American Institute of Mathematical Sciences

• Previous Article
New exact solutions for some fractional order differential equations via improved sub-equation method
• DCDS-S Home
• This Issue
• Next Article
Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative
June 2019, 12(3): 455-474. doi: 10.3934/dcdss.2019030

Modeling the transmission dynamics of avian influenza with saturation and psychological effect

 1 Department of Mathematics, City University of Science and Information, Technology, Peshawar, KP, 25000, Pakistan 2 Department of Mathematics, Abdul Wali Khan, University Mardan, KP, 23200, Pakistan 3 Department of Information Technology Education, University of Education, Winneba (Kumasi campus), Ghana

* Corresponding author: altafdir@gmail.com, makhan@cusit.edu.pk

Received  July 2017 Revised  November 2017 Published  September 2018

The present paper describes the mathematical analysis of an avian influenza model with saturation and psychological effect. The virus of avian influenza is not only a risk for birds but the population of human is also not safe from this. We proposed two models, one for birds and the other one for human. We consider saturated incidence rate and psychological effect in the model. The stability results for each model that is birds and human is investigated. The local and global dynamics for the disease free case of each model is proven when the basic reproduction number $\mathcal{R}_{0b}<1$ and $\mathcal{R}_0<1$. Further, the local and global stability of each model is investigated in the case when $\mathcal{R}_{0b}>1$ and $\mathcal{R}_0>1$. The mathematical results show that the considered saturation effect in population of birds and psychological effect in population of human does not effect the stability of equilibria, if the disease is prevalent then it can affect the number of infected humans. Numerical results are carried out in order to validate the theoretical results. Some numerical results for the proposed parameters are presented which can reduce the number of infective in the population of humans.

Citation: Muhammad Altaf Khan, Muhammad Farhan, Saeed Islam, Ebenezer Bonyah. Modeling the transmission dynamics of avian influenza with saturation and psychological effect. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 455-474. doi: 10.3934/dcdss.2019030
References:

show all references

References:
The behavior of infected individuals $I_h$, keeping $\alpha = m = 0.001$. Figure 1(a): varying $\beta_a$ and $\beta_h = 8\times 10^{-7}$ is fixed. Figure 1(b): varying $\beta_h$ and $\beta_a = 3\times 10^{-6}$ is fixed
The behavior of infected individuals $I_h$ when $\mathcal{R}_{0}>1$. Figure 2(a): $\alpha = m = 0$, Figure 2(b): $\alpha = m = 0.001$
The behavior of infected individuals $I_h$ and $\mathcal{R}_{0}>1$: Figure 3(a) when $\alpha = 0.001,~0.001,~0.01$ and $m = 0.001$ fixed. Figure 3(b) when $m = 0.001,~0.001,~0.01$ and $\alpha = 0.001$ fixed
The behavior of infected individuals $I_h$ and $\mathcal{R}_{0}<1$: Figure 4(a) when $\alpha = 0.001,~0.001,~0.01$ and $m = 0.001$ fixed. Figure 3(b) when $m = 0.001,~0.001,~0.01$ and $\alpha = 0.001$ fixed
The behavior of infected individuals $I_h$ and $\mathcal{R}_{0}<1$: $\alpha = 0.001,~0.01,~0.1$, $m = 0.001,~0.01,~0.1$
 [1] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [2] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [3] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [4] Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365 [5] Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621 [6] Junyuan Yang, Yuming Chen, Jiming Liu. Stability analysis of a two-strain epidemic model on complex networks with latency. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2851-2866. doi: 10.3934/dcdsb.2016076 [7] Scott W. Hansen. Controllability of a basic cochlea model. Evolution Equations & Control Theory, 2016, 5 (4) : 475-487. doi: 10.3934/eect.2016015 [8] Wenbo Cheng, Wanbiao Ma, Songbai Guo. A class of virus dynamic model with inhibitory effect on the growth of uninfected T cells caused by infected T cells and its stability analysis. Communications on Pure & Applied Analysis, 2016, 15 (3) : 795-806. doi: 10.3934/cpaa.2016.15.795 [9] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [10] Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119 [11] Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101 [12] Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045 [13] Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826 [14] Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635 [15] Jacopo De Simoi. Stability and instability results in a model of Fermi acceleration. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 719-750. doi: 10.3934/dcds.2009.25.719 [16] Ana I. Muñoz, José Ignacio Tello. Mathematical analysis and numerical simulation of a model of morphogenesis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1035-1059. doi: 10.3934/mbe.2011.8.1035 [17] Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 781-795. doi: 10.3934/dcdss.2020044 [18] Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Maurizio Grasselli, Roberto Natalini. A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results. Communications on Pure & Applied Analysis, 2019, 18 (2) : 977-998. doi: 10.3934/cpaa.2019048 [19] Sabri Bensid, Jesús Ildefonso Díaz. On the exact number of monotone solutions of a simplified Budyko climate model and their different stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1033-1047. doi: 10.3934/dcdsb.2019005 [20] Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002

2017 Impact Factor: 0.561

Tools

Article outline

Figures and Tables