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2013, 10(2): 463-481. doi: 10.3934/mbe.2013.10.463

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

Received  February 2012 Revised  August 2012 Published  January 2013

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. Mathematical Biosciences & Engineering, 2013, 10 (2) : 463-481. doi: 10.3934/mbe.2013.10.463
##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. [2] J. L. Aron and R. M. May, The population dynamics of malaria, in "Population Dynamics Of Infectious Diseases: Theory and Applications" (ed. R. M. Anderson), Chapman And Hall Press, (1982), 139-179. [3] F. Chanchod and N. F. Britton, Analysis of a vector-bias model on malaria, Bull. Math. Biol., 73 (2011), 639-657. doi: 10.1007/s11538-010-9545-0. [4] C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in "Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics" (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. [5] O. Diekmann, J. A. P Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, J. Math. Biol., 35 (1990), 503-522. [6] O. Diekmann, J. A. P. Heesterbeek and M. G. Robert, The construction of next generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2011), 873-885. [7] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. [8] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [9] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. [10] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproduction ratio, J. R. Soc. Interface, 2 (2005), 281-293. [11] W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607. [12] A. Korobeinikov, Lyapunov function and global properties for SIR and SEIR epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1007/s11538-008-9352-z. [13] A. Korobeinikov and P. K. Maini, A lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57. [14] Y. Lou and X-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8. [15] R. K. Miller, "Nonlinear Volterra Integral Equations," Benjamin, Menlo Park, California, 1971. [16] G. Macdonald, The analysis of sporozoite rate, Trop. Dis. Bull., 49 (1952), 569-585. [17] G. Macdonald, Epidemiological basis of malaria control, Bull. WHO, 15 (1956), 613-626. [18] G. Macdonald, "The Epidemiology And Control Of Malaria," Oxford University Press, London, 1957. [19] R. Ross, "The Prevention Of Malaria," J. Murray, London, 1910. [20] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for Malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [21] H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," 41. AMS, Providence, 1995. [22] A. M. Talman, O. Domarle, F. McKenzie, F. Ariey and V. Robert, Gametocytogenesis: the puberty of Plasmodium falciparum, Malaria Journal, 3 (2004), 24 pp. [23] H. R. Thieme, "Mathematics In Population Biology," Princeton University Press, Princeton, NJ, 2003. [24] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [25] P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205. [26] P. van den Drissche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

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##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. [2] J. L. Aron and R. M. May, The population dynamics of malaria, in "Population Dynamics Of Infectious Diseases: Theory and Applications" (ed. R. M. Anderson), Chapman And Hall Press, (1982), 139-179. [3] F. Chanchod and N. F. Britton, Analysis of a vector-bias model on malaria, Bull. Math. Biol., 73 (2011), 639-657. doi: 10.1007/s11538-010-9545-0. [4] C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in "Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics" (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. [5] O. Diekmann, J. A. P Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, J. Math. Biol., 35 (1990), 503-522. [6] O. Diekmann, J. A. P. Heesterbeek and M. G. Robert, The construction of next generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2011), 873-885. [7] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. [8] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [9] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, New York, 1993. [10] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproduction ratio, J. R. Soc. Interface, 2 (2005), 281-293. [11] W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607. [12] A. Korobeinikov, Lyapunov function and global properties for SIR and SEIR epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1007/s11538-008-9352-z. [13] A. Korobeinikov and P. K. Maini, A lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57. [14] Y. Lou and X-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8. [15] R. K. Miller, "Nonlinear Volterra Integral Equations," Benjamin, Menlo Park, California, 1971. [16] G. Macdonald, The analysis of sporozoite rate, Trop. Dis. Bull., 49 (1952), 569-585. [17] G. Macdonald, Epidemiological basis of malaria control, Bull. WHO, 15 (1956), 613-626. [18] G. Macdonald, "The Epidemiology And Control Of Malaria," Oxford University Press, London, 1957. [19] R. Ross, "The Prevention Of Malaria," J. Murray, London, 1910. [20] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for Malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [21] H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," 41. AMS, Providence, 1995. [22] A. M. Talman, O. Domarle, F. McKenzie, F. Ariey and V. Robert, Gametocytogenesis: the puberty of Plasmodium falciparum, Malaria Journal, 3 (2004), 24 pp. [23] H. R. Thieme, "Mathematics In Population Biology," Princeton University Press, Princeton, NJ, 2003. [24] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [25] P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205. [26] P. van den Drissche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
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