Study of Mathematical Modeling for Plant Disease Transmission: A Systematic Literature Review during 2012-2022

Sanubari Tansah Tresna, Nursanti Anggriani, Asep Kuswandi Supriatna

Abstract


Many models representing disease transmission have been constructed and analyzed mathematically. However, literature studies on the mathematical models for vector-borne disease are sparse, especially on the plant disease transmission model. This study aims to obtain information about the research conducted and find room for developing the model, including mathematical analysis, intervention used, and biological factors considered. We employ a Systematic Literature Review (SLR) to explore all of the studies on plant disease transmission modeling collected from four digital databases. First, the JabRef reference manager helps conduct the inclusion and exclusion processing. Then, we obtain 60 selected articles that passed the criterion. Next, the VOSviewer application is resulting a bibliometric analysis of the database containing chosen articles. Finally, we classify the model constructed based on the system used and elaborate on the intervention used. The results show that the existing researcher clusters are not linked to each other, and the models only consider usual interventions such as roguing and insecticide spraying. Hence, there is much room to build collaboration between the researcher and develop models for plant disease transmission by considering the other various intervention and biological factors in the model to improve further.

Keywords


Bibliometric analysis; Mathematical model; Plant disease; Prisma method

Full Text:

PDF

References


T. Zhao, Y. Xiao, and R. J. Smith, “Non-smooth plant disease models with economic thresholds,†Mathematical Biosciences, vol. 241, no. 1, pp. 34–48, 2013. DOI: 10.1016/j.mbs.2012.09.005

J. Wang, S. Gao, Y. Luo, and D. Xie, “Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments,†Abstract and Applied Analysis, vol. 2014, pp. 1–10, 2014. DOI: 10.1155/2014/841367

F. Zhang, Z. Qiu, B. Zhong, T. Feng, and A. Huang, “Modeling Citrus Huanglongbing transmission within an orchard and its optimal control,†Mathematical Biosciences and Engineering, vol. 17, no. 3, pp. 2048–2069, 2020. DOI: 10.3934/mbe.2020109

S. Gao, D. Yu, X. Meng, and F. Zhang, “Global dynamics of a stage-structured huanglongbing model with time delay,†Chaos, Solitons & Fractals, vol. 117, pp. 60–67, 2018. DOI: 10.1016/j.chaos.2018.10.008

N. Anggriani, M. Ndii, D. Arumi, N. Istifadah, and A. Supriatna, “Mathematical model for plant disease dynamics with curative and preventive treatments,†in AIP Conference Proceedings, vol. 2043, no. 1. AIP Publishing LLC, 2018, p. 020016. DOI: 10.1063/1.5080035

W. Suryaningrat, N. Anggriani, A. Supriatna, and N. Istifadah, “The optimal control of rice tungro disease with insecticide and biological agent,†in AIP Conference Proceedings, vol. 2264, no. 1. AIP Publishing LLC, 2020, p. 040002. DOI: 10.1063/5.0023569

K. M. Putri et al., “A maize foliar disease mathematical model with standard incidence rate,†in IOP Conference Series: Materials Science and Engineering, vol. 546, no. 5. IOP Publishing, 2019, p. 052085. DOI: 10.1088/1757- 899X/546/5/052085

O. C. Collins and K. J. Duffy, “Optimal control of foliar disease dynamics for multiple maize varieties,†Acta Agriculturae Scandinavica, Section B—Soil & Plant Science, vol. 68, no. 5, pp. 412–423, 2018. DOI: 10.1080/09064710.2017.1416156

B. Oduro, O. O. Apenteng, and H. Nkansah, “Assessing the effect of fungicide treatment on cocoa black pod disease in ghana: Insight from mathematical modeling,†Statistics, Optimization & Information Computing, vol. 8, no. 2, pp. 374–385, 2020. DOI: 10.19139/soic-2310-5070-686

R. Shi, H. Zhao, and S. Tang, “Global dynamic analysis of a vector-borne plant disease model,†Advances in Difference Equations, vol. 2014, no. 1, pp. 1–16, 2014. DOI: 10.1186/1687-1847-2014-59

Z. Zhonghua and S. Yaohong, “Stability and sensitivity analysis of a plant disease model with continuous cultural control strategy,†Journal of Applied Mathematics, vol. 2014, 2014. DOI: 10.1155/2014/207959

J. A. Stenberg, “A conceptual framework for integrated pest management,†Trends in plant science, vol. 22, no. 9, pp. 759–769, 2017. DOI: 10.1016/j.tplants.2017.06.010

