.; Zhu, H., Media/psychological impact on multiple outbreaks of emerging infectious diseases, Comput. Math. Methods Med., 8, 3, 153-164 (2007) · Zbl 1121.92060 |
| [48] | Manfredi, P.; D’Onofrio, A., Modeling the Interplay between Human Behavior and the Spread of Infectious Diseases (2013), Springer-Verlag: Springer-Verlag New York · Zbl 1276.92002 |
| [49] | Margevicius, R.; Joshi, H., The influence of education in reducing the HIV epidemic, Involve J. Math., 6, 2, 127-135 (2013) · Zbl 1271.92031 |
| [50] | Misra, A.; Sharma, A.; Shukla, J., Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Math. Comput. Model., 53, 5, 1221-1228 (2011) · Zbl 1217.34097 |
| [51] | Misra, A.; Sharma, A.; Singh, V., Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Syst., 19, 02, 389-402 (2011) · Zbl 1228.92067 |
| [52] | Philipson, T., Private vaccination and public healthan empirical examination for US measles, J. Hum. Resour., 31, 611-630 (1996) |
| [53] | Pontryagin, L., Mathematical Theory of Optimal Processes (1987), CRC Press: CRC Press Montreux |
| [54] | Reluga, T. C.; Bauch, C. T.; Galvani, A. P., Evolving public perceptions and stability in vaccine uptake, Math. Biosci., 204, 2, 185-198 (2006) · Zbl 1104.92042 |
| [55] | Russell, S., The economic burden of illness for households in developing countriesa review of studies focusing on malaria, tuberculosis, and human immunodeficiency virus/acquired immunodeficiency syndrome, Am. J. Trop. Med. Hyg., 71, Suppl. 2, 147-155 (2004) |
| [56] | Samanta, S.; Rana, S.; Sharma, A.; Misra, A.; Chattopadhyay, J., Effect of awareness programs by media on the epidemic outbreaksa mathematical model, Appl. Math. Comput., 219, 12, 6965-6977 (2013) · Zbl 1316.34052 |
| [57] | Tchuenche, J.; Bauch, C., Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomath., 2012, 10 (2012) · Zbl 1269.92052 |
| [58] | Tchuenche, J.; Dube, N.; Bhunu, C.; Bauch, C., The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11, Suppl. 1, S5 (2011) |
| [60] | Van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 1, 29-48 (2002) · Zbl 1015.92036 |
| [61] | Wang, A.; Xiao, Y., A Filippov system describing media effects on the spread of infectious diseases, Nonlinear Anal.: Hybrid Syst., 11, 84-97 (2014) · Zbl 1323.92215 |
| [62] | Wang, Y.; Cao, J.; Jin, Z.; Zhang, H.; Sun, G., Impact of media coverage on epidemic spreading in complex networks, Physica A: Stat. Mech. Appl., 392, 23, 5824-5835 (2013) · Zbl 1395.34055 |
| [63] | Wang, Q.; Zhao, L.; Huang, R.; Yang, Y.; Wu, J., Interaction of media and disease dynamics and its impact on emerging infection management, Discret. Contin. Dyn. Syst.—Ser. B, 20, 1, 215-230 (2015) · Zbl 1333.92065 |
| [65] | Wells, C.; Bauch, C., The impact of personal experiences with infection and vaccination on behaviour-incidence dynamics of seasonal influenza, Epidemics, 4, 3, 139-151 (2012) |
| [67] | Xiao, Y.; Zhao, T.; Tang, S., Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. Eng., 10, 2, 445-461 (2013) · Zbl 1259.92061 |
| [68] | Xiao, Y.; Tang, S.; Wu, J., Media impact switching surface during an infectious disease outbreak, Sci. Rep., 5, 7838 (2015) |
| [69] | Zaman, G.; Han Kang, Y.; Jung, I., Stability analysis and optimal vaccination of an SIR epidemic model, BioSystems, 93, 3, 240-249 (2008) |
| [70] | Zeiler, I.; Caulkins, J.; Grass, D.; Tragler, G., Keeping options openan optimal control model with trajectories that reach a DNSS point in positive time, SIAM J. Control Optim., 48, 6, 3698-3707 (2010) · Zbl 1203.49026 |
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.
