# Dynamic Analysis and Optimal Control of Fractional Order African Swine Fever Models with Media Coverage

^{*}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Models Formulation

#### 2.2. Methods

- (i)
- The next generation matrix method is used to obtain the basic reproduction number.
- (ii)
- The Descartess rule of signs is used to determine the existence of a positive equilibrium.
- (iii)
- The eigenvalue method, Routh-Hurwitz criteria and LaSalle’s invariance principle are used to prove the stability of two equilibriums.
- (iv)
- The Pontryagin’s maximum principle is used to derive the formula for the optimal solution of System (2).
- (v)
- The Adams-type predictor corrector method and MATLAB software are used for the numerical simulations.

## 3. Results

#### 3.1. Qualitative Analysis Results for System (1)

**Lemma**

**1.**

- (i)
- If ${\mathrm{D}}^{\alpha}g\left(t\right)\ge 0$, for $\forall t\in (a,b)$, then $g\left(t\right)$ is non-decreasing for each $t\in [a,b]$.
- (ii)
- If ${\mathrm{D}}^{\alpha}g\left(t\right)\le 0$, for $\forall t\in (a,b)$, then $g\left(t\right)$ is non-increasing for each $t\in [a,b]$.

**Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Proof of Theorem**

**4.**

**Theorem**

**5.**

- (i)
- When $\alpha =1$, the endemic equilibrium ${E}^{*}$ is locally asymptotically stable, provided that$$\begin{array}{c}{\kappa}_{i}>0,i=0,1,2,3\phantom{\rule{14.22636pt}{0ex}}and\phantom{\rule{14.22636pt}{0ex}}{\kappa}_{3}{\kappa}_{2}{\kappa}_{1}>{\kappa}_{1}^{2}+{\kappa}_{0}{\kappa}_{3}^{2}.\hfill \end{array}$$
- (ii)
- When $\alpha \in (0,1)$, the above conditions are sufficient but not necessary for the local asymptotic stability of the endemic equilibrium ${E}^{*}$. In fact, ${E}^{*}$ is still locally asymptotically stable if all eigenvalues ${\lambda}_{i}$ of Equation (A4) satisfy$$\left|\mathrm{arg}\left({\lambda}_{i}\right)\right|>\frac{\alpha \pi}{2}.$$

#### 3.2. Examples and Numerical Simulation Results for System (1)

**Example**

**1.**

- (i)
- In Figure 1, the initial value is ${X}_{0}$ = [164,000, 470, 100, 300], and α have different values ($\alpha =0.68,\phantom{\rule{2.84544pt}{0ex}}0.73,\phantom{\rule{4pt}{0ex}}0.88,\phantom{\rule{4pt}{0ex}}0.95,\phantom{\rule{4pt}{0ex}}1$). Figure 1 shows that if ${R}_{0}=0.0287<1$, then the disease-free equilibrium ${E}_{0}$ is always asymptotically stable for for all $\alpha \in [0.68,1]$.
- (ii)
- In Figure 2, the value of α is fixed to $0.85$, and different initial values are taken. ${X}_{0}$ = [164,000, 470, 100, 300], [164,000, 400, 170, 300], [163,000, 370, 500, 500], [160,000, 1070, 700, 600]. Figure 2 indicates that different initial values do not affect the stability of the disease-free equilibrium ${E}_{0}$ of system (1).

**Example**

**2.**

- (i)
- In Figure 3, the initial value is fixed to ${X}_{0}$ = [164,000, 470, 100, 300], and α have different values ($\alpha =0.73,\phantom{\rule{4pt}{0ex}}0.85,\phantom{\rule{4pt}{0ex}}0.93,\phantom{\rule{4pt}{0ex}}0.98,\phantom{\rule{4pt}{0ex}}1$). Figure 3 shows that if ${R}_{0}=13.0203>1$, then the endemic equilibrium ${E}^{*}$ is always asymptotically stable for all $\alpha \in [0.73,1]$.
- (ii)
- In Figure 4, the value of α is fixed to $\alpha =0.95$, and different initial values are taken as ${X}_{0}$ = [164,000, 470, 100, 300], [163,000, 370, 500, 500], [160,000, 1070, 700, 600]. Figure 4 indicates that different initial values do not affect the stability of the endemic equilibrium ${E}^{*}$ of system (1).