M. Chapwanya and Y. Dumont, “On crop vector-borne diseases. impact of virus lifespan and contact rate on the traveling-wave speed of infective fronts,†Ecological Complexity, vol. 34, pp. 119–133, 2018. DOI: 10.1016/j.ecocom.2017.08.002

F. Brauer, “Mathematical epidemiology: past, present, and future. infectious disease modelling. 2, 113–127 (2017). DOi: 10.1016/j.idm.2017.02.001

M. Jeger, F. Van Den Bosch, L. Madden, and J. Holt, “A model for analysing plant-virus transmission characteristics and epidemic development,†Mathematical Medicine and Biology: A Journal of the IMA, vol. 15, no. 1, pp. 1–18, 1998. DOI: 10.1093/imammb/15.1.1

M. Jeger et al., “Theory and plant epidemiology.†Plant Pathology, vol. 49, no. 6, pp. 651–658, 2000. DOI: 10.1046/j.1365-3059.2000.00522.x

F. Van den Bosch, N. McRoberts, F. Van den Berg, and L. Madden, “The basic reproduction number of plant pathogens: matrix approaches to complex dynamics,†Phytopathology, vol. 98, no. 2, pp. 239–249, 2008. DOI: 10.1094/PHYTO-98-2-0239

S. Kirtphaiboon, U. Humphries, A. Khan, and A. Yusuf, “Model of rice blast disease under tropical climate conditions,†Chaos, Solitons & Fractals, vol. 143, p. 110530, 2021. DOI: 10.1016/j.chaos.2020.110530

M. G. Amorocho and A. M. Loaiza, “A theoretical model for the prevention of banana moko (musa aab simmonds),†F1000Research, vol. 9, 2020. DOI: 10.12688/f1000research.27373.2

B. Nannyonga, L. S. Luboobi, P. Tushemerirwe, and M. Jabłonska-Sabuka, “Using contaminated tools fuels outbreaks of Banana Xanthomonas wilt : An optimal control study within plantations using Runge–Kutta fourth-order algorithms,†International Journal of Biomathematics, vol. 08, no. 05, p. 1550065, 2015. DOI: 10.1142/S1793524515500655

M. Chapwanya and Y. Dumont, “Application of mathematical epidemiology to crop vector-borne diseases: the cassava mosaic virus disease case,†in Infectious Diseases and Our Planet. Springer, 2021, pp. 57–95. DOI: 10.1007/978-3-030-50826-5_4

I. S. Onah, S. E. Aniaku, and O. M. Ezugorie, “Analysis and optimal control measures of diseases in cassava population,†Optimal Control Applications and Methods, 2022. DOI: 10.1002/oca.2901

G. Wake, N. Williams, and T. Pleasants, “A dynamical systems model for poly-cyclic foliar forest pathogens,†ANZIAM Journal, vol. 59, pp. C1–C14, 2017. DOI: 10.21914/anziamj.v59i0.12625

T. Zhang, X. Meng, Y. Song, and Z. Li, “Dynamical analysis of delayed plant disease models with continuous or impulsive cultural control strategies,†in Abstract and applied analysis, vol. 2012. Hindawi, 2012. DOI: 10.1155/2012/428453

G. Neofytou, Y. Kyrychko, and K. Blyuss, “Time-delayed model of immune response in plants,†Journal of theoretical biology, vol. 389, pp. 28–39, 2016. DOI: 10.1016/j.jtbi.2015.10.020

W. Li, L. Huang, Z. Guo, and J. Ji, “Global dynamic behavior of a plant disease model with ratio-dependent impulsive control strategy,†Mathematics and Computers in Simulation, vol. 177, pp. 120–139, 2020. DOI: 10.1016/j.matcom.2020.03.009

H. T. Alemneh, A. S. Kassa, and A. A. Godana, “An optimal control model with cost-effectiveness analysis of maize streak virus disease in maize plant,†Infectious Disease Modelling, vol. 6, pp. 169–182, 2021. DOI: 10.1016/j.idm.2020.12.001

H. T. Alemneh, O. D. Makinde, and D. Mwangi Theuri, “Ecoepidemiological model and analysis of msv disease transmission dynamics in maize plant,†International Journal of Mathematics and Mathematical Sciences, vol. 2019, 2019. DOI: 10.1155/2019/7965232

