Limits...
Quantifying the impact of decay in bed-net efficacy on malaria transmission.

Ngonghala CN, Del Valle SY, Zhao R, Mohammed-Awel J - J. Theor. Biol. (2014)

Bottom Line: The potential impact of ITNs on reducing malaria transmission is limited due to inconsistent or improper use, as well as physical decay in effectiveness.We develop a model for malaria spread that captures the decrease in ITN effectiveness due to physical and chemical decay, as well as human behavior as a function of time.These analyses show that the basic reproduction number R0, and the infectious human population are most sensitive to bed-net coverage and the biting rate of mosquitoes.

View Article: PubMed Central - PubMed

Affiliation: Department of Global Health and Social Medicine, Harvard Medical School, Boston, MA 02115, USA; National Institute for Mathematical and Biological Synthesis, Knoxville, TN 37996-1527, USA. Electronic address: Calistus_Ngonghala@hms.harvard.edu.

Show MeSH

Related in: MedlinePlus

Bifurcation diagrams for γ̃h = 1/30, ρh = 9 × 10−3 and different values of the human disease-induced death rate δh. On the graphs, solid red lines represent stable interior or endemic equilibrium solutions Ee, dashed red lines represent unstable endemic equilibrium solutions, solid blue lines represent stable disease-free equilibrium solutions Edf, and dashed blue lines represent unstable disease-free equilibrium solutions. Graphs (a)–(c) show the infectious human population Ih as a function of the basic reproduction number R0. Graph (a) shows a forward bifurcation for δh = 9.0 × 10−5, which might represent a mild version of malaria, Graph (b) is the bifurcation diagram for the case δh = 1.5 × 10−4, and Graph (c) shows a backward bifurcation for δh = 3.4 × 10−4, which might represent a severe version of malaria. On the second row, the basic reproduction number R0 is plotted against ITN coverage, b. The minimum level of ITN coverage required contain the malaria disease is given by the point of intersection of the solid green curve and (i) the dashed green line when there is backward bifurcation (Graphs (e) and (f)) and (ii) the solid purple line when there is no backward bifurcation (Graph (d)). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4374367&req=5

Figure 3: Bifurcation diagrams for γ̃h = 1/30, ρh = 9 × 10−3 and different values of the human disease-induced death rate δh. On the graphs, solid red lines represent stable interior or endemic equilibrium solutions Ee, dashed red lines represent unstable endemic equilibrium solutions, solid blue lines represent stable disease-free equilibrium solutions Edf, and dashed blue lines represent unstable disease-free equilibrium solutions. Graphs (a)–(c) show the infectious human population Ih as a function of the basic reproduction number R0. Graph (a) shows a forward bifurcation for δh = 9.0 × 10−5, which might represent a mild version of malaria, Graph (b) is the bifurcation diagram for the case δh = 1.5 × 10−4, and Graph (c) shows a backward bifurcation for δh = 3.4 × 10−4, which might represent a severe version of malaria. On the second row, the basic reproduction number R0 is plotted against ITN coverage, b. The minimum level of ITN coverage required contain the malaria disease is given by the point of intersection of the solid green curve and (i) the dashed green line when there is backward bifurcation (Graphs (e) and (f)) and (ii) the solid purple line when there is no backward bifurcation (Graph (d)). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Mentions: We now illustrate numerically the existence and stability of endemic equilibria and the occurrence of a backward bifurcation for system (2.4). Unless otherwise stated here and in subsequent sections, the parameters used in our simulations are the baseline parameters provided in Table 1. Fig. 3 illustrates the dynamics of system (2.4) for γ̃h = 1/30, ρh = 9 × 10−3 and within parameter regimes determined by the human disease-induced mortality rate δh, where a forward or a backward bifurcation occurs. Apart from the fact that a higher value of δh might be required for a backward bifurcation, the same qualitative results are obtained with the baseline values of γ̃h and ρh. Fig. 3(a) shows a forward bifurcation, wherein a stable disease-free equilibrium solution exists when R0 <1 and a stable endemic equilibrium solution exists when R0 >1. Fig. 3(b) shows a barely noticeable backward bifurcation when . Figure 3(c) indicates that for , there is an asymptotically stable disease-free equilibrium solution Edf and for , there is a backward bifurcation. This result implies that a locally stable disease-free equilibrium solution coexists with a locally stable endemic equilibrium and an unstable endemic equilibrium. These results confirm Theorems 3.2 and 3.3.


