On latencies in malaria infections and their impact on the disease dynamics

1.
Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7

2.
Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received:
01 February 2012
Accepted:
29 June 2018
Published:
01 January 2013


MSC :
Primary: 92D25, 92D30; Secondary: 37G99.


>
In this paper, we modify the classic RossMacdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions ($P_1(t)$ and $P_2(t)$) to reflect the fact that the latencies differ from individuals to individuals. We justify the wellposedness of the new model, identify the basic reproduction number $\mathcal{R}_0$ for the model and analyze the dynamics of the model. We show that when $\mathcal{R}_0 <1 the="" disease="" free="" equilibrium="" e_0="" is="" globally="" asymptotically="" stable="" meaning="" that="" the="" malaria="" disease="" will="" eventually="" die="" out="" and="" if="" mathcal="" r="" _0="">1$, $E_0$ becomes unstable.When $\mathcal{R}_0 >1$, we consider two specific forms for $P_1(t)$ and $P_2(t)$: (i) $P_1(t)$ and $P_2(t)$ are both exponential functions; (ii) $P_1(t)$ and $P_2(t)$ are both step functions.For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when $\mathcal{R}_0>1$ then the disease will persist; moreover if there is no recovery ($\gamma_1=0$), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.
Citation: Yanyu Xiao, Xingfu Zou. On latencies in malaria infections and their impact on the disease dynamics[J]. Mathematical Biosciences and Engineering, 2013, 10(2): 463481. doi: 10.3934/mbe.2013.10.463

Abstract
In this paper, we modify the classic RossMacdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions ($P_1(t)$ and $P_2(t)$) to reflect the fact that the latencies differ from individuals to individuals. We justify the wellposedness of the new model, identify the basic reproduction number $\mathcal{R}_0$ for the model and analyze the dynamics of the model. We show that when $\mathcal{R}_0 <1 the="" disease="" free="" equilibrium="" e_0="" is="" globally="" asymptotically="" stable="" meaning="" that="" the="" malaria="" disease="" will="" eventually="" die="" out="" and="" if="" mathcal="" r="" _0="">1$, $E_0$ becomes unstable.When $\mathcal{R}_0 >1$, we consider two specific forms for $P_1(t)$ and $P_2(t)$: (i) $P_1(t)$ and $P_2(t)$ are both exponential functions; (ii) $P_1(t)$ and $P_2(t)$ are both step functions.For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when $\mathcal{R}_0>1$ then the disease will persist; moreover if there is no recovery ($\gamma_1=0$), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.
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