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## Evolution of mathematical modeling applying in infectious disease studies

Mathematical modeling has been involved in epidemiological study of vector borne and infectious diseases and also acted as a tool in guiding and assessing empirical control measures for more than 40 years (Reiner et al., 2013; Stegeman et al., 2010). Mathematical modelling allowed researchers to assess the contribution of the complex network of livestock movements and trading patterns on the spread of diseases, estimate the risk of disease introduction in different scenarios and further develop contingency plans for possible outbreaks (Thrush and Peeler, 2006; Ciccolini et al., 2012; Napp et al., 2013).

Mathematical modeling applied in infectious disease dynamics is a systemic way of transforming data and assumptions regarding disease transmission into an estimation quantitatively of how an epidemic develops in space and time (Wu and Cowling, 2011).

However, this quantitative approach is not recent. The history of mathematical modeling in infectious disease studies dates back to as early as 1766 when Daniel Bernoulli developed a mathematical model to analyze the life expectancy with or without variolation which protects against the smallpox infection and Lambert worked based on Bernoulli’s by extending the model to incorporate age-dependent parameters. Then, Enko, in 1889, developed a probabilistic model to describe the epidemics of measles by evaluating number of contacts between infectious and susceptible individuals in populations (Siettos and Russo, 2013).

Nonetheless, the researches in this area has not yet systematically developed until the remarkable paper of Ross in 1911 which is actually the beginning of modern mathematical epidemiology. In this publication, Ross used a set of equations to approximate the dynamic transmission of malaria agent through the mosquitos. Subsequently, Kermack and McKendrick followed Ross’s work and established the deterministic compartmental epidemic modeling in the case of direct transmission diseases. In their example, they hypothesized a mass–action low for the disease transmission. This type of deterministic model is strongly analogous to so-called a susceptible infectious-recovery (SIR) model. The model implies that all individuals are homogeneously mixing in a closed system (Siettos and Russo, 2013) and the expected number of secondary cases per primary case in an entirely susceptible population was determined by basic reproductive number ( 0 R ) (Roberts, 2007). The disease can persist and spread in the population if 0 R > 1 and the disease disappearwhen 0 R < 1 (Capaldi, 2009).

However, the fundamental concept of homogeneous mixing of total population is not always true in reality. The concept of metapopulation is thus used to divide total population into different subpopulations and these subpopulations connect each other by some means of mobility flows (Apolloni et al., 2014). For instance, all backyard chickens in the province are account together as total population and the chickens raised in each village are subpopulation. The chickens in a subpopulation of village are possible to contact with chickens in other villages through animal movements which are managed by poultry traders or by other means. The contacts of each individual or subpopulation can also be explained in term of social network. Social network is a term obtained from social science which is defined as a group of elements and the connections between or among them (Martınez-Lopez et al., 2009). In term of infectious disease modeling, social network

can facilitate in contacting individuals or subpopulations as per considered assumptions or empirical evidences and make the model more realistic.

**Stochastic approaches in mathematical modeling of infectious diseases**

Stochastic model is more applicable than deterministic one in case of small size population (Chaharborj et al., 2013). For the same initial conditions, stochastic model illustrates better and more intuitively how disease spread in the population because each simulation always provides different results (Chalvet-Monfray, 2006).

Stochastic model used in mathematical epidemiology was firstly proposed by McKendrick (1926). This model was actually a stochastic version of the classic deterministic model of Kermack and McKendrick. Later on, an interesting stochastic model was proposed by Reed and Frost in a conference in 1926. However, this model has never been published (Andersson and Britton, 2000). The Reed-Frost model was a chain-binomial model which simply explains how an epidemic behaves over time. In 1931, Greenwood proposed another chain-binomial model which assumed a constant probability of infection. Then, Bartlett (1949) found the stochastic general epidemic model which proposes the removal of infected individuals from closed population (Billard and Zhao, 1993). Subsequently, Markov chain methods were developed in chain-binomial model by Bailey (1968) and Gani (1969, 1971). In a series of papers proposed by Becker (1977, 1980, 1981), Reed-Frost model and Greenwood model were combined into a general chain binomial model (Cairoli, 1988).

For the deterministic as for stochastic models, there is also the same distinction about the way the contamination is simulated according the hypothesis is made : law of mass action or the pseudo –law mass action.

**Law of mass action and pseudo-law of mass action**

Law of mass action was originated from chemical studies and it was firstly called fundamental law of chemical kinetics and, later on, it has been applied in many area of science including mathematical epidemiology. The law was proposed by Norwegian scientists Guldberg and Waag during 1864-1879. It was explained by the law that, for any homogeneous system, the rate of any simple chemical reaction is proportional to the probability that the reacting molecules will be found together in a small volume. Thus, for mathematical modeling, the law is applicable to rates of transition of individuals between two interacting compartments of the whole population, for example, the rate that susceptible population become infected after an adequate contact (Bubniakova, 2007).

