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## MaMBIES: Mathematical Modelling of Bioinvasions and Epidemic Spread

Introduction

** Rationale **

Environmental assessment

Heterogeneity

Natural and anthropogenic perturbations

**Mathematical methods**

Difference-differential equations

Partial differential equations

Effects of noise

**Applications**

Epidemics in farming ecosystems

Diseases in agriculture

Viruses in aquatic ecosystems

**Supported by CAPES-FAPERGS, DAAD, DFG, EU and JSPS**

**Cooperation** (in alphabetical order)

Frank M. Hilker

Institute of Environmental Systems Research, School of Mathematics/Computer Science, Osnabrück University, Germany

Alex James

Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

Michel Langlais

Mathématiques Appliquées de Bordeaux, Université Victor Segalen Bordeaux 2, France

Diomar Cristina Mistro and Luiz Alberto Díaz Rodrigues

Departamento de Matemática, Universidade Federal de Santa Maria RS , Brasil

Sergei V. Petrovskii

Department of Mathematics, University of Leicester, Leicester, UK

Jean-Christophe Poggiale

Aix-Marseille Université, Institut Méditerranéen d'Océanologie, Marseille, France

Michael Sieber

Research Group Ecology and Ecosystem Modelling, Institute of Biochemistry and Biology, University of Potsdam, Germany

Ivo Siekmann

NICTA, Victoria Research Laboratory, University of Melbourne, Australia

Ezio Venturino

Dipartimento di Matematica, Università degli Studi di Torino, Italia

**Summary**

The project is sought so as to include a spatial description of the evolution of diseases in ecosystems, in order to develop the mathematical theory of bioinvasions and epidemic spread as well as to conduct more realistic simulations with the ultimate aim of prescribing better safety measures and a reliable scenario to forcast the harvesting of crops for the single farmer.

**Introduction**

The transmission dynamics of infectious diseases is one of the oldest topics of mathematical biology. Already in 1760, Daniel Bernoulli provided the first known mathematical result of epidemiology, that is the defence of the practice of inoculation against smallpox (Brauer and Castillo-Chavez, 2001). The amount of works in this area exploded in the last decades. Different aspects are dealt with in the literature, from human health to environmental assessment. Moreover, it is a subject where mathematics is deeply involved, from simple model analyses to the development of unifying methods to deal with large classes of models. Spatial heterogeneity plays a crucial role in ecology and its consequences on the ecosystems functioning has been widely investigated. These consequences are very complex, starting from stabilising effects to generating chaos and complex patterns. This topic is currently the subject of renewed attention. The project aims at making a review of the different mathematical methods in epidemiological problems and at analysing the impact of the spread of pathogenic agents on the functioning of different ecosystems in space and time. Then, we will describe three applications. One concerns the functioning of aquatic ecosystems both salty and fresh water, another the Aujesky disease in hog raising farms in Piemonte while the third deals with the propagation of pathogens in plants (vine-yards in Piemonte and oranges in Sicily).

### Rationale

Models in epidemiology first attempted to explain the temporal dynamics of a disease in a homogeneous isolated population. Traditionally, they involve at least two types of variables which are the amount (density, etc.) of susceptible individuals (those who could contract the disease) and infected individuals, cf. the well- known Kermack and McKendrick models (Kermack and McKendrick, 1927, 1932, 1933). This kind of models gives some ideas explaining why a disease may be epidemic, endemic or not, or what are the main factors controlling the disease in the population. They also show that diseases can be a population control factor.

**Environmental assessment**

Mathematical models provide quantitative methods to study the effect of a pathogenic agent on the functioning of given ecosystems. They are also useful in the context of contamination of exploited resources, for instance in fisheries. The parasites can affect the exploited population (the fish in fisheries problems, cows for foot-and-mouth disease or mad-cow syndrome, pigs for swine fever, poultry for avian influenza, etc.) or its resources. What is the effect of a parasite spread in the population? How long and how severe will be the epidemic? How fast will the epidemic spread in the population? If we have access to a curative treatment, in which case will it be efficient? What will be the consequence of the treatment? If there are various treatments, which is the more efficient? What is the effect of a virus on a bacterial community and what is the consequence on the ecosystems functioning? All these questions will be evoked in the various applications and we shall propose, by the means of mathematical models, some criteria on which the inherent decisions to the management of the risks can lean. Notice that the questions we mentioned above could also be applied to the case of the spread of invasive species.

**Heterogeneity **

The spatial heterogeneity has important effects on ecosystems functioning and it is therefore essential to consider the spread of pathogenic agents in a heterogeneous environment. Spatially extended models have been proposed to analyse such effects (cf. Lloyd and May 1996, Shofield 2002, Malchow et al. 2004, 2005, Hilker et al. 2006a). These works aim at investigating properties like persistence of epidemics in heterogeneous environment or at comparing different kinds of models with respect to the estimated speed of propagation of epidemics or invasive species. Another source of heterogeneity is represented by temporal forcing like seasonal fluctuations for instance. These forcings have also qualitative and quantitative impacts on ecosystems functioning (cf. Grover 1988, Steffen and Malchow 1996, Steffen et al. 1997, Scheffer 1998, Anderies and Beisner 2000).

