On latencies in malaria infections and their impact on the disease dynamics
In this paper, we modify the classic Ross-Macdonald 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 latenci...
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AIMS Press
2012-12-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.463 |
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| author | Yanyu Xiao Xingfu Zou |
| author_facet | Yanyu Xiao Xingfu Zou |
| author_sort | Yanyu Xiao |
| collection | DOAJ |
| description | In this paper, we modify the classic Ross-Macdonald 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 well-posedness 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. |
| format | Article |
| id | doaj-art-0a6eb50cb72241ef849ba7317f135ec1 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2012-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-0a6eb50cb72241ef849ba7317f135ec12025-01-24T02:25:53ZengAIMS PressMathematical Biosciences and Engineering1551-00182012-12-0110246348110.3934/mbe.2013.10.463On latencies in malaria infections and their impact on the disease dynamicsYanyu Xiao0Xingfu Zou1Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7In this paper, we modify the classic Ross-Macdonald 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 well-posedness 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.https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.463lyapunov function/functionalstabilitylatencymalariapersistence.basicreproduction numberdelay |
| spellingShingle | Yanyu Xiao Xingfu Zou On latencies in malaria infections and their impact on the disease dynamics Mathematical Biosciences and Engineering lyapunov function/functional stability latency malaria persistence. basicreproduction number delay |
| title | On latencies in malaria infections and their impact on the disease dynamics |
| title_full | On latencies in malaria infections and their impact on the disease dynamics |
| title_fullStr | On latencies in malaria infections and their impact on the disease dynamics |
| title_full_unstemmed | On latencies in malaria infections and their impact on the disease dynamics |
| title_short | On latencies in malaria infections and their impact on the disease dynamics |
| title_sort | on latencies in malaria infections and their impact on the disease dynamics |
| topic | lyapunov function/functional stability latency malaria persistence. basicreproduction number delay |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.463 |
| work_keys_str_mv | AT yanyuxiao onlatenciesinmalariainfectionsandtheirimpactonthediseasedynamics AT xingfuzou onlatenciesinmalariainfectionsandtheirimpactonthediseasedynamics |