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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

Yanyu Xiao 1, and  Xingfu Zou 2,

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.
Keywords: Lyapunov function/functional, stability, latency, Malaria, persistence., basic reproduction number, delay.
Mathematics Subject Classification: Primary: 92D25, 92D30; Secondary: 37G9.
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.

show all references

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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