Quantifying the impact of decay in bed-net efficacy on malaria transmission.

Ngonghala CN, Del Valle SY, Zhao R, Mohammed-Awel J - J. Theor. Biol. (2014)

Bifurcation diagrams for γ̃h = 1/30, ρh = 9 × 10−3 and different values of the human disease-induced death rate δh. On the graphs, solid red lines represent stable interior or endemic equilibrium solutions Ee, dashed red lines represent unstable endemic equilibrium solutions, solid blue lines represent stable disease-free equilibrium solutions Edf, and dashed blue lines represent unstable disease-free equilibrium solutions. Graphs (a)–(c) show the infectious human population Ih as a function of the basic reproduction number R0. Graph (a) shows a forward bifurcation for δh = 9.0 × 10−5, which might represent a mild version of malaria, Graph (b) is the bifurcation diagram for the case δh = 1.5 × 10−4, and Graph (c) shows a backward bifurcation for δh = 3.4 × 10−4, which might represent a severe version of malaria. On the second row, the basic reproduction number R0 is plotted against ITN coverage, b. The minimum level of ITN coverage required contain the malaria disease is given by the point of intersection of the solid green curve and (i) the dashed green line when there is backward bifurcation (Graphs (e) and (f)) and (ii) the solid purple line when there is no backward bifurcation (Graph (d)). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4374367&req=5

Figure 3: Bifurcation diagrams for γ̃h = 1/30, ρh = 9 × 10−3 and different values of the human disease-induced death rate δh. On the graphs, solid red lines represent stable interior or endemic equilibrium solutions Ee, dashed red lines represent unstable endemic equilibrium solutions, solid blue lines represent stable disease-free equilibrium solutions Edf, and dashed blue lines represent unstable disease-free equilibrium solutions. Graphs (a)–(c) show the infectious human population Ih as a function of the basic reproduction number R0. Graph (a) shows a forward bifurcation for δh = 9.0 × 10−5, which might represent a mild version of malaria, Graph (b) is the bifurcation diagram for the case δh = 1.5 × 10−4, and Graph (c) shows a backward bifurcation for δh = 3.4 × 10−4, which might represent a severe version of malaria. On the second row, the basic reproduction number R0 is plotted against ITN coverage, b. The minimum level of ITN coverage required contain the malaria disease is given by the point of intersection of the solid green curve and (i) the dashed green line when there is backward bifurcation (Graphs (e) and (f)) and (ii) the solid purple line when there is no backward bifurcation (Graph (d)). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
Mentions: We now illustrate numerically the existence and stability of endemic equilibria and the occurrence of a backward bifurcation for system (2.4). Unless otherwise stated here and in subsequent sections, the parameters used in our simulations are the baseline parameters provided in Table 1. Fig. 3 illustrates the dynamics of system (2.4) for γ̃h = 1/30, ρh = 9 × 10−3 and within parameter regimes determined by the human disease-induced mortality rate δh, where a forward or a backward bifurcation occurs. Apart from the fact that a higher value of δh might be required for a backward bifurcation, the same qualitative results are obtained with the baseline values of γ̃h and ρh. Fig. 3(a) shows a forward bifurcation, wherein a stable disease-free equilibrium solution exists when R0 <1 and a stable endemic equilibrium solution exists when R0 >1. Fig. 3(b) shows a barely noticeable backward bifurcation when . Figure 3(c) indicates that for , there is an asymptotically stable disease-free equilibrium solution Edf and for , there is a backward bifurcation. This result implies that a locally stable disease-free equilibrium solution coexists with a locally stable endemic equilibrium and an unstable endemic equilibrium. These results confirm Theorems 3.2 and 3.3.

Bottom Line: The potential impact of ITNs on reducing malaria transmission is limited due to inconsistent or improper use, as well as physical decay in effectiveness.We develop a model for malaria spread that captures the decrease in ITN effectiveness due to physical and chemical decay, as well as human behavior as a function of time.These analyses show that the basic reproduction number R0, and the infectious human population are most sensitive to bed-net coverage and the biting rate of mosquitoes.

View Article: PubMed Central - PubMed

Affiliation: Department of Global Health and Social Medicine, Harvard Medical School, Boston, MA 02115, USA; National Institute for Mathematical and Biological Synthesis, Knoxville, TN 37996-1527, USA. Electronic address: Calistus_Ngonghala@hms.harvard.edu.

Show MeSH
Related in: MedlinePlus