**5. Compartmental mathematical models**

These models can be either Susceptible-Infectious-Recovered (SIR), Susceptible-Infectious-Recovered-Susceptible (SIRS), Susceptible-Exposed-Infectious-Recovered (SEIR), Susceptible-Infected (SI) or Susceptible-Infectious-Susceptible (SIS). Number of compartments in the model depend on the disease being studied and the objective of the study. In this approach, the progress of the disease is defined in terms of level of pathogen within the host.

#### **5.1. SIR model**

output. A model is usually expressed in terms of variables which are related to experimentally

A mathematical model helps in establishment of links between sets of epidemiological data through well-understood mathematical relationships. This is facilitated only through a thorough understanding of various factors associated with the disease like, incubation, transmission and mortality and also factors associated with the vector. Models may also help to characterize and integrate the cellular network and molecular data operating within the different compartments of the host immunity system for disease progression. At a mechanistic level, they help in finding answers to several biological questions which cannot be addressed experimentally. Different conceptual qualitative results and threshold values of paramount importance like basic reproduction ratio, contact numbers, replacement numbers and herd immunity, are derived from mathematical models, since the quantity and diversity of available data is limited. Conditions for local and global stability of various equilibria, relationships among these stability conditions and endemicity are also derived from the epidemiological models. Prediction and recommendations for control of an epidemic outbreak are therefore, the most important outcomes of mathematical modeling of a communicable disease and then they are known as decision models. These models facilitate economic evaluation of different courses of action for mitigation of a particular infection and finally help in selection of optimal

Spatio-temporal progression and temporal development of communicable diseases can be explained by compartmental, agent-based or contact network-based model. In the compart‐ mental model, the host population is divided into different states or compartments, depending on the level of infection in them. Rates of transfer e.g. transmission rate, removal rate between compartments are expressed as derivatives of the sizes of the compartments with respect to time and are assumed to be constant. Ordinary differential equations form the basic framework of compartmental models. In agent-based model, the region of interaction of people in a population is considered as a system of software agents interacting in time and space. It is a high fidelity model and involves complex parameterization and extensive computation. Modern concepts of network theory are employed in dissecting the transmission dynamics within heterogeneous population in a contact network model. Social networking among the individuals influence the possibility, extent and speed of epidemic spreading. This model is intermediate between compartmental model and agent-based model. In this model, the pathogen and the social network within the population are closely intertwined and represent‐ ed by a node for each individual. The framework of contact network highly depends on the

Accuracy, transparency and flexibility are the key elements which should be balanced to develop a mathematical model of high quality. For development of any type of mathematical model, mathematical modelers formulate a set of equations by feeding into them various type of factors like the length of time one is ill, the length of time one can infect others, the level of contagiousness of the disease, the number of uninfected individuals who can contract the disease, human behavior and any such known component of disease dynamics. Once a model with biological significance and relevance has been formulated, the model can be fitted to the

measurable quantities.

186 Trends in Infectious Diseases

control measure [8].

mode of transmission of the disease [2], [9].

