Simulation Approach to the Control of Insect-Borne Virus Diseases

Shun'ichi Miyai
Laboratory of Insect Pest Management
Tohoku National Agricultural Experiment Station
Akahira, Shimokuriyagawa, Morioka, Iwate 020-01, Japan, 1993-12-01

Abstracts in Other Languages: 中文, 日本語, 한국어

Introduction

Plant disease epidemiology can be studied quantitatively by using different types of models. In the past, rather empirical models have been used to describe the progress curves of plant diseases, but recently more mechanistic models have been developed to analyze disease dynamics. Mechanistic models can be classified into analytical and simulation models. Analytical models are usually represented by one or a few differential (or difference) equations, and can be used to describe the general behavior of epidemics. Madden et al. (1990) have developed some simple analytical models for viruses transmitted in a nonpersistent manner. On the other hand, simulation models have a more complex structure and are especially useful in mimicking quantitatively the course of epidemics, and the response of a specific pathosystem to external variables such as environmental effects, control practices, etc. Jeger (1986) has thoroughly compared analytical and simulation approaches in plant disease epidemiology.

Simulation models are undoubtedly necessary tools for understanding the dynamics of insect-borne virus diseases at a field level. Although great efforts have been made to develop simulation models for fungus diseases, considerably less research has been done on modelling virus disease epidemiology (Campbell and Madden 1990). Probably the disciplinary isolation of scientists studying various aspects of plant virus epidemiology is the main reason why modelling has been used so seldom. Simulation models so far developed for insect-borne virus epidemiology are listed in Table 1. The vector species concerned are leafhoppers, planthoppers and aphids. They model in various ways the effects of population dynamics of these vectors on disease spread.

Plant viruses are classified into persistent, semipersistent and nonpersistent categories, on the basis of the maximum length of time that vectors can retain inoculativity after acquisition (Berger and Ferris 1989). In spite of the fact that the largest group of viruses are nonpersistent, until recently very little modelling work had been done on nonpersistent viruses. Only the models of Ruesink and Irwin (1986) and Marcus and Raccah (1986) are for nonpersistent virus.

The models of Berger and Ferriss (1989) and Miyai (1991) are partially stochastic, whereas the other models are wholly deterministic. In deterministic simulation models, computations are based on the expected values of the parameters, and the model need be run only once with any particular set of parameter values. Simulation models that incorporate stochastic elements require a sequence of random numbers, and the model need be run as many times as we want by changing the sequence of random numbers.

All the models except those of Berger and Ferriss (1989) and Miyai (1991) ignore the spatial aspects of epidemic spread to concentrate on temporal aspects of the disease in a single population. This approach simplifies considerably the resultant model, at the expense of understanding the importance of spatial factors in disease transmission. However, the models of Ruesink and Irwin (1986) and Marcus and Raccah (1986) do take into consideration the spatial distribution of virus infection (random or clumped) caused by vectors moving within the field.

Berger and Ferriss (1989) and Miyai (1991) have developed models incorporating the movement of vectors from one plant to another. Although their modelling of plant-to-plant movement is primitive, it should be useful for the study of the relationship between movement of vectors and the spread of the virus.

As computer simulations are vast topics, in this paper I discuss only two examples of simulation models of insect-borne virus disease and their application.

Carter (1986) has provided an essential background for simulation modelling in relation to plant virus epidemiology. Kiritani et al. (1987) has reviewed modelling approaches in plant disease epidemiology with special reference to insect-borne virus diseases.

Examples of Simulation Models of Insect-Borne Virus Diseases

A Model for Aphid-Borne Nonpersistent Virus Disease

Marcus and Raccah (1986) constructed a discrete-time simulation model to study the spread of aphid-borne nonpersistent virus diseases within a field of crops. Although their original model is a rather general one which can handle multiple aphid species, in the following illustration it is restricted for simplicity in such a way that the model deals with only one species.

