 |
 |
 |
 |
 |
 |
Sains Malaysiana 53(12)(2024): 3425-3435
http://doi.org/10.17576/jsm-2024-5312-23
A Compartmental
Model for the Transmission Dynamics of Rabies Disease in Dog Population
(Suatu Model Petak untuk Transmisi Dinamik Penyakit Rabies dalam Populasi Anjing)
THANISHA
KALIAPAN1, NYUK SIAN CHONG1,2,*, ILYANI ABDULLAH1,2,
ZABIDIN SALLEH1,2 & JANE LABADIN3
1Faculty of Computer Science and
Mathematics, Universiti Malaysia Terengganu, 21030
Kuala Nerus, Terengganu, Malaysia
Diserahkan: 6 Februari 2024/Diterima: 21 Oktober 2024
Abstract
Dogs are
the main source of more than 90% of human rabies infections that pose a
significant threat to public health, primarily in Africa and Asia. However, it
is also one of the viral diseases that can be prevented by vaccination that
affects both warm-blooded animals and humans. There are two types of rabies
vaccines: pre-exposure prophylaxis and post-exposure prophylaxis (PEP). Mathematical models can be valuable tools for
predicting and controlling the spread of rabies disease. Thus, we introduce an
SEIV (Susceptible-Exposed-Infected-Vaccinated) model incorporate vaccination
control strategy to examine the transmission dynamics of rabies disease in dog
population. The basic reproduction number, , positively invariant and attracting region,
steady states, and the stability analysis of the model are investigated. We
find that there are two equilibria exist in the model, i.e., disease-free and
endemic equilibria. To prove the global stability of disease-free and endemic
equilibria, the theory of asymptotic autonomous system and geometric approach
have been applied, respectively. Hence, we find that the disease-free and
endemic equilibria are globally asymptotically stable if and , respectively. Numerical
simulations are performed to depict the dynamics of the model. As a conclusion,
we will be able to control the disease effectively if the vaccination rate is
sufficiently large.
Keywords:
Nonlinear ordinary differential equation; numerical simulation; rabies disease;
stability analysis; steady state
Abstrak
Anjing merupakan punca utama lebih daripada 90% jangkitan rabies manusia yang menimbulkan ancaman ketara kepada kesihatan awam, terutamanya di Afrika
dan Asia. Walau bagaimanapun, ia juga merupakan salah satu penyakit virus yang boleh dicegah dengan vaksinasi yang memberi kesan kepada haiwan berdarah panas dan manusia. Terdapat dua jenis vaksin rabies: profilaksis pra-pendedahan dan profilaksis pasca-pendedahan (PPP). Model matematik boleh menjadi alat yang berguna untuk meramal dan mengawal penyebaran penyakit rabies. Oleh itu, kami memperkenalkan suatu model SEIV (Rentan-Terdedah-Dijangkiti-Diberi vaksin)
yang menggabungkan strategi kawalan vaksinasi untuk mengkaji dinamik penularan penyakit rabies dalam populasi anjing. Nombor reproduksi asas, , rantau invarian positif dan menarik, keadaan pegun dan analisis kestabilan model telah dikaji. Kami mendapati bahawa terdapat dua titik keseimbangan yang wujud dalam model, iaitu, titik keseimbangan bebas penyakit dan endemik. Untuk membuktikan kestabilan global bagi titik keseimbangan bebas penyakit dan endemik, teori sistem autonomi asimptotik dan pendekatan geometri masing-masing telah diaplikasi. Oleh itu, kami mendapati bahawa titik keseimbangan bebas penyakit dan endemik masing-masing mencapai kestabilan secara asimptotik secara global jika dan . Simulasi berangka dijalankan untuk menggambarkan kedinamikan model. Kesimpulannya, kita akan dapat mengawal penyakit ini dengan berkesan sekiranya kadar vaksinasi cukup besar.
Kata kunci: Analisis kestabilan; keadaan pegun; penyakit rabies; persamaan pembezaan biasa tak linear; simulasi berangka
RUJUKAN
Abdulmajid, S. & Hassan, A.S. 2021. Analysis of
time delayed Rabies model in human and dog populations with controls. Afrika Matematika 32(5): 1067-1085.
Bornaa, C.S., Seidu, B.
& Daabo, M.I. 2020. Mathematical analysis of
rabies infection. Journal of Applied Mathematics 2020: 1804270.
https://doi.org/10.1155/2020/1804270
Bryce, C.M. 2021. Dogs as pets and pests:
Global patterns of canine abundance, activity, and health. Integrative and
Comparative Biology 61(1): 154-165.
CDC. 2023. Information for Veterinarians.
Rabies. https://www.cdc.gov/rabies/hcp/
veterinarians/index.html. Accessed 24
August 2024.
Colombi, D., Poletto,
C., Nakouné, E., Bourhy, H.