Number or density of individuals in each stage of infection is more important than the load of the pathogen per person in modeling of microparasitic infections. In the SIR model in order to emulate epidemics, population is classified as Susceptible, currently Infectious and Recovered. The total size of the host population is the summation of the three classes. When an individual is concerned, he is assumed to exist in either of the three states. Susceptible means that the individual has never had the disease and is susceptible to contraction of the disease by an infected individual, at random from the population. The mode of transmission of infection depends on the type of the pathogen. Depending on the relative magnitude of the latent period of the infection, the infected host can infect others and then it becomes Infectious. The total time spent in the infected state by an individual is a geometric random variable. When the infectious agent is removed from the system of the infected class or death occurs, they become Recovered with immunity and will never be infected again. In case of a dead individual, he cannot get infected or cannot infect anybody and thus is equivalent to a recovered individual with acquired immunity. Then, it can be assumed that the number of infected individuals tends to decrease towards zero and finally disappears from the network permanently. In the study of disease dynamics, six distinct and well-defined events can be assumed to occur : birth, death of a susceptible individual, death of an infected individual, death of a recovered individual, infection and recovery. Epidemic data supports the assumption that the per capita rate at which a given susceptible individual becomes infected is proportional to the prevalence of infection in the population. Therefore, according to SIR model, number of cases increases exponentially initially till there has been a sufficient decrease in the proportion of susceptible when the growth rate slows; this process goes on until the epidemic can no longer be maintained and the number of cases goes below a threshold level resulting in disease eradication. Human behavior can affect the disease dynamics in an individual as also in the whole population because it may influence the disease state of an individual, rate of infection or recovery rate and the contact network structure. Therefore, behavioral responses should be considered as an integral part of the study of dynamics of infectious diseases. An intriguing feature of childhood microparasitic infections is that the children are born in the susceptible category as there is no vertical transmission. Birth rate and vaccination affect the recruitment to the susceptible compartment. Moreover, in case of common cold, there is no permanent removal state because the individual enters the susceptible class soon after recovery. This widely used model is applicable for diseases which are contracted by an individual only once in its lifetime and either acquired lifelong immunity develops or death occurs as in case of measles, mumps, SARS, influenza. An important drawback of the model is that it ignores the random effects, specially at early stages of infection, when the number of both susceptible and infected classes is low. This model is unable to describe the spatial aspects of the spread of the disease. Moreover, in this modeling approach it is assumed that each individual has the same amount of contacts as every other individual. Therefore, if the rate of contact varies during course of infection, it cannot be appropriately included in the simple SIR model. Precision in prediction can be improved by incorporating realistic contact patterns or by using modeling approaches possessing higher fidelity.

Infectious and Infectious-Recovered is not very distinct because of variability in responses between different individuals and variability in pathogen levels over the infectious period [7].

Unearthing the Complexities of Mathematical Modeling of Infectious Disease Transmission Dynamics

http://dx.doi.org/10.5772/57387

189

This modeling approach is unrealistic in case of animal or human infections because here it is assumed that an infected individual remains in this state forever. Hence, the ultimate conse‐ quence of the presence of one infected individual in the population is the infection of the entire population. In the early stages of an infection, when the count of infected cases is low, the SI and SIR models behave in a similar fashion as there is limited number of recovered individuals.

In this model, individuals run stochastically through the cycle of Susceptible-Infected – Susceptible. Thus, here exist only two states because the infected individual, after recovery, again becomes susceptible to infection. Removal due to death or acquired immunization is not considered. Therefore, the number of infected individuals increases up to a stationary nonzero constant value as observed in case of sexually transmitted diseases [10]. The renewed susceptibility of an individual is due to vast antigenic variation associated with gonorrhea and other similar sexually transmitted diseases [7]. This model is based on assumptions like, different contagion probabilities between different pairs of people, probabilistic recovery from

the disease and multiple stages of infection, with varying disease properties [9].

**6. Deterministic and Stochastic approach to mathematical modeling**

If the objective of the study is to model disease propagation in a large population, with continuous variations of population sizes, deterministic model is the most appropriate one. Such model indicates same output or same fate of individuals subjected to same possible events with identical probabilities [6]. For a given set of parameter values, the deterministic model has one solution which can be fitted to data using various methods. The most common method minimizes the sum of squares of differences between observed data and model prediction.

Natural and biological discrete events of random occurrence in a small population cannot be explained by deterministic and continuous modeling and here comes the utility of stochastic modeling. Stochastic models capture the randomness of birth and death rates associated with disease dynamics and also the variability among disease strains. The earliest stages of an infection are stochastic. This occurs because encounter with a pathogen may either lead to an infection or elimination from the system. Stochasticity decreases as the number of cases increases. Emergent behaviors that have neither been defined nor expected are simulated in this modeling approach. Probability distributions associated with such systems are typically memory-less. If the objective of modeling of disease dynamics is to establish the conditions for disease eradication, or if irregular epidemics are to be modeled, stochastic model is the choice. Stochasticity can induce chance extinctions of the disease and it introduces variances and co-

**5.4. SI model**

**5.5. SIS model**

The SIR model can also be expressed by a stochastic version where the future course of the infection is independent of the past, if present is known completely [2], [6], [9-10].

#### **5.2. SIRS**

This model assumes that after recovery, the person becomes susceptible again as immunity wanes, i.e. recovered individuals possess short-term immunity [4], [14].

