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## Mathematical modelling of infectious disease

Infectious diseases can be classified into two broad categories: those caused by viruses and bacteria are microparasitic diseases, and those due to worms are macroparasitic. In this section we concentrate on infectious diseases where individuals are infected by pathogen microorganisms (like, for instance, viruses, bacteria, fungi or other microparasites). Some well known examples of such infectious diseases are: Influenza, SARS, Reublla, AIDS, Ebola …(Viral infectious diseases); Cholera, Plague, Typhus, Leprosy…(Bacterial infectious diseases); Malaria, Taeniasis… (Parasitic infectious diseases);… Mathematical modelling of infectious diseases is a tool to provides understanding of the underlying mechanisms that influence the spread of disease, explains epidemiological phenomena, and predict the future course in order to control an epidemic.

Here, we summarize some of the simple classical models for microparasitic infections.

In these models we assume the population has no births, deaths, migration and the total population size is a constant N. We also assume that every individual has an equal chance to meet any other member of the population and these interactions are random.

### Building the aggregated model

Now, we shall take advantage of the three time scales to build a reduced model governing the total fish density and the total fishing e↵ort. Aggregation methods were introduced in ecology by Iwasa et al [73, 74]. Here, we use time scale separation methods based on the central manifold theory and we refer to the following articles for aggregation methods [10, 11, 14]. Usually, the complete system involves only two time scales. Under this condition, the aggregation is realized by calculating the fast equilibrium and the aggregated model is obtained by substituting the fast equilibrium into the complete model.

In our present case, three time scales are considered. As a consequence, the aggregation is going to require two steps. In a first step, we shall look for the existence of a very fast equilibrium and we shall substitute it into the complete model. This will lead to an ”intermediate” model at the fast time scale. The second and last step will consist in looking for the existence of a fast equilibrium whose substitution in the intermediate model will lead to the aggregated and final slow model.

#### First step of aggregation: very fast fish movements

Let us set » = μ = 0 leading to the very fast model that describes the patch change from MPA to fishing area and inversely: At the very fast time scale, the sub-populations small and large clusters are constant, i.e. the next variables are first integrals: nS = nSM + nSF , nL = nLM + nLF .

**Comparison with one-step aggregation**

It would have been possible to decide to perform only a one-step aggregation. The first possibility is to assume that » = 0 in order to study the very fast dynamics, and then not assuming that μ = 0. This corresponds to the first step of the previous aggregation and leads to a three equation system, which is more difficult to analyse than the previous aggregated model. The other possibility is to assume that μ = 0 In order to study the fast dynamics, without assuming at any moment that » = 0. The system obtained after a two-step aggregation appears as an approximation for » = 0 of the one-step aggregation. The dynamics obtained is a slightly better approximation of the complete dynamics than the one obtained with the two-step aggregation method. Indeed, substituting the frequencies at the fast equilibrium which are solutions of equations (3.4.1) would lead to another one-step aggregated model that could be developed as a Taylor expansion with respect to « . The zero order term of this Taylor expansion would exactly correspond to the aggregated model (3.3.3) obtained by the two-step method but, with the advantage that the first order term would give a correction term of the order of » leading to a better approximation of the complete model. However, determining the frequencies at fast equilibrium is more difficult than with the two-step aggregation methods: it requires solving a four-dimension system of equations in order to determine the fast equilibrium (first four equations of system (3.4.1)). The two-step method requires solving more (three) systems of equations, but with a lower number of equations (only two equations).

To summarize, two aggregation methods have been proposed:

• The two-step method leads to an aggregated model with less approximation but in most cases, it could be easier to handle it as it can be switched into several systems of equations, very fast and fast.

• The one-step method allows to calculate some correction terms leading to a better approximation but, we need to deal with a single system of equations to get the fast equilibrium. The later system may be more difficult to handle analytically.

**Dynamical behavior of a stochastic SIRS epidemic model**

This section details paper [111]. The dynamics of disease spreading among a population have been investigated very widely in the frame of deterministic models e.g. [25], [39], [86], [105]. In such deterministic models, the environment is assumed to be constant. However, in most real situations, it is necessary to take into account random change of environmental conditions and their e↵ects on the spread of the disease. For instance, the disease can be more likely to spread in wet (cold) condition rather than in dry (hot) condition or any other characteristics of the environment that may change randomly. Therefore, it is important to consider the disease dynamics under the impact of randomness of environmental conditions. There are many papers about this topic in recent years e.g. [9], [63], [76], [84].

**The semigroup and the stability in distribution**

It is well-known that the pair (⇠t, S(t), I(t)) is a homogeneous Markov process with the state space V := E ⇥intR2 +. Let B(V) be the Borel 1−algebra on V and & be the Lebesgue measure on intR2 +. Denote by m the product measure on (V, B(V)) defined by m(+, A) = p&(A) and m(−, A) = q&(A). As shown in [51, Lemma 3.1], if the distribution of (⇠0, S(0), I(0)) is absolutely continuous with respect to the measure m, so is the distribution of (⇠t, S(t), I(t)). We can therefore define P(t)f to be the density function of (⇠t, S(t), I(t)) given that (⇠0, S(0), I(0)) has the density f.

If & > 0 and (5.1.13) holds, all positive solutions converge almost surely to the equilibrium (s+⇤ , i+⇤ ) = (s−⇤ , i−⇤ ). Otherwise, we have Theorem 5.1.14. If & > 0 and (5.1.13) does not hold, then (⇠t, S(t), I(t)) has a stationary distribution ⌫⇤, concentrated on E ⇥ (r\intR2 +). In addition, ⌫⇤ is the unique stationary distribution having the density f⇤, and lim t!1kP(t)f − f⇤k = 0 for any f 2 D.

**Table of contents :**

R´esum´e

Abstract

**1 Introduction **

1.1 Population dynamics

1.1.1 Single species growth

1.1.2 Biological interaction

1.1.3 Lotka-Volterra systems

1.2 Mathematical modelling of infectious disease

1.3 Models and obtained results

**2 Spatial heterogeneity, fast migration and coexistence of Intraguild Predation Dynamics **

2.1 Introduction

2.2 Model

2.3 Model reduction

2.3.1 Fast equilibrium

2.3.2 Aggregated model

2.4 Results and discussions

2.5 Conclusion and perspectives

**3 E↵ect of small versus large clusters of fish school on the yield of a purse-seine small pelagic fishery including a marine protected area **

3.1 Introduction

3.2 Complete Model

3.3 Building the aggregated model

3.3.1 First step of aggregation: very fast fish movements

3.3.2 Second step of aggregation: fast changes in clusters size .

3.4 Comparison with one-step aggregation

3.5 Harvest Optimization

3.6 Discussion and perspectives

**4 Dynamics of species in a model with two predators and one prey **

4.1 Introduction

4.2 Definitions and notation

4.3 The model with general coefficients

4.4 The model with periodic coefficients

4.5 Numerical examples and conclusion

**5 Population dynamics in random environments **

5.1 Dynamical behavior of a stochastic SIRS epidemic model

5.1.1 Introduction

5.1.2 Preliminary

5.1.3 Dynamical behavior of solutions

5.1.4 The semigroup and the stability in distribution

5.1.5 Simulation and discussion

5.2 Evolution of Lotka-Volterra predator-prey systems under telegraph noise

5.2.1 Model

5.2.2 Predator-prey system with carrying capacity

5.2.3 Predator-prey system with the absence of carrying capacity

5.2.4 Discussion and conclusion

**6 Conclusion **

**Bibliography**