**Example**

**3.**

- (i)
- (ii)

**Example**

**4.**

**Remark**

**2.**

- (i)
- Figure 1 and Figure 2 show that if ${R}_{0}<1$, then the disease-free equilibrium ${E}_{0}$ is always stable. If the basic reproduction number ${R}_{0}<1$, that is, the number of healthy pigs infected by a diseased pig during its average disease period does not exceed 1, then the disease will eventually disappear, and this result is consistent with reality. The value of α can affect the speed towards the equilibrium. The initial values will not affect the stability, which is in line with Theorems 1 and 4.
- (ii)
- Figure 3 and Figure 4 indicate that if ${R}_{0}>1$, then the disease-free equilibrium ${E}_{0}$ is unstable and the endemic equilibrium exists. If the basic reproduction number ${R}_{0}>1$, that is, if the number of healthy pigs infected by a diseased pig during its average disease period is more than 1, then the disease will break out in this region and become an endemic. The value of α will affect the speed towards the endemic equilibrium ${E}^{*}$. The initial values will not affect the stability, which is in accordance with Theorems 1 and 2.
- (iii)
- Figure 5 and Figure 6 show the sensitivity analysis for parameters d and φ. Through observation, it can be observed that the mortality rate d of pigs and the clearance rate φ of viruses have a significant impact on system (1). Therefore, it is reasonable for us to consider specific control measures in system (2) as removing diseased pigs and strengthening the disinfection and sterilization of pig breeding environments.
- (iv)

#### 3.3. Qualitative Analysis Results for System (2)

**Theorem**

**6.**

#### 3.4. Examples and Numerical Simulation Results for System (2)

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

- (i)
- In Figure 10, the initial value is fixed to ${X}_{0}$ = [164,000, 470, 100, 300]. For different values of α, this figure demonstrates the optimal solution of ${u}_{1}^{*}\left(t\right)$ and ${u}_{2}^{*}\left(t\right)$ when the upper limit of ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$ is relatively small (realistically reasonable).
- (ii)
- In Figure 11, the initial value is fixed to ${X}_{0}$ = [164,000, 470, 100, 300]. For different values of α, this figure shows the optimal solution of ${u}_{1}^{*}\left(t\right)$ and ${u}_{2}^{*}\left(t\right)$ when the upper limit of ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$ is relatively larger (realistically unreasonable).

**Remark**

**3.**

- (i)
- When ${R}_{0}=13.0203>1$, Figure 8 shows that for all $\alpha \in [0.77,1]$, the corresponding optimal solution tend to be stable with different speeds. This indicates that under the values of Example 3, ASF will outbreak in a certain region, and it will gradually become an endemic.
- (ii)
- A comparison between Figure 7 and Figure 9 shows that media coverage combined with control measures can suppress the spread of ASF more effectively. That is to say, if pig farmers take timely measures to eliminate suspected infected pigs and disinfect the environment of pig farms on a large scale after receiving media reports of the outbreak of ASF in the local area, they can greatly reduce the infection and help to prevent the spread of the epidemic.
- (iii)
- Since the magnitudes change dramatically for different parameters, choosing a suitable upper limit of ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$ is important.Figure 10 shows that if the upper limit of ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$ is relatively small, then the optimal control solutions ${u}_{1}^{*}\left(t\right)$ and ${u}_{2}^{*}\left(t\right)$ can be suitably solved. However, if the upper limit of ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$ is relatively big, then the optimal control solutions ${u}_{1}^{*}$ and ${u}_{2}^{*}$ cannot be solved suitably, as shown in Figure 11. In fact, obvious and chaotic oscillations occur.

## 4. Discussion

- There always exists a unique positive solution for any positive initial value, and the set $\Gamma $ is positively invariant for this system. This conclusion is essential from a biological perspective.
- The basic reproduction number ${R}_{0}$ is obtained.
- The sufficient conditions for the existence and stability of the disease-free equilibrium ${E}_{0}$ and endemic equilibrium ${E}^{*}$ are derived.
- From Figure 1, Figure 2, Figure 3 and Figure 4, it can be observed that the initial value is not crucial and it does not affect the stability. This means that the initial value of susceptible pigs and diseased pigs is not a key factor. However, the value of $\alpha $ is important, and it will affect the speed towards a stable state. This result indicates that the fractional order system is different from its corresponding integer order system.
- Figure 5 and Figure 6 show that both parameters d and $\phi $ are sensitive. In fact, d and $\phi $ have a significant effect on the basic reproduction number ${R}_{0}$. In practice, we can reduce the value of ${R}_{0}$ by increasing the mortality rate of diseased pigs or increasing disinfection measures in pig houses, thereby achieving the goal of preventing the continued spread of the disease.
- Figure 7 shows that media coverage is a very useful measure to control the disease.

- The formula of the optimal control solution ${u}_{1}^{*}\left(t\right)$ and ${u}_{2}^{*}\left(t\right)$ is obtained by using the Pontryagin’s maximum principle.
- Figure 9 indicates that media coverage combined with control measures (such as disinfection and sterilization) can suppress the spread of the disease more effectively.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Proof of Theorem**

**1.**

## Appendix B. Proof of Theorem 3

**Proof of Theorem**

**3.**

## Appendix C. Proof of Theorem 5

**Proof of Theorem**

**5.**

## Appendix D. Proof of Theorem 6

**Proof of Theorem**

**6.**

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**Figure 4.**Phase portrait of $S\left(t\right)$, ${I}_{s}\left(t\right)$, and ${I}_{a}\left(t\right)$ for different initial values. Here, ${R}_{0}=13.0203>1$.