The number of newly infected plants during the j-th time unit is denoted by I (j) and is given by:

I (j) = H (j) [1 - f (j)] (1)

where H (j) is the number of healthy plants at the beginning of the j-th time unit and f (j) is the probability that a given plant is not inoculated during the j-th time unit. The time unit may be one day, one week, etc.

Spread of the virus is assumed to occur both by immigration of aphids into the field and by interplant movement of aphids within the field. In their model, therefore, aphids originating outside the field are discriminated from the ones within the field. According to this distinction, f (j) is divided into two components, that is, one for incoming aphids and the other for internal aphids:

f (j) = f₁ (j) f₂ (j) (2)

in which f₁ (j) is the probability that a given plant is not inoculated by incoming aphids during the j-th time unit and f₂ (j) is the probability that a given plant is not inoculated by internal aphids during the j-th time unit.

The incoming aphids that have acquired virus from external source plants introduce the virus into the crop field. The average number of inoculations per plant caused by them during the j-th time unit is denoted by A₁ (j) and is given by:

A₁ (j) = m₁ (j) a₁ (j) v (j) b₁ (j) /Q (3)

where m₁ (j) is the total number of incoming aphids during the j-th time unit, a₁ (j) is the efficiency of virus acquisition from external source plants, b₁ (j) is the efficiency of virus inoculation to the host plant, v(j) is the proportion of source plants in the vegetation around the crop field during the j-th time unit, and Q is the total number of plants present within the crop field. Assuming that the new inoculations caused by incoming aphids are randomly distributed within the crop field, it follows that the number of inoculations has the Poisson distribution with the mean A₁ (j). Accordingly f₁ (j) is given by:

f₁ (j) = exp [-A₁ (j)] (4)

Internal aphids also contribute to an increase in the number of inoculated plants. When there is a single infectious plant within the crop field at the start of the j-th time unit, the average number of inoculations caused by them, R₂ (j), is given by:

R₂ (j) = m₂ (j) a₂ (j) b₂ (j) /Q (5)

where m₂ (j) is the total number of internal aphids moving within the crop field during the j-th time unit, a₂ (j) is the efficiency of virus acquisition from infectious plants, and b₂ (j) is the efficiency of virus inoculation to the host plant. Accordingly the average number of inoculations per plant caused by internal aphids, A₂ (j), is obtained by dividing R₂ (j) by the total number of plants present within the crop field:

A₂ (j) = R₂ (j)/Q (6)

When there are S(j) infectious plants at the beginning of the j-th time unit, the average number of inoculations per plant becomes A₂ (j) S (j). Assuming that the new inoculations caused by internal aphids are clumped around foci of infection in such a way that the number of inoculations per plant has the negative binomial distribution with the mean A₂ (j) S (j), it follows that:

f₂ (j) = {k/ [k + A₂ (j) S (j)]}^k (7)

where k is the other parameter of the negative binomial distribution which is called an index of clumping.

Substituting equations (4) and (7) in equation (2) gives:

f(j)={exp[-A₁ (j)]}{k/[k+A₂(j)S(j )]}^k (8)

Therefore equation (1) can be written as:

I (j)=H(j){1-[exp(-A₁(j ))][k/(k+A₂(j)S(j))]^k } (9)

The model takes account of a latent period of virus in a plant whose length is p time units. In a given time unit, therefore, a plant belongs to one of the three epidemiological classes: healthy (non-infected), latent infected and infectious. Aphids can acquire virus only from infectious plants. According to the definition of I (j) and a latent period, S (j) is obtained by:

S (j) = S I (i) (10)

Finally the increase in disease incidence is represented as:

X (j+1) = X (j) + I (j)/Q (11)

where X (j) is the proportion of both latent infected and infectious plants at the beginning of the j-th time unit. Figure 2 shows the results of simulations using k=0.5, p=1, Q=10000 and some combinations of A₁ (j) and R₂ (j) values.

Estimating roughly the parameters of the model from the laboratory and field experiments, Marcus and Raccah (1986) carried out some simulations to examine the effect of different control measures on consequent cucumber mosaic virus spread in a pepper field.

