Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment. (English) Zbl 1396.92086
Summary: A family of deterministic SEIRS epidemic dynamic models for malaria is presented. The family type is determined by a general functional response for the nonlinear incidence rate of the disease. Furthermore, the malaria models exhibit three random delays – the incubation periods of the plasmodium inside the female mosquito and human hosts, and also the period of effective acquired natural immunity against the disease. Insights about the effects of the delays and the nonlinear incidence rate of the disease on (1) eradication and (2) persistence of malaria in the human population are obtained via analyzing and interpreting the global asymptotic stability results of the disease-free and endemic equilibrium of the system. The basic reproduction numbers and other threshold values for malaria are calculated, and superior threshold conditions for the stability of the equilibria are found. Numerical simulation results are presented.
Keywords:
disease-free and endemic steady states; global asymptotic stability; basic reproduction number; Lyapunov functional techniqueReferences:
| [1] | Anderson, R. M.; May, R. M., Infectious Diseases of Humans: Dynamics and Control, (1991), Oxford University Press, Oxford |
| [2] | Cooke, K. L., Stability analysis for a vector disease model, Rocky Mountain J. Math., 9, 1, 31-42, (1979) · Zbl 0423.92029 |
| [3] | Capasso, V.; Serio, G. A., A generalization of the kermack-mckendrick deterministic epidemic model, Math Biosci., 42, 43, (1978) · Zbl 0398.92026 |
| [4] | Cooke, K. L.; van den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35, 2, 240-260, (1996) · Zbl 0865.92019 |
| [5] | Chitnis, N.; Hyman, J. M.; Cushing, J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70, 1272, (2008) · Zbl 1142.92025 |
| [6] | De la Sen, M.; Alonso-Quesada, S.; Ibeas, A., On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270, 1, 953-976, (2015) · Zbl 1410.92059 |
| [7] | Doolan, D. L.; Dobano, C.; Baird, J. K., Acquired immunity to malaria, Clin. Microbiol. Rev., 22, 1, 13-36, (2009) |
| [8] | Avila, E.; Buonomo, V. B., Analysis of a mosquito-borne disease transmission model with vector stages and nonlinear forces of infection, Ricerche Mat., 64, 2, 377-390, (2015) · Zbl 1335.92089 |
| [9] | Liu, W. M.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, 359, (1987) · Zbl 0621.92014 |
| [10] | Huo, H.-F.; Ma, Z.-P., Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul., 15, 2, 459-468, (2010) · Zbl 1221.34197 |
| [11] | Hyun, M. Y., Malaria transmission model for different levels of acquired immunity and temperature dependent parameters (vector), Rev. Saude Publica, 34, 3, 223-231, (2000) |
| [12] | Hviid, L., Naturally acquired immunity to plasmodium falciparum malaria, Acta Tropica, 95, 3, 270-275, (2005) |
| [13] | Mateusa, J. P.; Silvab, C. M., Existence of periodic solutions of a periodic SEIRS model with general incidence, Nonlinear Anal. Real World Appl., 34, 379-402, (2017) · Zbl 1354.34085 |
| [14] | Mateus, J. P.; Silva, C. M., A non-autonomous SEIRS model with general incidence rate, Appl. Math. Comput., 247, 169-189, (2014) · Zbl 1338.92133 |
| [15] | Crutcher, J. M.; Hoffman, S. L., Medical Microbiology, Malaria, (1996), University of Texas Medical Branch at Galveston, Galveston (TX) |
| [16] | Korobeinikov, A.; Maini, P. K., A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1, 1, 57-60, (2004) · Zbl 1062.92061 |
| [17] | Liu, Q.; Chen, Q., Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A, 428, 140-153, (2015) · Zbl 1400.92515 |
| [18] | Pang, L.; Ruan, S.; Liu, S.; Zhao, Z.; Zhang, X., Transmission dynamics and optimal control of measles epidemics, Appl. Math. Comput., 256, 131-147, (2015) · Zbl 1338.92135 |
| [19] | Ladde, G. S., Cellular systems-II. stability of campartmental systems, Math. Biosci., 30, 1-21, (1976) · Zbl 0327.92004 |
| [20] | Liu, W. M.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, 4, 359-380, (1987) · Zbl 0621.92014 |
| [21] | Moghadas, S. M.; Gumel, A. B., Global stability of a two-stage epidemic model with generalized nonlinear incidence, Math. Comput. Simul., 60, 107-118, (2002) · Zbl 1005.92031 |
| [22] | Macdonald, G., The analysis of infection rates in diseases in which superinfection occurs, Trop. Dis. Bull., 47, 907-915, (1950) |
