«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

## Optimal Control in Epidemic Models of Transmissive Diseases with SEI-SEIR Systems

Author(s)
R. M. Batalin, V. A. Terletskiy
Abstract

This paper discusses the age-dependent epidemic models of transmissive diseases. The models consists of coupled partial differential equations for human population and ordinary differential equations for vector population. Based on these models, the problems of optimal control of funding level of programs to prevent spread of infections were built. The combined minimization both count of infected human population and founding level of disease transition prevention programs was selected as a target of optimization. These two criterias conflict and to resolve this contradiction in this paper using the approach of weight coefficients. Unfortunately, the problem is nonlinear in this statement and development of methods, more effective than widely known gradient methods or methods based on Pontryagin maximum principle, turned out to be rather nontrivial. For this reason this paper assumes that number of infected vectors is a constant. This simplification allow us to generalized original nonlinear models in the optimal control problem with linear dynamic system. For this problem accurate formulas of increment of cost functional and numerical method based on these formulas are built. These methods are more effective than commonly known standard methods because allows to improve control by solving one Cauchy problem. Furthermore, these methods are capable of improving extreme and degenerate controls.

Keywords
optimal control, accurate formulas of increment of cost functional, agedependent epidemic models, transmissive diseases
UDC
517.977.56
References

1. Vasilev F.P. Methods of optimization. Moscow, Faktorial Press, 2002.

2. Srochko V.A. Iterative methods for solving optimal control problems. Moscow, FIZMALIT, 2000.

3. Terletskiy V.A. Generalized solution for one-dimensional semi-linear hyperbolic systems with mixing conditions. Izvestiya vuzov. Matematika., 2004, vol. 12, pp. 82-90.

4. Gupur G., Li Xue-Zhi, Zhu Guang-Tian. Threshold and Stability Results for an Age-Structured Epidemic Model. Computers and Mathematics with Applications, 2001, vol. 42, no 6, pp. 883-907.

5. Hoppensteadt F. An age dependent epidemic model. Journal of the Franklin Institute, 1974, vol. 297, no 5, pp. 325-333.

6. Hoppensteadt F. Mathematical Theories of Populations: Demographics, Genetics, and Epidemics. Society of Industrial and Applied Mathematics, Philadelphia, PA, 1975.

7. Inaba H., Threshold and stability results for an age-structured epidemic model. J. Math. Biol., 1990, vol. 28, no 4, pp. 411-434.

8. Macdonald G. The measurement of malaria transmission. Proc. R. Soc. Med., 1955, vol. 48, no 4, pp. 295–302.

9. Park T. Age-dependence in epidemic models of vector-borne infections. The University of Alabama, Huntsville, 2004.

10. Ross R. Report on the prevention of malaria in Mauritius. New York: E. P. Dutton & Company,;1908.

11. Ross R. The logical basis of the sanitary policy of mosquito reduction. Science, 1905, vol. 22, no 577, pp. 689–699.

12. Ross R. The prevention of malaria. London, John Murray, 1910.