A Model for Rice Dwarf Virus Disease

Rice dwart virus (RDV) is transmitted in a persistent manner by green rice leafhoppers (GRL). Nakasuji et al. (1985) developed a mathematical model of RDV disease epidemiology to explore the dynamic relationship between RDV and GRL.

In their model, the population of rice hills is divided into three epidemiological classes: (1) healthy (non-infected), (2) latent infected (that is, infected but not yet infectious) and (3) infectious rice hills. The numbers of these three classes at time t are represented by X(t), Y(t) and Z(t), respectively. Similarly, the vector (GRL) population is grouped into the same three classes: the numbers per rice hill of healthy (non-infected), latent infected, and infectious individuals at time t are denoted by U(t), V(t) and W(t), respectively, and the total number by N(t) [= U(t) + V(t) + W(t)]. For notational convenience t is omitted from the variables hereafter.

The net rate of the virus transmission to the healthy class of rice hills (that is, the rate at which healthy rice hills acquire a latent infection) is assumed to be proportional to both the density of infectious vectors, W, and the number of healthy rice hills, X, and, therefore, is given by aWX where a is a transmission parameter. On the other hand, the rice hills with latent infection are assumed to become infectious at the rate of s, given as the inverse of the mean latent period of the virus in rice plant. These assumptions yield the following set of three differential equations that express the instantaneous rates of change in X, Y and Z:

dX/dt = - aWX

dY/dt = aWX - sY (12)

dZ/dt = sY

The total number of the rice hills, R(=X+Y+Z), does not change, since no rice hills die of RDV infection and infectious rice hills are not removed from the population.

The population growth of the vector (GRL) in a paddy field is modelled using the following equation of a logistic type:

dN/dt = aN(1 - bN) - cN (13)

where a, b and c are all constant parameters representing the birth rate, the rate of the density-dependent decrease in natality, and the natural mortality, respectively.

The net rate of acquisition of the virus by the non-infected vectors is assumed to be proportional to both the density of non-infected vectors, U, and the number of infectious rice hills, Z, so it is given by bUZ where b is a virus acquisition parameter. The vectors with latent infection are assumed to become infectious at the rate of l, which can be obtained as the inverse of the mean latent period of the virus in the vector. Since the birth rate of infectious vectors is known to be lower than that of non-infected ones, the deleterious effect on the birth rate is incorporated replacing a with a', which represents the reduced birth rate of the infectious vector. The birth rate of the vector with latent infection is no different from that of the non-infected one. In the RDV pathosystem, the virus is not only persistent in the vector body, but also transferred to the progeny by the path of transovarial transmission. This rate of transovarial transmission is denoted by g. By putting together all these assumptions, the rates of change in U, V and W are represented by the following set of three differential equations:

dU/dt=(U+V)a(1-bN)+(1-g) Wa'(1-bN)

-cU-bUZ

dV/dt = bUZ-(c+l)V (14)

dW/dt = gWa'(1-bN) + lV-cW

Setting appropriately initial conditions of plant and vector populations, necessary parameter values, rice growing period (one cropping season of paddy) and susceptible period of rice plant to RDV, Nakasuji et al. (1985) conducted the model calculations using equations (12) and (14), with a time unit of one day, and obtained the curves which present changes in the proportion of infectious insects and the proportion of infectious rice hills in one cropping season (Fig. 1). These curves mimicked well, at least qualitatively, the empirical results obtained in the paddy fields.

A SIMPLE APPLICATION OF

SIMULATION MODELS

The simulation models explained above can be used to study the potential effects of global warming on the epidemiology of insect-borne virus diseases. The effect of elevated temperature is examined indirectly, by changing values of the parameters which are considered to be dependent on temperature.

