Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

Garba, Salisu M.; Safi, Mohammad A.; Gumel, Abba B.

doi:10.1016/j.nonrwa.2012.10.003

Cited by 22 publications

(16 citation statements)

References 34 publications

(86 reference statements)

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“…Item (c) of Theorem suggests the existence of multiple endemic equilibria when

R_{0} < 1

(which is typically a signature for the existence of the phenomenon of backward bifurcation. The phenomenon of backward bifurcation, which is characterized by the co‐existence of a stable DFE and a stable endemic equilibrium when the associated reproduction number of the model is less than unity, has been observed in numerous disease transmission models, such as those for (or with) vector‐borne diseases, imperfect vaccination, and multigroups . The presence of this phenomenon in the model is now formally explored.…”

Section: Mathematical Analysismentioning

confidence: 99%

Mathematics of a sex‐structured model for syphilis transmission dynamics

Gumel

Lubuma

Sharomi

et al. 2018

Math Methods in App Sciences

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Communicated by: J. Banasiak MSC Classification: 34A34; 37N25; 92B05; 92D30 Syphilis, a major sexually transmitted disease, continues to pose major public health burden in both underdeveloped and developed nations of the world.This study presents a new 2-group sex-structured model for assessing the community-level impact of treatment and condom use on the transmission dynamics and control of syphilis. Rigorous analysis of the model shows that it undergoes the phenomenon of backward bifurcation. In the absence of this phenomenon (which is shown to arise because of the reinfection of recovered individuals), the disease-free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model can have multiple endemic equilibria when the reproduction threshold exceeds unity. Numerical simulations of the model, using data relevant to the transmission dynamics of the disease in Nigeria, show that, with the assumed 80% condom efficacy, the disease will continue to persist (ie, remain endemic) in the population regardless of the level of compliance in condom usage by men. Furthermore, detailed optimal control analysis (using Pontraygin's maximum principle) reveals that, for situations where the cost of implementing the controls (treatment and condom-use) considered in this study is low, channelling resources to a treatment-only strategy is more effective than channelling them to a condom-use only strategy. Furthermore, as expected, the combined condom-treatment strategy provides a higher population-level impact than the treatment-only strategy or the condom-use only strategy. When the cost of implementing the controls is high, the 3 strategies are essentially equally as ineffective.

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“…Item (c) of Theorem suggests the existence of multiple endemic equilibria when

R_{0} < 1

Section: Mathematical Analysismentioning

confidence: 99%

Mathematics of a sex‐structured model for syphilis transmission dynamics

Gumel

Lubuma

Sharomi

et al. 2018

Math Methods in App Sciences

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“…In order to do this, we derive the following results. Proof To show the global stability of the disease-free equilibrium E 1 , we construct the following Lyapunov function, following the method used in [35]:…”

Section: Global Stability Analysismentioning

confidence: 99%

New mathematical model of vertical transmission and cure of vector-borne diseases and its numerical simulation

2018

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In this research article, a new mathematical model for the transmission dynamics of vector-borne diseases with vertical transmission and cure is developed. The non-negative solutions of the model are shown. To understand the dynamical behavior of the epidemic model, the theory of basic reproduction number is used. As this number increases, the disease invades the population and vice versa. The effect of vertical transmission and cure rate on the basic reproduction number is shown. The disease-free and endemic equilibria of the model are found and both their local and global stabilities are presented. Finally, numerical simulations are carried out graphically to show the dynamical behaviors. These results show that vertical transmission and cure have a valuable effect on the transmission dynamics of the disease.

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“…The proof, based on using a non‐linear Lyapunov function of Goh‐Volterra type, is presented in Appendix . Functions of such type have been used in mathematical epidemiology/ecology literature (see, for instance, Garba and Gumel and Garba et al. Simulations of the model showing convergence to EEP when

R_{0} > 1

is depicted in Figure A and B.…”

Section: Analysis Of the Submodelmentioning

confidence: 99%

Mathematical analysis of a model for the transmission dynamics of Trichomonas vaginalis (TV) and HIV coinfection

Garba

Mumba

2018

Math Methods in App Sciences

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A deterministic model for the transmission dynamics of HIV and Trichomonas vaginalis (TV) in a human population is designed and rigorously analysed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed by assuming that the coinfection of individuals with HIV and TV is negligible. Furthermore, in the absence of coinfection, the disease-free equilibrium of the model is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Numerical simulation of the model, using initial and demographic data, shows that increased incidence of TV in a population increases HIV incidence in the population. It is further shown that control strategies, such as the treatment, condom use, and counselling of individuals with TV symptoms, can lead to the effective control or elimination of the HIV in the population if their effectiveness level is high enough. The time to disease elimination is reduced if more than one strategy (hybrid strategy) is considered.

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Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

Cited by 22 publications

References 34 publications

Mathematics of a sex‐structured model for syphilis transmission dynamics

Mathematics of a sex‐structured model for syphilis transmission dynamics

New mathematical model of vertical transmission and cure of vector-borne diseases and its numerical simulation

Mathematical analysis of a model for the transmission dynamics of Trichomonas vaginalis (TV) and HIV coinfection

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