Modelling the dynamics of endemic malaria in growing populations

Discrete and Continuous Dynamical Systems - Series B

Volumn 4, Issue 4, 2004, Pages 1173-1202

  Ngwa, G A ^a,b  

^a ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS   (Italy)

^b UNIVERSITY OF BUEA   (Cameroon)

Author keywords

Diffusion approximation; Disease related deaths; Threshold parameter

Indexed keywords

EID: 4944266469     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2004.4.1173     Document Type: Article

References (60)

1. 1. Biometrika 58, (1973) 669-672.
2. R. M. Anderson and R. M. May, POPULATION BIOLOGY OF INFECTIOUS DISEASES: PART I Nature 280 (1979), 361-367.
3. R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control", Oxford University Press, Oxford, (1991).
4. J. L. Aron and R. M. May, THE POPULATION DYNAMICS OF MALARIA, In "Population Dynamics of Infectious Diseases" (R. M. Anderson Ed) Chapman and Hall, London, (1982).
5. Joan L. Aron, MATHEMATICAL MODELING OF IMMUNITY TO MALARIA, Math. Biosci. 90 (1988), 385-396.
6. Joan L. Aron, DYNAMICS OF ACQUIRED IMMUNITY BOOSTED BY EXPOSURE TO INFECTION, Math. Biosci. 64 (1983), 249-253.
7. Joan L. Aron, ACQUIRED IMMUNITY DEPENDENT UPON EXPOSURE IN AN SIRS EPIDEMIC MODEL, Math. Biosci. 88 (1988), 37-47.
8. N. T. J. Bailey, "Elements of Stochastic Processes with Applications to the Natural Sciences," Wiley Publications (1964).
9. T. Bradley, TITUS BRADLEY AND DEPARTMENT OF MICROBIOLOGY AND IMMUNOLOGY OF LEICESTER," MALARIA AND DRUG RESISTANCE" , http://www-micro.msb.le. ac.uk/224/Bradley/Bradley.html, (1996).
10. S. Busenberg and P. Van Den Driessche, ANALYSIS OF A DISEASE TRANSMISSION MODEL IN A POPULATION WITH VARYING SIZE, J. Math. Biol. 28 (1990), 257-270.
11. R. Carter, SPATIAL SIMULATION OF MALARIA TRANSMISSION AND ITS CONTROL BY MALARIA TRANSMISSION BLOCKING VACCINATION International Journal for Parasitology 32 (2002) 1617-1624.
12. W. R. Derrick and P. Van Den Driessche, HOMOCLINIC ORBITS IN A DISEASE TRANSMISSION MODEL WITH NONLINEAR INCIDENCE AND NONCONSTANT POPULATION, Discrete and Continuous Dynamical Systems-Series B Vol 1(2) (2003) pp 233-256
13. 0 IN MODELS FOR INFECTIOUS DISEASES IN HETEROGENEOUS POPULATIONS, J. Math. Biol. 28 (1990), 365-382.
14. K. Dietz, L. Molineaux, and A. Thomas, A MALARIA MODEL TESTED IN THE AFRICAN SAVANNA, Bull. WHO 50 (1974), 347-357.
15. K. Dietz, MODELS FOR PARASITIC DISEASE CONTROL, Bull Intern. Statist. Inst. 46 (1975), 531-544.
16. A. C. Elizabeth, Newton and Paul Reiter, A MODEL FOR THE TRANSMISSION OF DENGUE FEVER WITH AN EVALUATION OF THE IMPACT OF ULTRA-LOW VOLUME (ULV) INSECTICIDE APPLICATIONS ON DENGUE EPIDEMICS, Am. J. Trop. Med. Hyg. 47(6) (1992), 709-720.
17. J. Gani, ON THE PARTIAL DIFFERENTIAL EQUATION OF EPIDEMIC THEORY I, Biometrika, 52 (1965), 617-622.
18. L. Q. Gao and H. W. Hethcote, DISEASE TRANSMISSION MODELS WITH DENSITY-DEPENDENT DEMOGRAPHICS. J. Math. Biol. 30:693-716, (1992).