In the model developed by Marcus and Raccah (1986) for aphid-borne nonpersistent virus, equation (3) representing the average number of inoculations per plant caused by incoming vectors has five parameters. Among them, m₁ (j), a₁ (j) and b₁ (j) are considered to be influenced by temperature. Higher temperatures will tend to increase their values. As a result, the value of A₁ (j) will become larger as temperatures increase. Likewise, among the four parameters in equation (5) which gives the average number of inoculations caused by internal vectors, m₂ (j), a₂ (j), and b₂ (j) are considered to be affected by temperature. They all will tend to increase as temperatures become higher. Consequently, higher temperatures will increase the value of R₂ (j).

To evaluate the effect of elevated temperatures on virus disease progress, simulations were carried out by changing the values of A₁ (j) and R₂ (j). Fig. 2 shows the results of simulations using k=0.5, p=1, Q=10000 and some combinations of A₁ (j) and R₂ (j) values. At low temperatures, the values of A₁ and R₂ are small, so the percentage of infected plants is not very high. As temperatures rise, the values of A₁ (j) and R₂ (j) become larger, so the percentage of infection attained is very high.

In the same way, the model of rice dwarf virus epidemiology developed by Nakasuji et al. (1985) can be used to examine the effect of elevated temperatures on persistent virus disease. In equation (12) for the population of rice hills, there are two parameters, a and s. Both of these are considered to be affected by temperature. Higher temperatures will tend to increase their values. In equation (14) for vector population, there are seven parameters. Of these, at least two parameters, b and l, are considered to be affected by temperature. They will increase as temperatures become higher. To evaluate the effect of elevated temperature, simulations can be conducted by changing the values of these parameters one by one.

The simulation results indicate that the increase in global mean temperature due to the greenhouse effect will have a profound influence on the epidemiology of insect-borne virus diseases, by accelerating the rate of virus spread through temperature-dependent components in the pathosystem.

CONCLUSION

A considerable number of simulation models have been developed for plant disease epidemiology since the late 1960s and early 1970s. It was thought that these models could serve to organize available knowledge about a plant pathosystem, and provide a better understanding of that system. Although it is controversial whether or not simulation models have answered the initial expectations (Campbell and Madden 1990), I consider that the modelling approach has played a valuable role in investigating the multiple cause-effect relationships occurring in plant disease epidemiology.

Insect-borne virus disease epidemiology is concerned with the complex interactions among viruses, vectors, plants and the environment. Furthermore, several factors governing these interactions are very difficult to measure and almost impossible to monitor on a regular basis. Approaches using simulation models are therefore a reasonable way of evaluating how insect-borne virus disease spread is affected by particular factors, including those which may be difficult or impossible to measure experimentally. Simulation models are also a useful way of evaluating the effects of different control measures aimed to reduce the spread of virus infection.

