The paper presents the basic model for the transmission dynamics of West Nile virus （WNV）. The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally...The paper presents the basic model for the transmission dynamics of West Nile virus （WNV）. The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease- free equilibrium whenever the associated reproduction number （R0） is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation （where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity）. It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever R0 〉 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by △b and △v are negative. On the other hand, increasing the length of the incubation period increases disease burden if △b 〉 0 and △v 〉 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence （considered in [2]） does not alter the qualitative dynamics of the autonomous ：system （with respect to the elimination or persistence of the disease）.展开更多
The present paper investigates the theoretical analysis of the tuberculosis(TB)model in the discrete-time case.The model is parameterized by the TB infection cases in the Pakistani province of Khyber Pakhtunkhwa betwe...The present paper investigates the theoretical analysis of the tuberculosis(TB)model in the discrete-time case.The model is parameterized by the TB infection cases in the Pakistani province of Khyber Pakhtunkhwa between 2002 and 2017.The model is parameterized and the basic reproduction number is obtained and it is found R_(0)=1:5853.The stability analysis for the model is presented and it is shown that the discrete-time tuberculosis model is stable at the disease-free equilibrium whenever R_(0)<1 and further we establish the results for the endemic equilibria and prove that the model is globally asymptotically stable if R_(0)>1.A discrete fractional model in the sense of Caputo derivative is presented.The numerical results of the model with various parameters and their effect on the model are presented.A comparison of discrete-time method with continuous-time model is presented graphically.A discrete fractional approach is compared with the existing method in literature and some reasonable results are achieved.Finally,a summary of results and conclusion are presented.展开更多
基金the support in part of the University of Pretoria Research Development Programme (RDP)
文摘The paper presents the basic model for the transmission dynamics of West Nile virus （WNV）. The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease- free equilibrium whenever the associated reproduction number （R0） is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation （where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity）. It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever R0 〉 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by △b and △v are negative. On the other hand, increasing the length of the incubation period increases disease burden if △b 〉 0 and △v 〉 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence （considered in [2]） does not alter the qualitative dynamics of the autonomous ：system （with respect to the elimination or persistence of the disease）.
文摘The present paper investigates the theoretical analysis of the tuberculosis(TB)model in the discrete-time case.The model is parameterized by the TB infection cases in the Pakistani province of Khyber Pakhtunkhwa between 2002 and 2017.The model is parameterized and the basic reproduction number is obtained and it is found R_(0)=1:5853.The stability analysis for the model is presented and it is shown that the discrete-time tuberculosis model is stable at the disease-free equilibrium whenever R_(0)<1 and further we establish the results for the endemic equilibria and prove that the model is globally asymptotically stable if R_(0)>1.A discrete fractional model in the sense of Caputo derivative is presented.The numerical results of the model with various parameters and their effect on the model are presented.A comparison of discrete-time method with continuous-time model is presented graphically.A discrete fractional approach is compared with the existing method in literature and some reasonable results are achieved.Finally,a summary of results and conclusion are presented.
