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)
- (i)
- If , for , then is non-decreasing for each .
- (ii)
- If , for , then is non-increasing for each .
- (i)
- When , the endemic equilibrium is locally asymptotically stable, provided that
- (ii)
- When , the above conditions are sufficient but not necessary for the local asymptotic stability of the endemic equilibrium . In fact, is still locally asymptotically stable if all eigenvalues of Equation (A4) satisfy
3.2. Examples and Numerical Simulation Results for System (1)
- (i)
- (ii)
- In Figure 2, the value of α is fixed to , and different initial values are taken. = [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 of system (1).
- (i)
- (ii)
- (i)
- (ii)
- (i)
- Figure 1 and Figure 2 show that if , then the disease-free equilibrium is always stable. If the basic reproduction number , 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 , then the disease-free equilibrium is unstable and the endemic equilibrium exists. If the basic reproduction number , 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 . 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)
3.4. Examples and Numerical Simulation Results for System (2)
- (i)
- In Figure 10, the initial value is fixed to = [164,000, 470, 100, 300]. For different values of α, this figure demonstrates the optimal solution of and when the upper limit of , is relatively small (realistically reasonable).
- (ii)
- In Figure 11, the initial value is fixed to = [164,000, 470, 100, 300]. For different values of α, this figure shows the optimal solution of and when the upper limit of , is relatively larger (realistically unreasonable).
- (i)
- When , Figure 8 shows that for all , 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 , is important.Figure 10 shows that if the upper limit of , is relatively small, then the optimal control solutions and can be suitably solved. However, if the upper limit of , is relatively big, then the optimal control solutions and 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 is positively invariant for this system. This conclusion is essential from a biological perspective.
- The basic reproduction number is obtained.
- The sufficient conditions for the existence and stability of the disease-free equilibrium and endemic equilibrium 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 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 are sensitive. In fact, d and have a significant effect on the basic reproduction number . In practice, we can reduce the value of 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 and 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
Appendix B. Proof of Theorem 3
Appendix C. Proof of Theorem 5
Appendix D. Proof of Theorem 6
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Density of the susceptible population | ||||
Density of the symptomatic infectious population | ||||
Density of the asymptomatic infectious population | ||||
Density of ASFV in the environment | ||||
The recruitment rate of population | 670 | [12] | ||
ASFV transmission rate with direct contact of infectious population | [15] | |||
Virus transmission rate of contaminated pig products and materials | [15] | |||
m | The half-saturation constant | 30 | [27] | |
Reduced rate by asymptomatic population | 0.7001 | [15] | ||
p | The proportion of symptomatic infectious population | 0.7899 | [15] | |
Natural and disease related death rate of population | 0.006040 | |||
d | Natural death rate of population | 0.004060 | ||
h | The release rate of virus from symptomatic infectious population | 10.0575 | [15] | |
k | The release rate of virus from asymptomatic infectious population | 299.6462 | [15] | |
Virus clearance rate | [15] | |||
Measures to eliminate suspected disease population | [0, 1] | |||
Measures for disinfection and sterilization with disinfectant | [0, 1] |
Number and Sign of Roots | ||||
---|---|---|---|---|
3 negative | + | + | + | + |
3 positive | + | − | + | − |
1 positive | + | − | − | − |
1 positive | + | + | − | − |
1 positive | + | + | + | − |
1 positive | − | + | + | + |
2 positive | + | − | + | + |
2 positive | + | − | − | + |
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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
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 StyleShi, 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