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New modelling approach concerning integrated disease control and cost-effectivity. (English) Zbl 1078.92059
Summary: Two new models for controlling diseases, incorporating the best features of different control measures, are proposed and analyzed. These models would draw from poultry, livestock and government expertise to quickly, cooperatively and cost-effectively stop disease outbreaks. The combination strategy of pulse vaccination and treatment (or isolation) is implemented in both models if the number of infectives reaches the risk level (RL). Firstly, for one time impulsive effect we compare three different control strategies for both models in terms of cost. The theoretical and numerical results show that there is an optimal vaccination and treatment proportion such that integrated pulse vaccination and treatment (or isolation) reaches its minimum in terms of cost. Moreover, this minimum cost of integrated strategy is less than any cost of single pulse vaccination or single treatment. Secondly, a more realistic case for the second model is investigated based on periodic impulsive control strategies. The existence and stability of periodic solutions with the maximum value of the infectives no larger than RL is obtained. Further, the period $T$ of the periodic solution is calculated, which can be used to estimate how long the infectious population will take to return back to its pre-control level (RL) once integrated control tactics cease. This implies that we can control the disease if we implement the integrated disease control tactics every period $T$. For periodic control strategy, if we aim to control the disease such that the maximum number of infectives is relatively small, our results show that the periodic pulse vaccination is optimal in terms of cost.

34C25Periodic solutions of ODE
34A37Differential equations with impulses
49N90Applications of optimal control and differential games
93C95Applications of control theory
Full Text: DOI
[1] Abakuks, A.: An optimal isolation policy for an epidemic. J. appl. Probab. 10, 247-262 (1973) · Zbl 0261.92009
[2] Abakuks, A.: Optimal immunization policies for epidemics. Adv. appl. Probab. 6, 494-511 (1974) · Zbl 0288.92017
[3] Agur, Z.; Cojocaru, L.; Mazor, G.; Anderson, R. M.; Danon, Y.: Pulse mass measles vaccination across age cohorts. Proc. natl. Acad. sci. USA 90, 11698-11702 (1993)
[4] Anderson, R. M.: Discussion: ecology of pests and pathogens. Perspectives in ecological theory (1989)
[5] Anderson, R. M.; May, R. M.: Population dynamics of human helminth infections: control by chemotherapy. Nature 297, 557-563 (1982)
[6] Anderson, R. M.; May, R. M.: Directly transmitted infectious diseases: control by vaccination. Science 215, 1053-1060 (1982) · Zbl 1225.37099
[7] Anderson, R. M.; May, R. M.: Vaccination and hers immunity to infectious diseases. Nature 318, 323-329 (1985)
[8] Anderson, R. M.; May, R. M.: The invasion, persistence and spread of infectious diseases within animal and plant communities. Philos. trans. R. soc. (London) B 314, 533-570 (1986)
[9] Bailey, N. T. J.: The mathematical theory of infectious diseases and its applications. (1975) · Zbl 0334.92024
[10] Bainov, D.; Simeonov, P.: Impulsive differential equations: periodic solutions and applications. Pitman monographs and surveys in pure and applied mathematics 66 (1993) · Zbl 0815.34001
[11] Chellaboina, V. S.; Bhat, S. P.; Haddad, W. M.: An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlinear anal. TMA 53, 527-550 (2003) · Zbl 1082.37018
[12] Corless, R. M.: On the Lambert W function. Adv. comput. Math. 5, 329-359 (1996) · Zbl 0863.65008
[13] Enzootic pneumonia of pigs, in: Veterinary Medicine, eighth ed., Saunders, London, 1997, pp. 916 -- 923.
[14] Halloran, M. E.: Discussion: vaccine effect on susceptibility. Stat. med. 15, 2405-2420 (1996)
[15] Hethcote, H. W.: Optimal ages of vaccination for measles. Math. biosci. 89, 29-35 (1988) · Zbl 0643.92014
[16] Kaul, S.: On impulsive semi-dynamical systems. J. math. Anal. appl. 150, 120-128 (1990) · Zbl 0711.34015
[17] Kermack, W. O.; Mckendrick, A. G.: Contributions to the mathematical theory of epidemics. Proc. R. Soc. 141, 94-122 (1932) · Zbl 0007.31502
[18] Matveev, A. S.; Savkin, A. V.: Qualitative theory of hybrid dynamical systems. (2000) · Zbl 1052.93004
[19] Ross, R. F.: Mycoplasmal diseases. Diseases of swine, 495-501 (1999)
[20] Schmitz, X. Shu-Fang Hsu: Effects of treatment or/and vaccination on HIV transmission in homosexuals with genetic heterogeneity. Math. biosci. 167, 1-18 (2000) · Zbl 0979.92023
[21] Shulgin, B.; Stone, L.; Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull. math. Biol. 60, 1-26 (1998) · Zbl 0941.92026
[22] Simeonov, P.; Bainov, D.: Orbital stability of periodic solutions of autonomous systems with impulse effect. Int. J. Systems sci. 19, 2561-2585 (1988) · Zbl 0669.34044
[23] Tang, S. Y.; Chen, L. S.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. math. Biol. 44, 185-199 (2002) · Zbl 0990.92033
[24] Tang, S. Y.; Chen, L. S.: Multiple attractors in stage-structured population models with birth pulses. Bull. math. Biol. 65, 479-495 (2003)
[25] Tang, S. Y.; Chen, L. S.: The effect of seasonal harvesting on stage-structured population models. J. math. Biol. 48, 357-374 (2004) · Zbl 1058.92051
[26] Tang, S. Y.; Chen, L. S.: Modelling and analysis of integrated pest management strategy. Discrete contin. Dyn. syst. B 4, 759-768 (2004) · Zbl 1114.92074
[27] Tang, S. Y.; Cheke, A. R.: State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. math. Biol. 50, 257-292 (2005) · Zbl 1080.92067