19. W. C. Gardiner, "Hand Book of Stochastic Methods for Physics, Chemistry and the Natural Sciences", Second Edition, Springer, (1985).
20. M. Gatton, W. Hogarth, A. Saul, P. Dayananda, A MODEL FOR PREDICTING THE TRANSMISSION RATE OF MALARIA FROM SEROLOGICAL DATA, J. math. Biol. 34 (1996),878-888.
21. H. M. Giles and D. A. Warrel, "Bruce-Chwatt's Essential Malariology". 3rd ed, Heinemann Medical Books, (1993).
22. D. Greenhalgh, HOPF BIFURCATION IN EPIDEMIC MODELS WITH LATENT PERIOD AND NONPERMANENT IMMUNITY, Math. Comput. Modelling 25(2) (1997), 85-107.
23. S. Gupta, K. Trenholme, M. Cox, R. Anderson and K. Day, THE EFFECT OF ANTIGENIC DIVERSITY ON ENDEMIC PREVALENCE, In "Epidemic Models: Their Structure and Relation to Data", (ed. D. Mollison), Cambridge University Press, 323-336 (1995).
24. J. K. Hale, "Ordinary Differential Equations". John Wiley, (1969).
25. H. W. Hethcote, H. W. Stech and P. van den Driessche, PERIODICITY AND STABILITY IN EPIDEMIC MODELS: A SURVEY, In: "Differential equations and Applications in Ecology, epidemics, and population Problems", (S. Busenberg and K.L Cooke, Eds), Academic Press, San Diego, pp 65-82, (1982).
26. H. W. Hethcote, A THOUSAND AND ONE EPIDEMIC MODELS. In: Frontiers in Mathematical Biology (Simon A. Levin Editor). Lecture Notes in Biomathematics 100 Springer 1984. pp 504-515
27. R. J. Kryscio and C. Lefèvre, ON THE EXTINCTION OF THE SIS STOCHASTIC LOGISTIC EPIDEMIC, J. Appl. Probab., 26 (1989), 685-694.
28. T. G. Kurtz, "Approximation of population processes," SIAM, Philadelphia, (1981).
29. G. Macdonald, THE ANALYSIS OF INFECTION RATES IN DISEASES IN WHICH SUPERINFECTION OCCURS, Trop. Dis. Bull. 47 (1950), 907-915.
30. G. Macdonald, THE ANALYSIS OF SPOROZOITE RATE, Trop. Dis. Bull. 49 (1952), 569-585.
31. G. Macdonald, "The Epidemiology and Control of Malaria," Oxford University Press, London, (1957).
32. P. Martens, R. S. Kovats, S. Nijhof, P. de Vries, M. T. J. Livermore, D. J. Bradley, J. Cox and A. J. McMichael, CLIMATE CHANGE AN FUTURE POPULATIONS AT RISK OF MALARIA Global Environmental Change 9 (1999) S89-S107.
33. C. Matessi, A THEORETICAL APPROACH TO THE DYNAMICS OF SINGLE POPULATIONS. In: Lecture Notes in Biomathematics (S. Levin Ed.) 39: Vito Volterra Symposium of Mathematical Models in Biology, pp 223-247, (1979).
34. D. Mollison, THE STRUCTURE OF EPIDEMIC MODELS, In "Epidemic Models: Their Structure and Relation to Data", (ed. D. Mollison), Cambridge: Cambridge University Press, 17-33 (1995).
35. L. Molineaux and G. Gramiccia, "The Garki Project: Research on the Epidemiology and Control of Malaria in the Sudan Savanna of West Africa", WHO, Geneva, (1980).
36. I. Nåsell, EXTINCTION AND QUASI-STATIONARITY IN THE VERHULST LOGISTIC MODEL, Applied Probability trust. To appear.
37. I. Nåsell, ENDEMICITY, PERSISTENCE AND QUASI-STATIONARITY, IMA proceedings. To Appear.