| [23] | Du, N. H.; Nhu, N. N., Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Appl. Math. Lett., 64, 223-230, (2017) · Zbl 1354.92090 |
| [24] | Ngwa, G. A.; Shu, W., A mathematical model for endemic malaria with variable human and mosquito population, Math. Comput. Model., 32, 747-763 · Zbl 0998.92035 |
| [25] | Ngwa, G. A.; Niger, A. M.; Gumel, A. B., Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector, Appl. Math. Comput., 217, 3286, (2010) · Zbl 1203.92041 |
| [26] | Ngonghala, C. N.; Ngwa, G. A.; Teboh-Ewungkem, M. I., Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission, Math. Biosci., 240, 1, 45-62, (2012) · Zbl 1319.92057 |
| [27] | Liu, Q.; Jiang, D.; Shi, N.; Hayat, T.; Alsaedi, A., Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence, Commun. Nonlinear Sci. Numer. Simul., 40, 89-99, (2016) · Zbl 1510.92215 |
| [28] | Ross, R., The Prevention of Malaria, (1911), John Murray, London |
| [29] | Gao, S.; Teng, Z.; Xie, D., The effects of pulse vaccination on SEIR model with two time delays, Appl. Math. Comput., 201, 1-2, 282-292, (2008) · Zbl 1143.92024 |
| [30] | Pradeep, B. G. Sampath Aruna; Ma, W., Global stability analysis for vector transmission disease dynamic model with non-linear incidence and two time delays, J. Interdiscip. Math., 18, 4, 395-415, (2015) |
| [31] | Syafruddin, S.; Noorani, M. S. Md., Lyapunov function of SIR and SEIR model for transmission of dengue fever disease, Int. J. Simul. Process Model. (IJSPM), 8, 2-3, 177-184, (2013) |
| [32] | Teboh-Ewungkem, M. I.; Yuster, T., A within-vector mathematical model of plasmodium falciparum and implications of incomplete fertilization on optimal gametocyte sex ratio, J. Theory Biol., 264, 273, (2010) · Zbl 1406.92065 |
| [33] | Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42, 931-947, (2000) · Zbl 0967.34070 |
| [34] | Capasso, V., Mathematical Structures of Epidemic Systems, 97, (1993), Springer · Zbl 0798.92024 |
| [35] | Wanduku, D., Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbation, Appl. Math. Comput., 294, 49-76, (2017) · Zbl 1411.34103 |
| [36] | Wanduku, D.; Ladde, G. S., Global properties of a two-scale network stochastic delayed human epidemic dynamic model, Nonlinear Anal. Real World Appl., 13, 794-816, (2012) · Zbl 1238.60046 |
| [37] | Wanduku, D.; Ladde, G. S., The global analysis of a stochastic two-scale network human epidemic dynamic model with varying immunity period, J. Appl. Math. Phys., 5, 1150-1173, (2017) |
| [38] | Wanduku, D.; Ladde, G. S., Global stability of two-scale network human epidemic dynamic model, Neural Parallel Sci. Comput., 19, 65-90, (2011) · Zbl 1228.92069 |
| [39] | Wanduku, D.; Ladde, G. S., Fundamental properties of a two-scale network stochastic human epidemic dynamic model, Neural Parallel Sci. Comput., 19, 229-270, (2011) · Zbl 1260.92083 |
| [40] | Wanduku, D.; Ladde, G. S., Global stability of a two-scale network SIR delayed epidemic dynamic model, Proc. Dynam. Syst. Appl., 6, 437-441, (2012) · Zbl 1332.92077 |
| [41] | Wanduku, D., Two-scale network epidemic dynamic model for vector borne diseases, Proc. Dynam. Syst. Appl., 6, 228-232, (2016) · Zbl 1390.92150 |
| [42] | Xuerong, M., Stochastic Differential Equations and Applications, (2008), Horwood Publishing |
| [43] | Xiao, D.; Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208, 2, 419-429, (2007) · Zbl 1119.92042 |
| [44] | Kyrychko, Y. N.; Blyussb, K. B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. Real World Appl., 6, 3, 495-507, (2005) · Zbl 1144.34374 |
| [45] | Xue, Y.; Duan, X., Dynamic analysis of an SIR epidemic model with nonlinear incidence rate and double delays, Int. J. Inform. Syst. Sci., 7, 1, 92-102, (2011) · Zbl 1498.34196 |
| [46] | Muroya, Y.; Enatsu, Y.; Nakata, Y., Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate, J. Math. Anal. Appl., 377, 1, 1-14, (2011) · Zbl 1242.92053 |
| [47] | Jianga, Z.; Mab, W.; Wei, J., Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model, Math. Comput. Simul., 122, 35-54, (2016) · Zbl 1519.92263 |
| [48] | Bai, Z.; Zhou, Y., Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Anal. Real World Appl., 13, 3, 1060-1068, (2012) · Zbl 1239.34038 |
| [49] | http://www.who.int/denguecontrol/human/en/. |
| [50] | https://www.cdc.gov/malaria/about/disease.html. |
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