**Figure 7.**Time series of system (1) with media coverage (in blue) or without media coverage (in purple).

**Figure 8.**Optimal solutions for system (2) with different values of $\alpha $. Here, ${R}_{0}=13.0203>1$.

**Figure 9.**Optimal solutions for system (2) with control measures or without control measures.

**Figure 10.**The optimal solutions of ${u}_{1}^{*}\left(t\right)$ and ${u}_{2}^{*}\left(t\right)$ for system (2) with different values of $\alpha $. Here, the upper limit of ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$ is relatively small (realistically reasonable).

**Figure 11.**The optimal solutions of ${u}_{1}^{*}\left(t\right)$ and ${u}_{2}^{*}\left(t\right)$ for system (2) with different values of $\alpha $. Here, the upper limit of ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$ is relatively big (realistically unreasonable).

$\mathbf{Variables}$ | $\mathbf{Description}$ | |||
---|---|---|---|---|

$S\left(t\right)$ | Density of the susceptible population | |||

${I}_{s}\left(t\right)$ | Density of the symptomatic infectious population | |||

${I}_{a}\left(t\right)$ | Density of the asymptomatic infectious population | |||

$V\left(t\right)$ | Density of ASFV in the environment | |||

$\mathbf{Parameters}$ | $\mathbf{Description}$ | $\mathbf{Value}$ | $\mathbf{Units}$ | $\mathbf{Refs}$ |

$\Lambda $ | The recruitment rate of population | 670 | ${\mathrm{day}}^{-1}$ | [12] |

${\beta}_{0}$ | ASFV transmission rate with direct contact of infectious population | $5.46\times {10}^{-10}$ | ${\mathrm{day}}^{-1}$ | [15] |

${\beta}_{2}$ | Virus transmission rate of contaminated pig products and materials | $3.80\times {10}^{-11}$ | ${\mathrm{day}}^{-1}$ | [15] |

m | The half-saturation constant | 30 | $--$ | [27] |

$\eta $ | Reduced rate by asymptomatic population | 0.7001 | ${\mathrm{day}}^{-1}$ | [15] |

p | The proportion of symptomatic infectious population | 0.7899 | $--$ | [15] |

${d}_{1}$ | Natural and disease related death rate of population | 0.006040 | ${\mathrm{day}}^{-1}$ | $--$ |

d | Natural death rate of population | 0.004060 | ${\mathrm{day}}^{-1}$ | $--$ |

h | The release rate of virus from symptomatic infectious population | 10.0575 | ${\mathrm{day}}^{-1}$ | [15] |

k | The release rate of virus from asymptomatic infectious population | 299.6462 | ${\mathrm{day}}^{-1}$ | [15] |

$\phi $ | Virus clearance rate | $0.3264$ | ${\mathrm{day}}^{-1}$ | [15] |

${u}_{1}\left(t\right)$ | Measures to eliminate suspected disease population | [0, 1] | ${\mathrm{day}}^{-1}$ | $--$ |

${u}_{2}\left(t\right)$ | Measures for disinfection and sterilization with disinfectant | [0, 1] | ${\mathrm{day}}^{-1}$ | $--$ |

**Table 2.**Descartes Sign Rule for Equation (6).

Number and Sign of Roots | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{0}$ |
---|---|---|---|---|

3 negative | + | + | + | + |

3 positive | + | − | + | − |

1 positive | + | − | − | − |

1 positive | + | + | − | − |

1 positive | + | + | + | − |

1 positive | − | + | + | + |

2 positive | + | − | + | + |

2 positive | + | − | − | + |

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## Share and Cite

**MDPI and ACS Style**

Shi, R.; Zhang, Y.; Wang, C.
Dynamic Analysis and Optimal Control of Fractional Order African Swine Fever Models with Media Coverage. *Animals* **2023**, *13*, 2252.
https://doi.org/10.3390/ani13142252

**AMA Style**

Shi R, Zhang Y, Wang C.
Dynamic Analysis and Optimal Control of Fractional Order African Swine Fever Models with Media Coverage. *Animals*. 2023; 13(14):2252.
https://doi.org/10.3390/ani13142252

**Chicago/Turabian Style**

Shi, Ruiqing, Yihong Zhang, and Cuihong Wang.
2023. "Dynamic Analysis and Optimal Control of Fractional Order African Swine Fever Models with Media Coverage" *Animals* 13, no. 14: 2252.
https://doi.org/10.3390/ani13142252