Schistosomiasis Model Incorporating Snail Predator as Biological Control Agent
Abstract
:1. Introduction
2. Mathematical Preliminaries
- A1:
- is the linearization matrix of (1) around equilibrium 0, and ω is evaluated at 0. Zero is a simple eigenvalue of , and the other eigenvalues of have a negative real part.
- A2:
- Matrix has a right eigenvector and a left eigenvector corresponding to the zero eigenvalue. Let be the component of f andThe dynamics of System (1) around 0 is totally determined by the signs of A and B.
- (i)
- . When with , 0 is asymptotically stable, and there is a positive unstable equilibrium. When with , 0 is unstable, and there is a negative asymptotically stable equilibrium;
- (ii)
- . When with , 0 is unstable. When with , 0 is asymptotically stable, and there is a positive unstable equilibrium;
- (iii)
- . When with , 0 is unstable, and there is a negative asymptotically stable equilibrium. When with , 0 is stable and a positive unstable equilibrium appears;
- (iv)
- . When ω changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and asymptotically stable.
3. Model Formulation and Basic Properties
3.1. Model Formulation
3.2. Invariant Region
3.3. Equilibrium Points and Basic Reproduction Number
- Disease-free equilibrium point .always exists in .
- Endemic equilibrium point .It is clear that exists in if .
- Disease-free equilibrium point .exists in if .
- Endemic equilibrium point .It is clear that exists in if and .
4. Stability Analysis
5. Numerical Simulations
5.1. Predator Becomes Extinct
5.2. Predator Survives
5.3. Impact of Snail Predator as Biological Control Agent
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
References
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Parameter | Description |
---|---|
Recruitment rate of humans | |
Effectiveness of education in reducing contact between humans and cercariae | |
Education coverage | |
Cercaria infection rate on susceptible humans | |
Recovery rate of infectious humans | |
Natural death rate of humans | |
Latent period | |
Recruitment rate of snails | |
Miracidia infection rate on susceptible snails | |
Natural death rate of snails | |
Molluscicide-related death rate of snails | |
Predation rate | |
Conversion rate | |
Cercaria production rate | |
Parasite-egg hatch rate | |
Number of eggs per gram of stool | |
Average weight of human stool per day | |
Natural death rate of snail predators | |
Natural death rate of cercariae | |
Natural death rate of miracidiae |
Column 1 | Column 2 | Column 3 | Column 4 | |
---|---|---|---|---|
1 | 0 | |||
0 | ||||
0 | 0 | |||
0 | 0 | |||
0 | 0 | 0 | ||
0 | 0 | 0 |
Column 1 | Column 2 | Column 3 | Column 4 | |
---|---|---|---|---|
1 | 0 | |||
0 | 0 | |||
0 | 0 | |||
0 | 0 | 0 | ||
0 | 0 | 0 |
Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | |
---|---|---|---|---|---|
1 | 0 | ||||
0 | |||||
0 | 0 | ||||
0 | 0 | ||||
0 | 0 | 0 | |||
0 | 0 | 0 | |||
0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 |
Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | |
---|---|---|---|---|---|
1 | 0 | ||||
0 | 0 | ||||
0 | 0 | ||||
0 | 0 | 0 | |||
0 | 0 | 0 | |||
0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 |
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Nur, W.; Trisilowati; Suryanto, A.; Kusumawinahyu, W.M. Schistosomiasis Model Incorporating Snail Predator as Biological Control Agent. Mathematics 2021, 9, 1858. https://doi.org/10.3390/math9161858
Nur W, Trisilowati, Suryanto A, Kusumawinahyu WM. Schistosomiasis Model Incorporating Snail Predator as Biological Control Agent. Mathematics. 2021; 9(16):1858. https://doi.org/10.3390/math9161858
Chicago/Turabian StyleNur, Wahyudin, Trisilowati, Agus Suryanto, and Wuryansari Muharini Kusumawinahyu. 2021. "Schistosomiasis Model Incorporating Snail Predator as Biological Control Agent" Mathematics 9, no. 16: 1858. https://doi.org/10.3390/math9161858