A. U. Awan, M. Ozair, Q. Din, and T. Hussain, “Stability analysis of pine wilt disease model by periodic use of insecticides,†Journal of Biological Dynamics, vol. 10, no. 1, pp. 506–524, 2016. DOI: 10.1080/17513758.2016.1225828

R. Anguelov, R. Bekker, and Y. Dumont, “Bi-stable dynamics of a host-pathogen model,†BIOMATH, vol. 8, no. 1, 2019. DOI: 10.11145/j.biomath.2019.01.029

B. M. Chen-Charpentier and M. Jackson, “Direct and indirect optimal control applied to plant virus propagation with seasonality and delays,†Journal of Computational and Applied Mathematics, vol. 380, p. 112983, 2020. DOI: 10.1016/j.cam.2020.112983

L. J. Francl, “The disease triangle: a plant pathological paradigm revisited,†The Plant Health Instructor, vol. 10, 2001. DOI: 10.1094/PHI-T-2001-0517-01

V. Bischoff, K. Farias, J. P. Menzen, and G. Pessin, “Technological support for detection and prediction of plant diseases: A systematic mapping study,†Computers and Electronics in Agriculture, vol. 181, p. 105922, 2021. DOI: 10.1016/j.compag.2020.105922

F. U. Rehman, M. Kalsoom, M. Adnan, N. Naz, T. Ahmad Nasir, H. Ali, T. Shafique, G. Murtaza, S. Anwar, and M. A. Arshad, “Soybean mosaic dis- ease (smd): a review,†Egyptian Journal of Basic and Applied Sciences, vol. 8, no. 1, pp. 12–16, 2021. DOI: 10.1080/2314808X.2021.1881245

W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,†Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character, vol. 115, no. 772, pp. 700–721, 1927. DOI: 10.1098/rspa.1927.0118

F. Coutinhoa, M. Burattinia, L. Lopeza, and E. Massada, “Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue,†Bulletin of mathematical biology, vol. 68, no. 8, pp. 2263– 2282, 2006. DOI: 10.1007/s11538-006-9108-6

K. F. Nipa, S. R.-J. Jang, and L. J. Allen, “The effect of demographic and environmental variability on disease outbreak for a dengue model with a seasonally varying vector population,†Mathematical Biosciences, vol. 331, p. 108516, 2021. DOI: 10.1016/j.mbs.2020.108516

M. Li and H. Zhao, “Dynamics of a dengue fever model with vertical transmission and time-periodic in spatially heterogeneous environments,†Mathematical Methods in the Applied Sciences, vol. 44, no. 14, pp. 11 350–11 375, 2021. DOI: 10.1002/mma.7496

J. Yang and X. Wang, “Existence of a nonautonomous sir epidemic model with age structure,†Advances in Difference Equations, vol. 2010, no. 1, p. 212858, 2010.

W. Mengist, T. Soromessa, and G. Legese, “Method for conducting systematic literature review and meta-analysis for environmental science research,â€MethodsX, vol. 7, p. 100777, 2020. DOI: 10.1016/j.mex.2019.100777

E. Stovold, D. Beecher, R. Foxlee, and A. Noel-Storr, “Study flow diagrams in cochrane systematic review updates: an adapted prisma flow diagram,†Systematic reviews, vol. 3, no. 1, pp. 1–5, 2014. DOI: 10.1186/2046-4053-3-54

D. Moher, A. Liberati, J. Tetzlaff, D. G. Altman, and P. Group*, “Preferred reporting items for systematic reviews and meta-analyses: the prisma statement,†Annals of internal medicine, vol. 151, no. 4, pp. 264–269, 2009. DOI: 10.7326/0003-4819-151-4-200908180-00135

O. Ellegaard and J. A. Wallin, “The bibliometric analysis of scholarly production: How great is the impact?†Scientometrics, vol. 105, no. 3, pp. 1809– 1831, 2015. DOI: 10.1007/s11192-015-1645-z

V. E. NJ and W. L., “Manual for vosviewer version 1.5.2,†Leiden: The Netherlands, 2012

M. J. Jeger and M. Pautasso, “Plant disease and global change: the importance of long-term data sets,†New Phytologist, vol. 177, no. 1, pp. 8–11, 2008. DOI: 10.1111/j.1469-8137.2007.02312.x

S. D. Khirade and A. Patil, “Plant disease detection using image processing,†in 2015 International conference on computing communication control and automation. IEEE, 2015, pp. 768–771. DOI: 10.1109/ICCUBEA.2015.153

