# American Institute of Mathematical Sciences

• Previous Article
Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays
• MBE Home
• This Issue
• Next Article
Dynamics of an infectious diseases with media/psychology induced non-smooth incidence
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, (1991). Google Scholar [2] J. L. Aron and R. M. May, The population dynamics of malaria,, in, (1982), 139. Google Scholar [3] F. Chanchod and N. F. Britton, Analysis of a vector-bias model on malaria,, Bull. Math. Biol., 73 (2011), 639. doi: 10.1007/s11538-010-9545-0. Google Scholar [4] C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models,, in, (1995), 33. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar [9] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993). Google Scholar [10] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproduction ratio,, J. R. Soc. Interface, 2 (2005), 281. Google Scholar [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. doi: 10.1002/cpa.3160380607. Google Scholar [12] A. Korobeinikov, Lyapunov function and global properties for SIR and SEIR epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1007/s11538-008-9352-z. Google Scholar [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. doi: 10.3934/mbe.2004.1.57. Google Scholar [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. doi: 10.1007/s00285-010-0346-8. Google Scholar [15] R. K. Miller, "Nonlinear Volterra Integral Equations,", Benjamin, (1971). Google Scholar [16] G. Macdonald, The analysis of sporozoite rate,, Trop. Dis. Bull., 49 (1952), 569. Google Scholar [17] G. Macdonald, Epidemiological basis of malaria control,, Bull. WHO, 15 (1956), 613. Google Scholar [18] G. Macdonald, "The Epidemiology And Control Of Malaria,", Oxford University Press, (1957). Google Scholar [19] R. Ross, "The Prevention Of Malaria,", J. Murray, (1910). Google Scholar [20] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for Malaria transmission,, Bull. Math. Biol., 70 (2008), 1098. doi: 10.1007/s11538-007-9292-z. Google Scholar [21] H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", \textbf{41}. AMS, 41 (1995). Google Scholar [22] A. M. Talman, O. Domarle, F. McKenzie, F. Ariey and V. Robert, Gametocytogenesis: the puberty of Plasmodium falciparum,, Malaria Journal, 3 (2004). Google Scholar [23] H. R. Thieme, "Mathematics In Population Biology,", Princeton University Press, (2003). Google Scholar [24] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar [25] P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205. doi: 10.3934/mbe.2007.4.205. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

show all references

##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1991). Google Scholar [2] J. L. Aron and R. M. May, The population dynamics of malaria,, in, (1982), 139. Google Scholar [3] F. Chanchod and N. F. Britton, Analysis of a vector-bias model on malaria,, Bull. Math. Biol., 73 (2011), 639. doi: 10.1007/s11538-010-9545-0. Google Scholar [4] C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models,, in, (1995), 33. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar [9] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993). Google Scholar [10] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproduction ratio,, J. R. Soc. Interface, 2 (2005), 281. Google Scholar [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. doi: 10.1002/cpa.3160380607. Google Scholar [12] A. Korobeinikov, Lyapunov function and global properties for SIR and SEIR epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1007/s11538-008-9352-z. Google Scholar [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. doi: 10.3934/mbe.2004.1.57. Google Scholar [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. doi: 10.1007/s00285-010-0346-8. Google Scholar [15] R. K. Miller, "Nonlinear Volterra Integral Equations,", Benjamin, (1971). Google Scholar [16] G. Macdonald, The analysis of sporozoite rate,, Trop. Dis. Bull., 49 (1952), 569. Google Scholar [17] G. Macdonald, Epidemiological basis of malaria control,, Bull. WHO, 15 (1956), 613. Google Scholar [18] G. Macdonald, "The Epidemiology And Control Of Malaria,", Oxford University Press, (1957). Google Scholar [19] R. Ross, "The Prevention Of Malaria,", J. Murray, (1910). Google Scholar [20] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for Malaria transmission,, Bull. Math. Biol., 70 (2008), 1098. doi: 10.1007/s11538-007-9292-z. Google Scholar [21] H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", \textbf{41}. AMS, 41 (1995). Google Scholar [22] A. M. Talman, O. Domarle, F. McKenzie, F. Ariey and V. Robert, Gametocytogenesis: the puberty of Plasmodium falciparum,, Malaria Journal, 3 (2004). Google Scholar [23] H. R. Thieme, "Mathematics In Population Biology,", Princeton University Press, (2003). Google Scholar [24] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar [25] P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205. doi: 10.3934/mbe.2007.4.205. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar
 [1] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [2] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [3] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [4] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [5] Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653 [6] Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338 [7] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [8] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [9] Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187 [10] Frédéric Grognard, Frédéric Mazenc, Alain Rapaport. Polytopic Lyapunov functions for persistence analysis of competing species. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 73-93. doi: 10.3934/dcdsb.2007.8.73 [11] Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115 [12] Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995 [13] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [14] Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457 [15] Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297 [16] Junyuan Yang, Yuming Chen, Jiming Liu. Stability analysis of a two-strain epidemic model on complex networks with latency. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2851-2866. doi: 10.3934/dcdsb.2016076 [17] Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807 [18] Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE models for populations with many stages. Mathematical Biosciences & Engineering, 2015, 12 (4) : 661-686. doi: 10.3934/mbe.2015.12.661 [19] Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671 [20] Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

2018 Impact Factor: 1.313