# Analysis of a Hand-Foot-Mouth Disease Model with Standard Incidence Rate and Estimation for Basic Reproduction Number

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Analysis

#### 2.1. The Model

**Theorem**

**1.**

#### 2.2. Local Stability Analysis

**Theorem**

**2.**

**Proof.**

**J**(${E}_{0}$), at the disease free equilibrium ${E}_{0}$ is

**I**is the unity matrix. From Equation (9), it is clear that the two characteristic roots of $\mathbf{J}\left({E}_{0}\right)$ is ${\lambda}_{1}=-\mu $, ${\lambda}_{2}=-(\mu +\delta )$, that are both negative. The other two characteristic roots are determined by the following equation

**Theorem**

**3.**

**Proof.**

#### 2.3. Numerical Simulation of Stability

## 3. Estimation of Basic Reproduction Number

#### 3.1. Estimation Formula

#### 3.2. Application of the Estimation Formula

^{st}e-week to the ${j}^{th}$ e-week of one year. Hence we select the value of j that has the largest $R-square$ value in that year and choose the corresponding value of b as the estimation of $\lambda $ of that year.

#### 3.3. Real Time Estimation of ${R}_{0}$

## 4. Discussion

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The dash line is the variation of s, while the solid line is the variation of i. (

**a**) The local stability of the disease free equilibrium when ${R}_{0}<1$. s tends to 1 and i tends to 0 as $t\to +\infty $; (

**b**) The local stability of the positive equilibrium when ${R}_{0}>1$. s tends to ${s}^{*}=1/{R}_{0}=0.5853$ and i tends to ${i}^{*}\approx 0.1488>0$ as $t\to +\infty $.

**Figure 2.**The newly infected cases of years 2015 and 2016 in Singapore notified by Ministry of Health of Singapore. The red line is the variations of the newly infected cases of year 2015, and the black line is that of 2016.

**Figure 3.**The accumulative infected cases fitted by the curve $y=aexp\left(bx\right)$. (

**a**) The year 2015; (

**b**) The year 2016.

**Figure 4.**The real time estimation of the basic reproduction number of hand-foot-mouth disease (HFMD) in Singapore of years 2015 and 2016. The time zone is from the 6th e-week of 2015 to the last e-week of 2016.

Parameter or Variable | Biological Implications | Value |
---|---|---|

S | number of susceptible individuals | variable |

E | number of exposed individuals | variable |

I | number of infective individuals | variable |

R | number of recovered individuals | variable |

N | total number of individuals, $=S+E+I+R$ | variable |

b | recruitment rate | demographical |

$\beta $ | transmission efficiency rate | estimated |

$\mu $ | natural death rate | demographical |

$\delta $ | loss of immunity rate | 0.07 [1,4] |

$1/\sigma $ | latent duration | estimated |

$\gamma $ | recovery rate | estimated |

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**MDPI and ACS Style**

Wu, C.
Analysis of a Hand-Foot-Mouth Disease Model with Standard Incidence Rate and Estimation for Basic Reproduction Number. *Math. Comput. Appl.* **2017**, *22*, 29.
https://doi.org/10.3390/mca22020029

**AMA Style**

Wu C.
Analysis of a Hand-Foot-Mouth Disease Model with Standard Incidence Rate and Estimation for Basic Reproduction Number. *Mathematical and Computational Applications*. 2017; 22(2):29.
https://doi.org/10.3390/mca22020029

**Chicago/Turabian Style**

Wu, Chunqing.
2017. "Analysis of a Hand-Foot-Mouth Disease Model with Standard Incidence Rate and Estimation for Basic Reproduction Number" *Mathematical and Computational Applications* 22, no. 2: 29.
https://doi.org/10.3390/mca22020029