& Colizza, V. 2020. Long-range movements coupled
with heterogeneous incubation period sustain dog rabies at the national scale
in Africa. PLoS Neglected Tropical Diseases 14(5): e0008317.
Coppel, W.A. & Vleck,
F.S.V. 1968. Stability and asymptotic behavior of
differential equations. The American Mathematical Monthly 75(3): 319. doi:10.2307/2315010
DeJesus, E.X. & Kaufman, C. 1987.
Routh-Hurwitz criterion in the examination of eigenvalues of a system of
nonlinear ordinary differential equations. Physical Review A 35(12):
5288.
Diekmann, O., Heesterbeek,
J.A.P. & Metz, J.A.J. 1990. On the definition and the computation of the
basic reproduction ratio R 0 in models for infectious diseases in heterogeneous
populations. Journal of Mathematical Biology 28: 365-382.
Eze, O.C., Mbah,
G.E., Nnaji, D.U. & Onyiaji, N.E. 2020.
Mathematical modelling of transmission dynamics of rabies virus. International
Journal of Mathematics Trends and Technology 66(7): 2231-5373.
Freedman, H.I., Ruan,
S. & Tang, M. 1994. Uniform persistence and flows near a closed positively
invariant set. Journal of Dynamics and Differential Equations 6:
583-600.
Gold, S., Donnelly, C.A., Woodroffe, R.
& Nouvellet, P. 2021. Modelling the influence of
naturally acquired immunity from subclinical infection on outbreak dynamics and
persistence of rabies in domestic dogs. PLoS Neglected Tropical Diseases 15(7): e0009581.
Gold, S., Donnelly, C.A., Nouvellet, P. & Woodroffe, R. 2020. Rabies
virus-neutralising antibodies in healthy, unvaccinated individuals: What do
they mean for rabies epidemiology? PLoS Neglected Tropical Diseases 14(2): e0007933.
Ho, M.T., Datta, A. & Bhattacharyya,
S.P. 1998. An elementary derivation of the Routh-Hurwitz criterion. IEEE
Transactions on Automatic Control 43(3): 405-409.
Huang, J., Ruan,
S., Shu, Y. & Wu, X. 2018. Modeling the
transmission dynamics of rabies for dog, chinese ferret badger and human interactions in Zhejiang Province, China. Bulletin
of Mathematical Biology 81: 939-962.
Hutson, V. & Schmitt, K. 1992.
Permanence and the dynamics of biological systems. Mathematical Biosciences 111(1): 1-71.
Khatwani, K. 1981. On routh-hurwitz criterion. IEEE Transactions on Automatic Control 26(2): 583-584.
Li, M.Y. & Muldowney, J.S. 1996. A
geometric approach to global-stability problems. SIAM Journal on
Mathematical Analysis 27(4): 1070-1083.
Lv, M.M., Sun, X.D., Jin,
Z., Wu, H.R., Li, M.T., Sun, G-Q., Pei, X., Wu, Y-T., Liu, P., Li, L. &
Zhang, J. 2023. Dynamic analysis of rabies transmission and elimination in
mainland China. One Health 17: 100615.
Martin Jr., R.H. 1974. Logarithmic norms
and projections applied to linear differential systems. Journal of
Mathematical Analysis and Applications 45(2): 432-454.
Musa, S.S., Zhao, S., He, D. & Liu, C.
2020. The long-term periodic patterns of global rabies epidemics among animals:
A modeling analysis. International Journal of
Bifurcation and Chaos 30(03): 2050047.
Nagarajan, T. & Rupprecht,
C.E. 2020. History of rabies and rabies vaccines. In Rabies and Rabies
Vaccines, edited by Ertl, H. Springer, Cham. pp.
11-43.
Renald, E.K., Kuznetsov,
D. & Kreppel, K. 2020. Desirable dog-rabies
control methods in an urban setting in Africa - A mathematical model. International
Journal of Mathematical Sciences and Computing 6(1): 49-67.
Ruan, S. 2017. Spatiotemporal epidemic models
for rabies among animals. Infectious Disease Modelling 2(3): 277-287.
Thieme, H.R. 1992. Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous
differential equations. Journal of Mathematical Biology 30(7): 755-763.
Van den Driessche,
P. & Watmough, J. 2002. Reproduction numbers and
sub-threshold endemic equilibria for compartmental models of disease
transmission. Mathematical Biosciences 180(1-2): 29-48.
World Health Organization (WHO). 2023a. Rabies.
https://www.who.int/ news-room/fact-sheets/detail/rabies. Accessed: 24 August
2024.
World Health Organization (WHO). 2023b. Zero
by 30: The Global Strategic Plan to End Human Deaths from Dog-Mediated Rabies
by 2030.
https://www.who.int/southeastasia/health-topics/rabies/rabies-zero-deaths-by-2030.
Accessed: 24 August 2024.
Wolkowicz, G.S. 1996. The theory of the chemostat:
Dynamics of microbial competition. Bulletin of Mathematical Biology 3(58): 595-598.
|
|||
|