D. Palma, F. Blanchini, and P. L. Montessoro, “A system-theoretic approach for image-based infectious plant disease severity estimation,†PloS one, vol. 17, no. 7, p. e0272002, 2022. DOI: 10.1371/journal.pone.0272002

S. P. Mohanty, D. P. Hughes, and M. Salathé, “Using deep learning for image-based plant disease detection,†Frontiers in plant science, vol. 7, p. 1419, 2016. DOI: 10.1371/journal.pone.0272002

S. Liu, M. Huang, and J. Wang, “Bifurcation control of a delayed fractional mosaic disease model for jatropha curcas with farming awareness,†Complexity, vol. 2020, 2020. DOI: 10.1155/2020/2380451

M. Vellappandi, P. Kumar, V. Govindaraj, and W. Albalawi, “An optimal control problem for mosaic disease via caputo fractional derivative,†Alexandria Engineering Journal, vol. 61, no. 10, pp. 8027–8037, 2022. DOI: 10.1016/j.aej.2022.01.055

P. Kumar, V. S. Erturk, and H. Almusawa, “Mathematical structure of mosaic disease using microbial biostimulants via caputo and atangana–baleanu derivatives,†Results in Physics, vol. 24, p. 104186, 2021. DOI: 10.1016/j.rinp.2021.104186

M. Jablonska-Sabuka, R. Kalaria, and T. Kauranne, “A dynamical model for epidemic outbursts by begomovirus population clusters,†Ecological Modelling, vol. 297, pp. 60–68, 2015. DOI: 10.1016/j.ecolmodel.2014.11.008

W. Suryaningrat, N. Anggriani, and A. K. Supriatna, “Mathematical analysis and numerical simulation of spatial-temporal model for rice tungro disease spread,†Commun. Math. Biol. Neurosci., vol. 2022, pp. Article–ID, 2022. DOI: 10.28919/cmbn/7160

F. Nakasuji, S. Miyai, H. Kawamoto, and K. Kiritani, “Mathematical epidemiology of rice dwarf virus transmitted by green rice leafhoppers: a differential equation model,†Journal of applied ecology, pp. 839–847, 1985. DOI: 10.2307/2403233

Y. Liu, C. Zeng, J. Guo, and Z. Liao, “Global dynamics of a new huanglongbing transmission model with quarantine measures,†Applied Mathematics, vol. 13, no. 1, pp. 1–18, 2022. DOI: 10.4236/am.2022.131001

D. Degefa, O. Makinde, and D. Temesgen, “Modeling potato virus y disease dynamics in a mixed-cropping system,†International Journal of Modelling and Simulation, vol. 42, no. 3, pp. 370–387, 2022. DOI: 10.1080/02286203.2021.1919818

S. Adhurya, F. A. Basir, and S. Ray, “Stage-structure model for the dynamics of whitefly transmitted plant viral disease: an optimal control approach,†Computational and Applied Mathematics, vol. 41, no. 4, pp. 1–22, 2022. DOI: 10.1007/s40314-022-01864-9

F. Zhang, Z. Qiu, A. Huang, and X. Zhao, “Optimal control and cost-effectiveness analysis of a huanglongbing model with comprehensive interventions,†Applied Mathematical Modelling, vol. 90, pp. 719–741, 2021. DOI: 10.1016/j.apm.2020.09.033

F. Hamelin, B. Bowen, P. Bernhard, and V. Bokil, “Optimal control of plant disease epidemics with clean seed usage,†Bulletin of mathematical biology, vol. 83, no. 5, pp. 1–24, 2021. DOI: 10.1007/s11538-021-00872-w

M. Ozair, T. Hussain, K. A. Abro, S. Jameel, and A. U. Awan, “Role of pine wilt disease based on optimal control strategy at multiple scales: A case study of korea,†Journal of Biosciences, vol. 46, no. 4, pp. 1–16, 2021. DOI: 10.1007/s12038-021-00208-9

M. Brunetti, V. Capasso, M. Montagna, and E. Venturino, “A mathematical model for xylella fastidiosa epidemics in the mediterranean regions. promoting good agronomic practices for their effective control.†Ecological Modelling, vol. 432, p. 109204, 2020. DOI: 10.1016/j.ecolmodel.2020.109204

M. Ozair, T. Hussain, A. U. Awan, A. Aslam, R. A. Khan, F. Ali, and F. Tasneem, “Bio-inspired analytical heuristics to study pine wilt disease model,†Scientific Reports, vol. 10, no. 1, pp. 1–15, 2020. DOI: 10.1038/s41598-020- 60088-1

W. Li, L. Huang, and J. Wang, “Dynamic analysis of discontinuous plant disease models with a non-smooth separation line,†Nonlinear Dynamics, vol. 99, no. 2, pp. 1675–1697, 2020. DOI: 10.1007/s11071-019-05384-w

S. Ray and F. Al Basir, “Impact of incubation delay in plant–vector interaction,†Mathematics and Computers in Simulation, vol. 170, pp. 16–31, 2020.