References

• Berger, P.H. and R.S. Ferriss. 1989. Mechanisms of arthropod transmission of plant viruses: Implications for the spread of disease. In: Spatial Components of Plant Disease Epidemics, M.J. Jeger (ed.). Prentice Hall, Englewood Cliffs, U.S.A., pp. 40-84.
• Campbell, C.L. and L.V. Madden. 1990. Introduction to Plant Disease Epidemiology. John Wiley & Sons, New York, U.S.A.
• Carter, N. 1986. Simulation modelling. In: Plant Virus Epidemics: Monitoring, Modelling and Predicting Outbreaks, G.D. McLean, R.G. Garrett and W.G. Ruesink (eds.). Academic Press, Sydney, Australia, pp. 193-215.
• Fishman, S., R. Marcus, H. Talpaz, M. Bar-Joseph, Y. Oren, R. Salomon and M. Zohar. 1983. Epidemiological and economic models for spread and control of citrus tristeza virus disease. Phytoparasitica 11: 39-49.
• Gutierrez, A.P., D.E. Havenstein, H.A. Nix and P.A. Moore. 1974. The ecology of Aphis craccivora Koch and subterranean clover stunt virus in south-east Australia II. A model of cowpea aphid populations in temperate pastures. Journal of Applied Ecology 11: 1-20.
• Jerger, M.J. 1986. The potential of analytic compared with simulation approaches to modeling in plant disease epidemiology. In: Plant Disease Epidemiology: Population Dynamics and Management, Vol. 1, K.J. Leonard and W.E.Fry (eds.). Macmillan Publishing Company, New York, U.S.A., pp. 255-281.
• Kiritani, K., F. Nakasuji and S. Miyai. 1987. Systems approaches for management of insect-borne rice diseases. In: Current Topics in Vector Research, Vol. 3, K.F. Harris (ed.). Springer-Verlag, New York, U.S.A., pp. 57-80.
• Kisimoto, R. and Y. Yamada. 1986. A planthopper - rice virus epidemiology model: Rice stripe and small brown planthopper, Laodelphax striatellus Fallén. In: Plant Virus Epidemics: Monitoring, Modelling and Predicting Outbreaks, G.D. McLean, R.G. Garrett and W.G. Ruesink (eds.). Academic Press, Sydney, Australia, pp. 327-344.
• Madden, L.V., B. Raccah and T.P. Pirone. 1990. Modeling plant disease increase as a function of vector numbers: Nonpersistent viruses. Researches on Population Ecology 32: 47-65.
• Marcus, R. and B. Raccah. 1986. Model for the spread of non-persistent virus diseases. Journal of Applied Statistics 13: 167-175.
• Miyai, S. 1991. Modelling the effects of aphid vector (Aulacorthum solani) movement on the spread of soybean dwarf virus. In: Proceedings of International Seminar on Migration and Dispersal of Agricultural Insects. National Institute of Agro-Environmental Sciences, Tsukuba, Japan, pp. 145-154.
• Miyai, S. and N. Hokyo. 1992. Modelling approach to simulate the progress of insect-borne rice virus diseases in paddy fields. In: Ecological Processes in Agro-Ecosystems, M. Shiyomi, E. Yano, H. Koizumi, D.A. Andow and N. Hokyo (eds.). National Institute of Agro-Environmental Sciences, Tsukuba, Japan, pp. 139-154.
• Miyai, S., K. Kiritani and F. Nakasuji. 1986. Models of epidemics of rice dwarf. In: Plant Virus Epidemics: Monitoring, Modelling and Predicting Outbreaks, G.D. McLean, R.G. Garrett and W.G. Ruesink (eds.). Academic Press, Sydney, Australia, pp. 459-480.
• Muramatsu, Y. 1979. A system dynamic model for epidemic of rice stripe virus. Bulletin of the Shizuoka Agricultural Experiment Station 24: 1-13. (In Japanese).
• Nakasuji, F. and K. Kiritani. 1972. Descriptive models for the system of the natural spread of infection of rice dwarf virus (RDV) by the green rice leafhopper, Nephotettix cincticeps Uhler (Hemiptera: Deltocephalidae). Researches on Population Ecology 14: 18-35.
• Nakasuji, F., S. Miyai, H. Kawamoto and K. Kiritani. 1985. Mathematical epidemiology of rice dwarf virus transmitted by green rice leafhoppers: A differential equation model. Journal of Applied Ecology 22: 839-847.
• Ruesink, W.G. and M.E. Irwin. 1986. Soybean mosaic virus epidemiology: A model and some implications. In: Plant Virus Epidemics: Monitoring, Modelling and Predicting Outbreaks, G.D. McLean, R.G. Garrett and W.G. Ruesink (eds.). Academic Press, Sydney, Australia, pp. 295-313.

Key words: Epidemiology, insect-borne virus, model, simulation

j-p-1

i=1

Index of Images

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  Figure 1 Calculated Values of the Percentages of Infectious Vectors and of Infectious Rice Hills

  Source:Nakasuji<I>etal</I>.1985
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  Figure 2 Simulated Effects of Temperature on Nonpersistent Virus Disease Spread

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  Table 1 Simulation Models Developed for Insect-Borne Plant Virus Diseases

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