# Stability analysis and optimal control of a hand-foot-mouth disease (HFMD) model

• Jun-Yuan Yang
• Yuming Chen
• Feng-Qin Zhang
Original Research

## Abstract

In this paper, we propose a system of ordinary differential equations to model the hand-foot-mouth disease (HFMD). We derive the expression of the basic reproduction number . When , the system only has the disease free equilibrium, which is globally asymptotically stable; otherwise, the system is persistent. By sensitivity analysis, we identify the control parameters. Then we formulate an optimal control problem to find the optimal control strategy. These results are applied to the spread of HFMD in Mainland China. The basic reproduction number tells us that it is outbreak in China.

## Keywords

Hand-foot-mouth disease Optimal control Stability Persistence Sensitivity analysis Least-squares approach

## Mathematics Subject Classification (2010)

34D20 92D30 49J15

## Notes

### Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments, which greatly improve the presentation of the paper. This work was done when Yang was a postdoctoral fellow at the Department of Mathematics, Wilfrid Laurier University. He would like to thank the Department for the hospitality. Research is supported partially by the NSF of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young Sciences Foundation of Shanxi (2011021001-1), the Foundation of University (YQ-2011045, JY-2011036), the Natural Science and Engineering Research Council of Canada (NSERC), the Early Researcher Award Program of Ontario, the One Hundred Talents Project of Shanxi Province, and the Program of Key Disciplines in Shanxi Province (20111028).

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