#### **5.3. SEIR**

It is an extended SIR model where a new compartment or state is added. It is known as Exposed or E which is positioned between the susceptible and infectious compartments. The Exposed individual is infected but not infectious, i.e. the disease remains in latent state [15].This concept can also be explained on the basis of the level of pathogen within the host and immunological status of the host. When the host is susceptible, it indicates that no pathogen is present and only a low level of non-specific immunity exists within the host. As soon as the susceptible encounters an infectious individual, he becomes infected. The pathogen increases in number and the infected host may not show any signs of infection and thus he enters the Exposed compartment. As soon as the pathogen burden is sufficiently high, the Exposed host becomes Infectious and disease is transmitted to another susceptible individual. When the Infectious individual can no longer transmit infection as the pathogen is cleared from his immunity system, he belongs to the Recovered category. The class distinction between ExposedInfectious and Infectious-Recovered is not very distinct because of variability in responses between different individuals and variability in pathogen levels over the infectious period [7].

### **5.4. SI model**

in the population. Therefore, according to SIR model, number of cases increases exponentially initially till there has been a sufficient decrease in the proportion of susceptible when the growth rate slows; this process goes on until the epidemic can no longer be maintained and the number of cases goes below a threshold level resulting in disease eradication. Human behavior can affect the disease dynamics in an individual as also in the whole population because it may influence the disease state of an individual, rate of infection or recovery rate and the contact network structure. Therefore, behavioral responses should be considered as an integral part of the study of dynamics of infectious diseases. An intriguing feature of childhood microparasitic infections is that the children are born in the susceptible category as there is no vertical transmission. Birth rate and vaccination affect the recruitment to the susceptible compartment. Moreover, in case of common cold, there is no permanent removal state because the individual enters the susceptible class soon after recovery. This widely used model is applicable for diseases which are contracted by an individual only once in its lifetime and either acquired lifelong immunity develops or death occurs as in case of measles, mumps, SARS, influenza. An important drawback of the model is that it ignores the random effects, specially at early stages of infection, when the number of both susceptible and infected classes is low. This model is unable to describe the spatial aspects of the spread of the disease. Moreover, in this modeling approach it is assumed that each individual has the same amount of contacts as every other individual. Therefore, if the rate of contact varies during course of infection, it cannot be appropriately included in the simple SIR model. Precision in prediction can be improved by incorporating realistic contact patterns or by using modeling approaches

The SIR model can also be expressed by a stochastic version where the future course of the

This model assumes that after recovery, the person becomes susceptible again as immunity

It is an extended SIR model where a new compartment or state is added. It is known as Exposed or E which is positioned between the susceptible and infectious compartments. The Exposed individual is infected but not infectious, i.e. the disease remains in latent state [15].This concept can also be explained on the basis of the level of pathogen within the host and immunological status of the host. When the host is susceptible, it indicates that no pathogen is present and only a low level of non-specific immunity exists within the host. As soon as the susceptible encounters an infectious individual, he becomes infected. The pathogen increases in number and the infected host may not show any signs of infection and thus he enters the Exposed compartment. As soon as the pathogen burden is sufficiently high, the Exposed host becomes Infectious and disease is transmitted to another susceptible individual. When the Infectious individual can no longer transmit infection as the pathogen is cleared from his immunity system, he belongs to the Recovered category. The class distinction between Exposed-

infection is independent of the past, if present is known completely [2], [6], [9-10].

wanes, i.e. recovered individuals possess short-term immunity [4], [14].

possessing higher fidelity.

188 Trends in Infectious Diseases

**5.2. SIRS**

**5.3. SEIR**

This modeling approach is unrealistic in case of animal or human infections because here it is assumed that an infected individual remains in this state forever. Hence, the ultimate conse‐ quence of the presence of one infected individual in the population is the infection of the entire population. In the early stages of an infection, when the count of infected cases is low, the SI and SIR models behave in a similar fashion as there is limited number of recovered individuals.

### **5.5. SIS model**

In this model, individuals run stochastically through the cycle of Susceptible-Infected – Susceptible. Thus, here exist only two states because the infected individual, after recovery, again becomes susceptible to infection. Removal due to death or acquired immunization is not considered. Therefore, the number of infected individuals increases up to a stationary nonzero constant value as observed in case of sexually transmitted diseases [10]. The renewed susceptibility of an individual is due to vast antigenic variation associated with gonorrhea and other similar sexually transmitted diseases [7]. This model is based on assumptions like, different contagion probabilities between different pairs of people, probabilistic recovery from the disease and multiple stages of infection, with varying disease properties [9].