38. I. Nåsell, MEASLES OUTBREAKS ARE NOT CHAOTIC, IMA Proceedings. To appear.
39. I. Nåsell, ON THE QUASI-STATIONARY DISTRIBUTION OF THE STOCHASTIC LOGISTIC MODEL, Mathematical Biosciences, 156 (1999a), 21-40.
40. I. Nåsell, ON THE TIME TO EXTINCTION OF RECURRENT EPIDEMICS, J. R. Statist. Soc. B 61, Part 2, (1999b) 309-330.
41. I. Nåsell, THE QUASI-STATIONARY DISTRIBUTION OF THE CLOSED ENDEMIC SIS MODEL, Adv. Appl. Prob. 28 (1996), 895-932.
42. Jerry Nedelman, INOCULATION AND RECOVERY RATES IN THE MALARIA MODEL OF DIETZ, MOLINEAUX, AND THOMAS, Math. Biosci. 89 (1983), 209-233.
43. Jerry Nedelman, INTRODUCTORY REVIEW SOME NEW THOUGHTS ABOUT SOME OLD MALARIA MODELS, Math. Biosci. 73 (1985), 158-182.
44. G. A. Ngwa and W. S. Shu, A MATHEMATICAL MODEL FOR ENDEMIC MALARIA WITH VARIABLE HUMAN AND MOSQUITO POPULATIONS, Mathematical and Computer Modelling 32 (2000), 747-763.
45. R. M. Nisbet and W. S. C. Gurney, "Modelling Fluctuating Populations", John Wiley and Sons, Chichester, (1982).
46. R. Pearl and L. J. Reed, ON THE RATE OF GROWTH OF THE POPULATION OF THE UNITED STATES SINCE 1790 AND ITS MATHEMATICAL REPRESENTATION, Proc. National Academy of Sciences, 6 (1920), 275-288.
47. E. Renshaw, "Modelling Biological Populations in Space and Time", Cambridge studies in Mathematical Biology: 11. Cambridge University Press, (1991).
48. S. A. Ritchie and C. L. Montague, SIMULATED POPULATIONS OF THE BLACK SALT MARSH MOSQUITO Aedes taeniorhynchus IN A FLORIDA MANGROVE FOREST Ecological Modelling 77 (1995) 123-141.
49. R. Ross, "THE PREVENTION OF MALARIA", John Murray, London, (1911).
50. P. Shears, EPIDEMIOLOGY AND INFECTION IN FAMINE AND DISASTERS. Epidemiol. Infect. 107:241-251, (1991).
51. B. Singer and Joel E. Cohen, ESTIMATING MALARIA INCIDENCE AND RECOVERY RATES FROM PANEL SURVEYS, Math. Biosci. 49 (1980), 273-305.
52. V. Siskind, SOLUTION OF THE GENERAL STOCHASTIC EPIDEMIC, Biometrika, 52 (1965), 613-616.
53. J-F Trape and C. Rogier, COMBATING MALARIA MORBIDITY AND MORTALITY BY REDUCING TRANSMISSION, Parasitology Today, 12(6) (1996), 236-240.
54. P. F. Verhulst, NOTICE SUR LA LOI QUE LA POPULATION SUIT DANS SON ACROISSEMENT, Correspondence Mathématique et Physique 10 (1838), 113-121.
55. P. Waltman, A THRESHOLD CRITERION FOR THE SPREAD OF INFECTION IN A TWO POPULATION MODEL. Math. Biosci. 21 pp 119-125 (1974).
56. WHO, SELECTED HEALTH PROBLEMS. WORLD HEALTH STATISTICS QUARTERLY, 44 (1991), 189-197.
57. WHO Report, DIVISION OF CONTROL OF TROPICAL DISEASES, update Sept. 1999.
58. WHO, THE URBAN CRISES. WORLD HEALTH STATISTICS QUARTERLY, 44 (1991), 208-233.
59. WHO, THE ROLE AND PARTICIPATION OF EUROPEAN COUNTRIES IN THE FIGHT AGAINST MALARIA, World report on a conference, WHO, Copenhagen, (1980).
60. T. V. William, A. T. Saul, H. P. William and P. F. Brian, "Numerical Recipes in Fortran: The Art of Scientific Computation", Cambridge University Press (1992).