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Mathematical Model of Three Age-Structured Transmission Dynamics of Chikungunya Virus.

Agusto FB, Easley S, Freeman K, Thomas M - Comput Math Methods Med (2016)

Bottom Line: Some of the theoretical and epidemiological findings indicate that the stable disease-free equilibrium is globally asymptotically stable when the associated reproduction number is less than unity.Furthermore, the model undergoes, in the presence of disease induced mortality, the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity.This is further emphasized by the sensitivity analysis results, which shows that the dominant parameters of the model are not altered by the inclusion of age structure.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, KS 66045, USA.

ABSTRACT
We developed a new age-structured deterministic model for the transmission dynamics of chikungunya virus. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicate that the stable disease-free equilibrium is globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model undergoes, in the presence of disease induced mortality, the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the qualitative dynamics of the model are not altered by the inclusion of age structure. This is further emphasized by the sensitivity analysis results, which shows that the dominant parameters of the model are not altered by the inclusion of age structure. However, the numerical simulations show the flaw of the exclusion of age in the transmission dynamics of chikungunya with regard to control implementations. The exclusion of age structure fails to show the age distribution needed for an effective age based control strategy, leading to a one size fits all blanket control for the entire population.

No MeSH data available.


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Systematic flow diagram of the age-structured chikungunya model (10).
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fig1: Systematic flow diagram of the age-structured chikungunya model (10).

Mentions: The population of the susceptible mosquitoes (SM) is generated by the recruitment rate πM and reduced following effective contact with an infected human. All mosquitoes classes are reduced by natural death at a rate μM. The equation for this class is given as follows:(7)dSMdt=πM−βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−μMSM.Mosquitoes in the exposed class EM are generated following the infection of the susceptible mosquitoes. They progress to the infected class at a rate σM. The equation for the exposed mosquitoes dynamics is given as follows:(8)dEMdt=βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−μMSM−σMEM.The infected mosquitoes class are populated from the exposed mosquitoes. The equation for this class is given as follows:(9)dIMdt=σMEM−μMIM.Combining the aforementioned derivations and assumptions the model for the transmission dynamics of chikungunya virus in a population is given by the following deterministic system of nonlinear differential equations:(10)dSJdt=πJ−βJbMSJIMNH−αSJ−μJSJ,dEJdt=βJbMSJIMNH−αEJ−σJ+μJEJ,dIAJdt=εJσJEJ−αIAJ−γAJ+μJIAJ,dISJdt=1−εJσJEJ−αISJ−γSJ+μJISJ,dRJdt=γAJIAJ+γSJISJ−αRJ−μJRJ,dSAdt=αSJ−βAbMSAIMNH−ξSA−μASA,dEAdt=αEJ+βAbMSAIMNH−ξEA−σA+μAEA,dIAAdt=αIAJ+εAσAEA−ξIAA−γAA+μAIAA,dISAdt=αISJ+1−εAσAEA−ξISA−γSA+μAISA,dRAdt=αRJ+γAAIAA+γSAISA−ξRA−μARA,dSSdt=ξSA−βSbMSSIMNH−μSSS,dESdt=ξEA+βSbMSSIMNH−σS+μSES,dIASdt=ξIAA+εSσSES−γAS+μSIAS,dISSdt=ξISA+1−εSσSES−γSS+μSISS,dRSdt=ξRA+γASIAS+γSSISS−μSRS,dSMdt=πM−βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−μMSM,dEMdt=βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−σM+μMEM,dIMdt=σMEM−μMIM.The flow diagram of the age-structured chikungunya model (10) is depicted in Figure 1 and the associated variables and parameters are described in Table 1. Model (10) is an extension of some of the chikungunya transmission models (e.g., those in [21, 25–30]) by (inter alia):


Mathematical Model of Three Age-Structured Transmission Dynamics of Chikungunya Virus.

Agusto FB, Easley S, Freeman K, Thomas M - Comput Math Methods Med (2016)

Systematic flow diagram of the age-structured chikungunya model (10).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4835678&req=5

fig1: Systematic flow diagram of the age-structured chikungunya model (10).
Mentions: The population of the susceptible mosquitoes (SM) is generated by the recruitment rate πM and reduced following effective contact with an infected human. All mosquitoes classes are reduced by natural death at a rate μM. The equation for this class is given as follows:(7)dSMdt=πM−βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−μMSM.Mosquitoes in the exposed class EM are generated following the infection of the susceptible mosquitoes. They progress to the infected class at a rate σM. The equation for the exposed mosquitoes dynamics is given as follows:(8)dEMdt=βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−μMSM−σMEM.The infected mosquitoes class are populated from the exposed mosquitoes. The equation for this class is given as follows:(9)dIMdt=σMEM−μMIM.Combining the aforementioned derivations and assumptions the model for the transmission dynamics of chikungunya virus in a population is given by the following deterministic system of nonlinear differential equations:(10)dSJdt=πJ−βJbMSJIMNH−αSJ−μJSJ,dEJdt=βJbMSJIMNH−αEJ−σJ+μJEJ,dIAJdt=εJσJEJ−αIAJ−γAJ+μJIAJ,dISJdt=1−εJσJEJ−αISJ−γSJ+μJISJ,dRJdt=γAJIAJ+γSJISJ−αRJ−μJRJ,dSAdt=αSJ−βAbMSAIMNH−ξSA−μASA,dEAdt=αEJ+βAbMSAIMNH−ξEA−σA+μAEA,dIAAdt=αIAJ+εAσAEA−ξIAA−γAA+μAIAA,dISAdt=αISJ+1−εAσAEA−ξISA−γSA+μAISA,dRAdt=αRJ+γAAIAA+γSAISA−ξRA−μARA,dSSdt=ξSA−βSbMSSIMNH−μSSS,dESdt=ξEA+βSbMSSIMNH−σS+μSES,dIASdt=ξIAA+εSσSES−γAS+μSIAS,dISSdt=ξISA+1−εSσSES−γSS+μSISS,dRSdt=ξRA+γASIAS+γSSISS−μSRS,dSMdt=πM−βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−μMSM,dEMdt=βMbMIAJ+ISJ+IAA+ISA+IAS+ISSNHSM−σM+μMEM,dIMdt=σMEM−μMIM.The flow diagram of the age-structured chikungunya model (10) is depicted in Figure 1 and the associated variables and parameters are described in Table 1. Model (10) is an extension of some of the chikungunya transmission models (e.g., those in [21, 25–30]) by (inter alia):

Bottom Line: Some of the theoretical and epidemiological findings indicate that the stable disease-free equilibrium is globally asymptotically stable when the associated reproduction number is less than unity.Furthermore, the model undergoes, in the presence of disease induced mortality, the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity.This is further emphasized by the sensitivity analysis results, which shows that the dominant parameters of the model are not altered by the inclusion of age structure.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, KS 66045, USA.

ABSTRACT
We developed a new age-structured deterministic model for the transmission dynamics of chikungunya virus. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicate that the stable disease-free equilibrium is globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model undergoes, in the presence of disease induced mortality, the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the qualitative dynamics of the model are not altered by the inclusion of age structure. This is further emphasized by the sensitivity analysis results, which shows that the dominant parameters of the model are not altered by the inclusion of age structure. However, the numerical simulations show the flaw of the exclusion of age in the transmission dynamics of chikungunya with regard to control implementations. The exclusion of age structure fails to show the age distribution needed for an effective age based control strategy, leading to a one size fits all blanket control for the entire population.

No MeSH data available.


Related in: MedlinePlus