DOI: 10.1016/j.matcom.2019.09.001

J. Wang, F. Feng, Z. Guo, H. Lv, and J. Wang, “Threshold dynamics of a vector-borne epidemic model for huanglongbing with impulsive con trol,†Applied Mathematics, vol. 10, no. 4, pp. 196–211, 2019. DOI: 10.4236/am.2019.104015

V. A. Bokil, L. Allen, M. J. Jeger, and S. Lenhart, “Optimal control of a vectored plant disease model for a crop with continuous replanting,†Journal of Biological Dynamics, vol. 13, no. sup1, pp. 325–353, 2019. DOI: 10.1080/17513758.2019.1622808

N. Anggriani, M. Ndii, N. Istifadah, and A. Supriatna, “Disease dynamics with curative and preventive treatments in a two-stage plant disease model,†in AIP Conference Proceedings, vol. 2043, no. 1. AIP Publishing LLC, 2018, p. 020010. DOI: 10.1063/1.5080029

C. Chen, C. Li, and Y. Kang, “Modelling the effects of cutting off infected branches and replanting on fire-blight transmission using filippov systems,†Journal of theoretical biology, vol. 439, pp. 127–140, 2018. DOI: 10.1016/j.jtbi.2017.11.017

C. Chen and X. Chen, “Rich sliding motion and dynamics in a filippov plant- disease system,†International Journal of Bifurcation and Chaos, vol. 28, no. 01, p. 1850012, 2018. DOI: 10.1142/S0218127418500128

J. Liu and T. Zhang, “Stability and hopf bifurcation analysis of a plant virus propagation model with two delays,†Discrete Dynamics in Nature and Society, vol. 2018, 2018. DOI: 10.1155/2018/7126135

N. Anggriani, M. Mardiyah, N. Istifadah, and A. Supriatna, “Optimal control issues in plant disease with host demographic factor and botanical fungicides,†in IOP conference series: Materials Science and Engineering, vol. 332, no. 1. IOP Publishing, 2018, p. 012036. DOI: 10.1088/1757- 899X/332/1/012036

M. Rafiq, A. Raza et al., “Numerical modeling of transmission dynamics of vector-borne plant pathogen,†in 2017 14th International Bhurban Conference on Applied Sciences and Technology (IBCAST). IEEE, 2017, pp. 214–219. DOI: 10.1109/IBCAST.2017.7868057

L. Luo, S. Gao, Y. Ge, and Y. Luo, “Transmission dynamics of a huanglong bing model with cross protection,†Advances in Difference Equations, vol. 2017, no. 1, pp. 1–21, 2017. DOI: 10.1186/s13662-017-1392-y

Y. Chen and J. Yang, “Global stability of an sei model for plant diseases,†Mathematica Slovaca, vol. 66, no. 1, pp. 305–311, 2016. DOI: 10.1515/ms- 2015-0137

N. S. A. Latif, G. C. Wake, T. Reglinski, and P. A. Elmer, “Modelling induced resistance to plant diseases,†Journal of theoretical biology, vol. 347, pp. 144– 150, 2014. DOI: 10.1016/j.jtbi.2013.12.023

R. Anguelov, J. Lubuma, and Y. Dumont, “Mathematical analysis of vector-borne diseases on plants,†in 2012 IEEE 4th international symposium on plant growth modeling, simulation, visualization and applications. IEEE, 2012, pp. 22–29. DOI: 10.1109/PMA.2012.6524808

H. H. Mustafa, N. A. Samat, Z. Mohamed, and F. A. Kassim, “Stage progression model for soil-borne plant disease in oil palm plantation,†Journal of Xi’an Shiyou University, vol. 15, no. 1, pp. 53–64, 2021

G. Qiu, S. Tang, and M. He, “Analysis of a high-dimensional mathematical model for plant virus transmission with continuous and impulsive roguing control,†Discrete Dynamics in Nature and Society, vol. 2021, 2021. DOI: 10.1155/2021/6177132