Spatial heterogeneities do not necessarily have to be imposed by external influences but can also arise as consequence of intrinsic processes in the ecosystem. The interaction of bacteria growth processes with diffusion processes can lead to spatially inhomogeneous distributions of nutrients and bacteria in marine sediments (Baurmann et al., 2004a,b). In this project we aim at studying virus abundance and its effects on the formation of spatially inhomogeneous patterns in microbial food webs in marine sediments.

Heterogeneity leads to experimental difficulties. For instance, it is difficult to study experimentally the effect of turbulence at a short time scale on a phytoplankton population growth, since the methods used to introduce the turbulence bring some bias in the results. Thus mathematical methods are essential to approach the role of heterogeneity in ecological and epidemiological problems.

**Natural and anthropogenic perturbations**

The last influencing factor which will be considered in this project is that of external perturbations, which have natural origins like phytoplankton blooms or rivers swelling or anthropogenic origins like fishing, chemical treatments against epidemics or farming management. These perturbations act on the living organisms which respond in different ways. A good understanding of these responses needs a good formulation of the model which should reproduce the response but not mimic it. In other words, it is rather easy to introduce a mathematical function in a model which makes it respond as expected by the modeller, but it is not the aim of a model. The response should be obtained on the basis of the mechanisms which can explain why the organisms respond this way. This is a challenge of modelling, which generally needs the addition of different variables in order to describe the organisms functioning. This leads also to mathematical complexity and, consequently, adapted mathematical methods are needed to deal with these models.

### Mathematical methods

**Difference-differential equations**

We shall use two different approaches to represent the spatial heterogeneity in our models. One will consider space as a finite set of different homogeneous patches. On each of them, a set of coupled ordinary differential equations describes the dynamics of parasites or pathogenic agents and their hosts. The patches are linked by the mean of movement equations or simply by diffusive exchange. This approach is rather simple and avoids some mathematical problems like the existence of solutions or numerical analysis problems. A first complication is the consideration of the inner-patch spatio-temporal dynamics, again coupled to the others (cf. Malchow et al. 2002). The second approach deals with a continuous spatial structure and involves partial differential equations. As we shall explain in the next section, the spatial heterogeneity implies mathematical difficulties in partial differential equations models.

As mentioned in the Rationale Section, spatially extended ecosystem models in a heterogeneous environment, in which pathogenic agents are introduced, involve a large number of variables, and those models are difficult to be dealt with from the mathematical point of view. To bypass this problem we may use mathematical properties in order to reduce the dimension of the mathematical systems or also develop and use numerical methods.

All the models mentioned above are deterministic. From an applied viewpoint, it would be interesting to check if the results are robust under random perturbations.

**Partial differential equations**

We shall use two different approaches to represent the spatial heterogeneity in our models. One will consider space as a finite set of different homogeneous patches. On each of them, a set of coupled ordinary differential equations describes the dynamics of parasites or pathogenic agents and their hosts. The patches are linked by the mean of movement equations or simply by diffusive exchange. This approach is rather simple and avoids some mathematical problems like the existence of solutions or numerical analysis problems. A first complication is the consideration of the inner-patch spatio-temporal dynamics, again coupled to the others (cf. Malchow et al. 2002). The second approach deals with a continuous spatial structure and involves partial differential equations. As we shall explain in the next section, the spatial heterogeneity implies mathematical difficulties in partial differential equations models.

As mentioned in the Rationale Section, spatially extended ecosystem models in a heterogeneous environment, in which pathogenic agents are introduced, involve a large number of variables, and those models are difficult to be dealt with from the mathematical point of view. To bypass this problem we may use mathematical properties in order to reduce the dimension of the mathematical systems or also develop and use numerical methods.

All the models mentioned above are deterministic. From an applied viewpoint, it would be interesting to check if the results are robust under random perturbations.

**Effects of noise**

For the description of epidemic spread or, in general, biological invasions, the existence of random fluctuations cannot be ignored but must be part of the models. The noise can be due to environmental or demographic variability. Much has already been done for dynamical systems only time-dependent (cf. an early highlight by Horsthemke and Lefever 1984) but spatiotemporal dynamics must be considered here. This can lead to stochastic partial differential equations like Master equations at the species level or Langevin equations at the population level (Malchow and Schimansky-Geier 1985, Garcia-Ojalvo and Sancho 1999, Allen 2003). Direct stochastic simulations like molecular dynamics can be used as well but here the focus is on equation-based models, particularly on systems with fluctuating parameters but mainly with external multiplicative white and non-white noise.

