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

  • Received: 01 February 2012 Accepted: 29 June 2018 Published: 01 January 2013
  • MSC : Primary: 92D25, 92D30; Secondary: 37G99.

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    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.

    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): 463-481. doi: 10.3934/mbe.2013.10.463

    Related Papers:

  • 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.


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    [1] Oxford University Press, Oxford, 1991.
    [2] in "Population Dynamics Of Infectious Diseases: Theory and Applications" (ed. R. M. Anderson), Chapman And Hall Press, (1982), 139-179.
    [3] Bull. Math. Biol., 73 (2011), 639-657.
    [4] in "Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics" (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50.
    [5] J. Math. Biol., 35 (1990), 503-522.
    [6] J. R. Soc. Interface, 7 (2011), 873-885.
    [7] Can. Appl. Math. Q., 14 (2006), 259-284.
    [8] Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
    [9] Springer-Verlag, New York, 1993.
    [10] J. R. Soc. Interface, 2 (2005), 281-293.
    [11] Comm. Pure Appl. Math., 38 (1985), 733-753.
    [12] Math. Med. Biol., 21 (2004), 75-83.
    [13] Math. Biosci. Eng., 1 (2004), 57-60.
    [14] J. Math. Biol., 62 (2011), 543-568.
    [15] Benjamin, Menlo Park, California, 1971.
    [16] Trop. Dis. Bull., 49 (1952), 569-585.
    [17] Bull. WHO, 15 (1956), 613-626.
    [18] Oxford University Press, London, 1957.
    [19] J. Murray, London, 1910.
    [20] Bull. Math. Biol., 70 (2008), 1098-1114.
    [21] 41. AMS, Providence, 1995.
    [22] Malaria Journal, 3 (2004), 24 pp.
    [23] Princeton University Press, Princeton, NJ, 2003.
    [24] SIAM J. Math. Anal., 24 (1993), 407-435.
    [25] Math. Biosci. Eng., 4 (2007), 205-219.
    [26] Math. Biosci., 180 (2002), 29-48.
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