R. Atlihan, N. F. Britton, S. Demir, A. Papasidero, M. R. Risvanli, M. Seminara, and E. Venturino, “Can symbiotic fungi protect plants from insect pests? a simple mathematical model,†Computational and Mathematical Methods, vol. 3, no. 6, p. e1121, 2021. DOI: 10.1002/cmm4.1121

F. P. Agouanet, I. Tankam-Chedjou, R. M. Etoua, and J. J. Tewa, “Mathematical modelling of banana black sigatoka disease with delay and seasonality,†Applied Mathematical Modelling, vol. 99, pp. 380–399, 2021. DOI: 10.1016/j.apm.2021.06.030

Y. Luo, F. Zhang, Y. Liu, and S. Gao, “Analysis and optimal control of a huanglongbing mathematical model with resistant vector,†Infectious Disease Modeling, vol. 6, pp. 782–804, 2021. DOI: 10.1016/j.idm.2021.05.004

J. N. Nakakawa, J. Y. Mugisha, M. W. Shaw, and E. Karamura, “Banana xan-thomonas wilt dynamics with mixed cultivars in a periodic environment,†International Journal of Biomathematics, vol. 13, no. 01, p. 2050005, 2020. DOI: 10.1142/S1793524520500059

W. Li, L. Huang, and J. Wang, “Global dynamics of filippov-type plant disease models with an interaction ratio threshold,†Mathematical Methods in the Applied Sciences, vol. 43, no. 11, pp. 6995–7008, 2020. DOI: 10.1002/mma.6450

F. Al Basir, S. Adhurya, M. Banerjee, E. Venturino, and S. Ray, “Modelling the effect of incubation and latent periods on the dynamics of vector-borne plant viral diseases,†Bulletin of Mathematical Biology, vol. 82, no. 7, pp. 1–22, 2020. DOI: 10.1007/s11538-020-00767-2

P. Baptista, I. M. Bulai, T. Gomes, and E. Venturino, “Modeling the interactions among phythopatogens and phyllosphere microorganisms for the biological disease control of olea europaea l.†Mathematical biosciences, vol. 308, pp. 42–58, 2019. DOI: 10.1016/j.mbs.2018.12.002

V. Krivan, M. Lewis, B. J. Bentz, S. Bewick, S. M. Lenhart, and A. Liebhold, “A dynamical model for bark beetle outbreaks,†Journal of Theoretical Biology, vol. 407, pp. 25–37, 2016. DOI: 10.1016/j.jtbi.2016.07.009

T. Zhao and Y. Xiao, “Plant disease models with nonlinear impulsive cultural control strategies for vegetatively propagated plants,†Mathematics and Computers in Simulation, vol. 107, pp. 61–91, 2015. DOI: 10.1016/j.matcom.2014.03.009

B. Buonomo and M. Cerasuolo, “Stability and bifurcation in plant–pathogens interactions,†Applied Mathematics and Computation, vol. 232, pp. 858–871, 2014. DOI: 10.1016/j.amc.2014.01.127

S. Lenhart and J. T. Workman, Optimal control applied to biological models. Chapman and Hall/CRC, 2007. DOI: 10.1201/9781420011418

B. M. Chen-Charpentier and I. Diakite, “A mathematical model of bone remodeling with delays,†Journal of computational and applied mathematics, vol. 291, pp. 76–84, 2016. DOI: 10.1016/j.cam.2014.11.025

M. J. Jeger, “The epidemiology of plant virus disease: Towards a new syn- thesis,†Plants, vol. 9, no. 12, p. 1768, 2020. DOI: 10.3390/plants9121768

M. Jackson and B. M. Chen-Charpentier, “Modeling plant virus propagation with delays,†Journal of Computational and Applied Mathematics, vol. 309, pp. 611–621, 2017. DOI: 10.1016/j.cam.2016.04.024




DOI: https://doi.org/10.34312/jjbm.v4i1.18443

Copyright (c) 2023 Sanubari Tansah Tresna, Nursanti Anggriani, Asep Kuswandi Supriatna

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.


Jambura Journal of Biomathematics (JJBM) has been indexed by:


                          EDITORIAL OFFICE OF JAMBURA JOURNAL OF BIOMATHEMATICS

 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango 96554, Gorontalo, Indonesia
 Email: editorial.jjbm@ung.ac.id
 +6281356190818 (Call/SMS/WA)
 Jambura Journal of Biomathematics (JJBM) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.  Powered by Public Knowledge Project OJS.