Another interesting aspect dealt with in this project is that marine ecosystems can exhibit different coexisting equilibrium states corresponding to different composition of species. This is particularly important in systems where species are competing for a small number of limiting nutrients. In such cases the influence of noise is of great importance. Already a small amount of noise can kick the system out of a metastable equilibrium with a certain composition of species and will then approach another metastable equilibrium with a different composition of species. Thus the system can jump between different states.

### Applications

**Epidemics in farming ecosystems**

In close connection with some veterinarians of the Cuneo province, the mathematical modelling of the spread of the Aujesky disease affecting the economically relevant hog farming activities in the province will be developed. The study is primarily aimed at assessing the role that measures of biological safety can play in containing the spread of the disease and maybe suggest further measures for its eventual eradication, in agreement with the EU directions. In northern countries these have already been successfully implemented and the impact of the disease is being controlled. The model for the moment is limited to the sole timeframe. While this approach may lead to establishing the long lasting consequences of the adopted prevention policies, it lacks any description of the spatial positioning of the farms.

**Diseases in agriculture**

A second very important area for the agricultural economics of Piemonte is the role that vineyards play for wine production. The maintenance and sustainability of the vineyard ecosystem is important in view also of possible and foreseeable changes in the climate, due to the greenhouse effect and global warming. The models look at the interplay between insects living in the vineyards and in the grassland and woods at their edge. The effects of spraying for the pest control need to be taken in consideration to assess whether the ecosystem will persist or the final result will endanger the spiders, thus depleting the vines from their natural predators and ultimately adversely affecting the harvest. Another general question concerns the fungi growing on the vines and the changes the latter will undergo under climatic different conditions, more humid or with higher temperatures.

In the above two lines of research the models being used at the moment involve only ordinary differential equations, i.e. just a description of the time evolution of the system. While the latter would be enough for achieving the general goal of producing software that the single farmer can use for decision-making, under several different foreseeable wheather scenarios, the possibility of including space in the description of the system will make the resulting simulations more fine-tuned for the individual user.

The study of a disease called Tristeza, affecting the orange trees in Sicily is relevant for its implications on the economy of the island. A model is currently developed for the spread of the disease in order to control it. The approach is based on cellular automata, but it can also be reformulated in a continuous setting by means of diffusion equations.

**Viruses in aquatic ecosystems**

Furthermore, we would also like to look at plankton models particularly in lakes, which are created by old excavations now abandoned. Microbes like viruses or bacteria play a large role in the functioning of aquatic ecosystems, in the whole water column. They control the biogeochemical cycles. On the other hand, any perturbation (natural or anthropogenic) affects these organisms. Aquatic ecosystem models are often developed to study large space and time scales and are not always adequate to describe microbial dynamics in heterogeneous environment, since in this case, the small scale heterogeneity can have a large influence. A good description of this influence would have benefits for subjects like biodiversity conservation, climate change, harvestable resources and others.

The abundance and the effects of viruses on the microbial food web have been exhibited in various situations (Bergh et al., 1989, Bratbak et al., 1990, Proctor and Fuhrman, 1990). Some authors show that viruses increase the mortality of prokaryotes in the oceans and in certain marine environments like in coastal areas (Fuhrman and Noble, 1995). They also may increase the cycling of dissolved organic matter (DOM) within the heterotrophic bacterial population, lowering the amount of matter and energy that is passed on to higher trophic levels (Bratbak et al., 1990, Fuhrman, 1992).

In this project, different rather simple heterogeneous ecosystems models will be developed. Virus abundance and effects on microbial food web will be described and analysed. We will also analyse the impact on higher trophic levels like fish populations.

Simple prey-predator models of phytoplankton-zooplankton interaction with virally infected phytoplankton will be considered. The role of lysogeny and lysis as well as possible switches between them needs to be studied (Tian and Burrage 2004). Mathematical models of these processes are still rare. The already classical paper is by Beltrami and Carroll (1994); more recent work is of Chattopadhyay and Pal (2002), or Malchow et al. (2004, 2005) or Hilker and Malchow (2006b). The primary productivity, extinction risk and survival probability of prey and predator have to be estimated for those different dynamic behaviours. These conceptual models can be easily extended by introducing nutrients and planktivorous fish. Such extensions allow the study of disease transmission along food chains. In space, patchy nutrients lead to a correspondingly patchy phytoplankton distribution. The spatiotemporal dynamics may yield not only stationary structures but different kinds of population waves including invasive waves of infection. All the mentioned structures also need to be considered under physical forcing and noise. An important question is whether an artificial (anthropogenic) infection can be used to control plankton and fish, i.e., for the termination of harmful algal blooms or for the protection of an exploited fish population.