| Citation: |
E. Takoutsing, S. Bowong, D. Yemele, A. Temgoua. DYNAMICS OF AN INTRA-HOST MODEL OF MALARIA WITH A CONSTANT DRUG EFFICIENCY[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2037-2069. doi: 10.11948/20160266
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DYNAMICS OF AN INTRA-HOST MODEL OF MALARIA WITH A CONSTANT DRUG EFFICIENCY
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Abstract
In this paper, we investigate the dynamics of an intra-host model of malaria with logistic red blood growth, treatment and immune response. We provide a theoretical study of the model. We derive the basic reproduction number $\mathcal R_f$ which determines the extinction and the persistence of malaria within the body of a host. We compute equilibria and study their stability. More precisely, we show that there exists a threshold parameter $\zeta$ such that if $\mathcal R_f\leq\zeta\leq1$, the disease-free equilibrium is globally asymptotically stable. However, if $\mathcal R_f>1$, there exist two malaria infection equilibria which are locally asymptotically stable: one malaria infection equilibrium without immune response and one malaria infection equilibrium with immune response. The sensitivity analysis of the model has been performed in order to determine the impact of related parameters on outbreak severity. The theory is supported by numerical simulations. We also derive a spatio-temporal model, using Diffusion-Reaction equations to model parasites dispersal. Finally, we provide numerical simulations for parasites spreading, and test different treatment scenarios.-
Keywords:
- Malaria /
- intra-host models /
- drug efficiency /
- stability /
- sensitivity analysis
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References
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E. Takoutsing, S. Bowong, D. Yemele, A. Temgoua. DYNAMICS OF AN INTRA-HOST MODEL OF MALARIA WITH A CONSTANT DRUG EFFICIENCY[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2037-2069. doi: 10.11948/20160266
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Figure 1. Time plot of densities of (a) RBCs
$ x(t) $ , (b) PRBCs$ y(t) $ , (c) free merozoites$ m(t) $ , (d) immune effectors$ I(t) $ and (e) gametocytes$ g(t) $ using various initial conditions when$ \beta = 10 $ and$ f = 0.6 $ (so that$ \zeta = 0.4295 $ ,$ \mathcal R_f = 0.3818<\zeta $ ). All other parameter values are given in Table 1. -
Figure 2. Time plot of densities of (a) RBCs
$ x(t) $ , (b) PRBCs$ y(t) $ , (c) free merozoites$ m(t) $ , (d) immune effectors$ I(t) $ and (e) gametocytes$ g(t) $ using various initial conditions when$ \beta = 16 $ ,$ f_c = 0.7188 $ and$ f = 0.8 $ (so that$ \zeta = 0.4227 $ ,$ \mathcal R_0 = 1.5030>1 $ ,$ \mathcal R_f = 0.3006<\zeta $ and$ f>f_c $ ). All other parameter values are given in Table 1. -
Figure 3. Time plot of densities of (a) RBCs
$ x(t) $ , (b) PRBCs$ y(t) $ , (c) free merozoites$ m(t) $ , (d) gametocytes$ g(t) $ using various initial conditions when$ \beta = 8 $ and$ f = 0.2 $ (so that$ \tau_0 = 0.2362<1 $ ,$ \tilde{\mathcal R}_{0} = 1.3913>1 $ ). All other parameter values are given in Table 1. -
Figure 4. In (a) and (b) we see that there is a curve in the
$ (U, V) $ which plane along with$ F(U, V) = 1 $ and$ G(U, V) = 1 $ , respectively. (c) illustrates that there is a unique point at which$ F(U, V) = G(U, V) = 1 $ . This point determines an endemic equilibrium$ E^* $ . (d) shows the contour curves$ F(U, V) = 1 $ and$ G(U, V) = 1 $ , and there is a unique intersection point. We choose$ \beta = 16 $ ,$ u = 0.5 $ and$ f = 0.2 $ (so that$ \mathcal R_f = 1.1307>1 $ ). All other parameter values are as in Table 1. -
Figure 5. Time plot of densities of (a) RBCs
$ x(t) $ , (b) PRBCs$ y(t) $ , (c) free merozoites$ m(t) $ , (d) immune effectors$ I(t) $ and (e) gametocytes$ g(t) $ using various initial conditions when$ \beta = 16 $ and$ f = 0.2 $ (so that$ \mathcal R_f = 1.12024>1 $ ). All other parameter values are given in Table 1. - Figure 6. Values of sensitivity indexes (PRCC) of uninfected RBCs, PRBCs, free merozoites, immune effectors and gametocytes.
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Figure 7. Spatio-temporal evolution of model system (3.1) after 10 days without any treatment, that is
$ f = 0 $ (so that$ \mathcal R_f = \mathcal R_0 = 1.5030 $ and$ f_c = 0.7188 $ ) All other parameter values as in Table 1. -
Figure 8. Spatio-temporal evolution of model system (3.1) after 10 days with insufficient drug efficiency when
$ f = 0.20 $ and$ f_c = 0.7188 $ (so that$ \mathcal R_f = 1.2024 $ and$ f<f_c $ ). All other parameter values as in Table 1. -
Figure 9. Spatio-temporal evolution of model system (3.1) after 10 days with a sufficient drug efficiency when
$ f = 0.8 $ and$ f_c = 0.7188 $ (so that$ \mathcal R_0 = 1.4134 $ ,$ \mathcal R_f = 0.3006 $ and$ f>f_c $ ). All other parameter values as in Table 1.