**Mathematical modeling of backyard chicken trade networks**

The dynamic flow of backyard chickens in space and time was simulated in a compartmental stochastic dynamic model. Backyard chickens in each village were considered as a whole homogeneous population, as these chickens are traditionally raised without any cages or fences and they thus mix together during daytime (Choprakarn and Wongpichet, 2007).

The model was divided into 2 main parts: village and trader. Each village was then subdivided into 3 compartments: (i) young chickens (defined as chickens aged < 5 months old), (ii) ready–to–sell chickens and (iii) stocked chickens. The village part of the model was run simultaneously in all 1,045 villages located in the Phitsanulok province.

Traders were also categorized into 3 types including (i) trader–slaughterhouse (TS), who collects chickens from villages and slaughters himself; (ii) household trader (HT), who collects chickens from villages and sells to other traders; and (iii) trader of trader (TT), who aggregates chickens collected by HTs and slaughters them.

In backyard chicken trade, backyard chickens in ready–to–sell chicken compartment had possibility to flow into either compartments of TS or HT. All chickens collected in HT compartments were then aggregated into the compartments of TT. At the end, all chickens from the compartments of TS and TT were slaughtered and sent to market vendors, restaurants and consumers. The conceptual model explaining what happens in both village and trader part and how they connected is shown in Figure 6.

### Trader type B:

Number of chickens collected by trader type B was according to CNY during Chinese New Year period (day 17–31). In other periods, the traders acted like trader type A. t CNY;t17,31 ; otherwise (1.5.9) (1.5.10).

Where CNY is lambda of number of chickens collected per village per day during Chinese New Year period (day 17–31).

#### Social network analysis of backyard chicken trade

Social network analysis was used in this study to illustrate the connectivity of village–trader networks. By constructing a sociogram, the network was well visualized and more understandable for ones who are not familiar with mathematical equations.

The social networks of village–trader connectivity in backyard chicken trade were simulated for 1,000 simulations. A node in these networks referred to a village and a tie referred to trading activities that occurred between a village and a trader. Since trader was a part of the village, the trader was also counted as a node. Degree centrality which is the number of immediate ties that each node has (Martinez–Lopez et al., 2009) was measured in each network. In our study, degree centrality was defined as the number of connections between villages and traders. In this model, villages could not connect with one another without traders.

All statistical and geographical analyses were performed using statistical computing language R version 3.0.1 (R development Core Team, 2013). Package ‘fitdistrplus’ was used in fitting the distribution. Package ‘sp’ was used in creating spatial coordinates. Package ‘igraph’ was used in illustrating the sociogram and package ‘spdep’ was used in locating the villages within the defined range.

**Table of contents :**

Acknowledgements

Summary

Preface

Introductory Chapter

Introduction

Outbreaks of highly pathogenic avian influenza (HPAI) H5N1

Global and regional situations

Situation in Thailand

Control measures applied by Thai government

Problem statement

Objectives

Mathematical modeling of infectious diseases

Evolution of mathematical modeling applying in infectious disease studies

Stochastic approaches in mathematical modeling of infectious diseases

Law of mass action and pseudo-law of mass action

Mathematical modeling of Avian Influenza

**Chapter 1: Mathematical modeling of trade networks of backyard chickens in Phitsanulok province, Thailand **

1.1. Introduction

1.2. Backyard chicken production chain

1.2.1. Poultry production systems in Thailand

1.2.2. Backyard chicken rearing system

1.2.3. Backyard chicken trade system

1.2.4. Effects of ritual festivals on backyard chicken trade

1.3. Data sources

1.3.1. Study site

1.3.2. Backyard chicken population

1.3.3. Field survey on backyard chicken trade

1.4. Mathematical modelling and social network analysis

1.4.1. Mathematical modeling of backyard chicken trade networks

1.4.2. Social network analysis of backyard chicken trade

1.5. Results

1.5.1. Backyard chicken movement characteristics and model inputs

1.5.2. Simulation output

1.6. Discussion and conclusion

**Chapter 2: Mathematical modeling of infectious spread of HPAI H5N1 along trade networks of backyard chickens in Phitsanulok province, Thailand **

2.1 Introduction

2.2 Model parameters and assumptions

2.3 Mathematical Model

2.3.1 Total population

2.3.2 Susceptible compartments

2.3.3 Infectious compartments

2.3.4 Removed compartments

2.4 Sensitivity analysis of parameters related to HPAI H5N1 outbreaks .

2.4.1 Mortality rate due to avian influenza infection

2.4.2 Disease transmission rate

2.4.3 Transmission rate by visit of infected trader

2.4.4 Zone of initial infected village

2.5 Mathematical modeling of different scenarios of control measures

2.5.1 Allow all traders

2.5.2 Allow only local traders (TS and HT)

2.5.3 Ban all traders

2.6 Results

2.6.1 Sensitivity analysis

2.6.2 Baseline model output

2.6.3 Control measure simulations

2.7 Discussion and conclusion

Chapter of conclusion and perspectives

**Bibliography**