Dynamical Model for Virus Transmission in Plants with Two Time Delays

Authors
Tayebe Waezizadeh 1
Tayebe Parsaei 1
Fereshte Fourozesh 2
1 Shahid Bahonar university of Kerman
2 Bam university
10.52547/mmr.6.3.487
Abstract
Introduction

One of the major challenges in supporting a growing human population is supplies of food. Plants play a major rule in providing human food. Hence, it is important to study plant diseases and provide appropriate models for describing the relationship between plant infection and its growth and reproduction. One of effective models that describes this relationship is mathematical model. One of the important aspects that the mathematical model can presented is the dynamic of the plant’s immune system.

In this paper, a mathematical model for diffusion of infection in the host plant is introduced. The model is based on a differential equation system with two time delays. In this model, the host population of cells is divided into the classes of susceptible cells consisting of mature cells and are susceptible to infection, infected cells that spread the infection, recovered cells that are no longer infectious and are proliferating cells that become susceptible after reaching maturity.

We consider two time delays, and , in equations. The proliferating cells have the average maturity time , after which they are recruited to the susceptible class. is the average time of antiviral effects.

In the next sections of this paper, stability conditions of equilibrium points are investigated. In the last section, we consider a plant in two different modes, organic and non- organic. Then the solution curves are plotted with different time delays and compare solutions together.

Material and methods

In this scheme, first we explain the conditions of plant. Then, a mathematical model with two time delays is introduced. As follows, the dynamical behavior of the model is investigated. At the end of paper, we consider a plant with two different modes and plot the solution curves.

Results and discussion

We introduce a mathematical model which explain conditions of plant cells. In this model the independent variable is time, so the model is ODE with two time delays. As follows, using some theorems in dynamical systems, the dynamical behavior of the model is investigated. Using these results, we can provide good conditions for a plant that epidemic does not happen. At the end, we use of Matlab software to plot the solution curves in two different conditions. The curves explain the behavior of plant cells when they are infectious.

Conclusion

The following conclusions were drawn from this research.

A mathematical model which is introduced in this paper is more realistic than the previous models because, the grow rate of a plant is considered to be logistic.
Theorems show that how we can control the virus to prevent epidemic outbreak.
We plot solution curves for two different plants (organic and non-organic). Solution curves show that how the conditions of plant cells change by changing the parameters.
Keywords
Mathematical model
Equilibrium point
Stability
Hopf bifurcation
1. Strange R. N., Scott P. R., "Plant disease: a threat to global food security", Annu Rev Phytopathol, 43 (2005) 83-116.## 2. Tilman D., Balzer C., Hill G., Befort B.L., "Global food demand and the sustainable intensification of agriculture". Proc. Natl. Acad. Sci. USA, 108 (2011) 20260-20264. ## 3. Jeger M. J., Holt J., Van Den Bosch F., Madden L. V., "Epidemiology of insect-Transmitted plant viruses: modelling disease dynamics and control interventions", Physiological Entomology, 29 (2004) 291-304. ## 4. Purcell A. H., Almeida R. P. P., "Insects as vectors of disease agents, in Dekker, M. (Ed.), Encyclopedia of plant and crop science", Taylor and Francis, London (2005). ## 5. Zhang T., Meng X., Li Y., Song Z., "Dynamical analysis of delayed plant disease models with continuous or impulsive cultural control strategies", Abstr. Appl. Anal., 2012 (2012) 1-25. ## 6. Van den Bosch, F., de Roos A. M., "The dynamics of infectious diseases in orchards with rogueing and replanting as control strategy", J. Math. Biol., 35 (1996) 129-157. ## 7. Chan M.S., Jeger M.J., "An analytical model of plant virus disease dynamics with rogueing and replanting", J Appl Ecol., 31 (1994) 413-427. ## 8. Zhang X.S., Holt J., Colvin J., "Mathematical models of host plant infection by helper-dependent virus complexes: why are helper viruses always avirulent? Phytopath. 90 (2000) 85-93. ## 9. Zhang X.S., Holt J., Colvin, J., "A general model of plant-virus disease infection incorporating vector aggregation "Plant Path. 49 (2000) 435-444. ## 10. Jones J. D.G., Dangl J. L., "The plant immune system", Nature, 444 (2006) 323-329. ## 11. Ma Z., Li J., "Dynamical Modeling and Analysis of Epidemics", World Scientific Publishing Co. Pte. Ltd (2009). ## 12. Waezizadeh, T., "Seir epidemic model with two time delays", Journal of Dynamical Systems and Geometric Theories, 14 (2016) 189-200. ## 13. Kuang Y., "Delay Differential Equation with Application in Population Dynamics", Academic Press INC (1993). ## 14. Farkas M. "Dynamical Models in Biology", Academic Press (2001). ## 15. Kuznetsov Y. A., "Elements of Applied Bifurcation theory", Second Edition, Springer. Verlag New York (2000). ## 16. Costa A. T., Bravo J. P., Makiyama R. K., Vasconcellos Nunes A., Maia I. G., "Viral Counter Defines X Antiviral Immunity in Plants: Mechanisms for Survival, in Romanowski, V., Current Issues in Molecular Virology- viral Genetics and Biothechnological Applications", In Tech. (2013). ## 17. Blyuss K. B., Kyrychko Y. N., "Stability and Bifurcations in an Epidemic Model with Varying Immunity Period", Bull. Math. Biol. Bulltein of Mathematical Biology. 72 (2010) 490-505. ## 18. Blyuss K. B., Yuan Y., Belair J., "Threshold Dynamics in an SEIRS Model with Latency and Temporary Immunity", J. Math. Biol., 69 (2014) 875-904. ## 19. Neofytou G., Kyrychko Y. N., Blyuss K. B., "Time- Delayed Model of Immune Response in Plants", J. Theor. Biol. 398 (2016) 28-39. ## 20. Maxwell D. J., Partridge J. C., Roberts N. W., Boonham N., Foster G. D., "The Effects of Plant Virus Infection on Polarization Reflection from Leaves", Plos One, 11(2016) 1-18. ## 
Mathematical Research
Volume 6, Issue 3 - Serial Number 14
Autumn 2020
Pages 487-500