Dynamic analysis and optimal control of a fractional order model for hand-foot-mouth Disease
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Dynamic analysis and optimal control of a fractional order model for hand-foot-mouth Disease Ruiqing Shi1
· Ting Lu1
Received: 24 March 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, a fractional order mathematical model is constructed to describe the transmission of hand-foot-mouth disease. Two cases are considered: constant control and optimal control. In the former case, the existence and uniqueness of positive solutions are proved; then the sufficient conditions for the existence and stability of two equilibriums are obtained. In the latter case, the existence of an optimal control solution is obtained; next, the optimality control conditions are derived by using Pontryagin’s Maximum Principle. After that, some numerical simulations are performed to verify the theoretical prediction. Keywords Fractional order · Hand-foot-mouth disease · Basic reproduction number · Stability · Optimal control Mathematics Subject Classification 92B05
1 Introduction Hand-foot-mouth disease (HFMD) is an infectious disease caused by Enterovirus. There are more than 20 types of enterovirus caused HFMD, among which coxsackie virus A16 (Cox A16) and enterovirus 71 (EV 71) are the most common [1]. Most infected individuals are children under five years old and the incidence is extremely high in children under three years old [2]. In most cases, the disease present with oral pain, anorexia, low fever, and herpes or ulcer in hand, foot, mouth and other parts [3]. Most of the children heal spontaneously for about a week, while few children may cause complications such as myocarditis, pulmonary edema and aseptic meningitis.
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Ruiqing Shi [email protected] Ting Lu [email protected]
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School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, China
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R. Shi, T. Lu
The disease develops rapidly in some children with low resistance, leading directly to death under the threat of the disease [4]. There are two main routes for the transmission of HFMD: contact transmission (contact with infected individual or contact with tools that have been used by infected people) and respiratory transmission (virus are easily exposed to the air and then enter the body through breathing when infectious coughs or sneezes). As we all known, HFMD epidemic has appeared in many countries and regions, such as Asian [5–7], Europe [8,9], America [10]. In mainland China, the first case of HFMD occurred in Shanghai [11]. It grew rapidly and spread across many provinces or cities [12–14], especially Shandong [15], Jiangxi [16], Beijing [17]. According to relevant reports, HFMD epidemic outbreaks every two or three years and the peak season for disease is in summer. Many scholars have investigated the transmission and control of many disease through constructing mathematical models [18–25]. In 2008, a simple SIR model for HFMD was proposed to estimate the number of the infected population and the duration of the illness [21]. Then Roy and Halder developed a more reasonable HFMD
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