**1. Introduction**

Epidemiology is considered as the study of causes of occurrence and transmission of diseases in human population. It deals with the properties of epidemics in the equilibrium or long-time steady state. It involves prediction and monitoring of the spread of both naturally occurring infection and infection caused by bioterrorism, within a population based on the data regard‐ ing course of infection in a single isolated individual. It enables identification of measures for improving the health of the community as a whole. Epidemics can pass through the population at an extremely fast rate, may persist for a long time at low levels, may show cyclic patterns or there may be sudden flare-ups. Therefore, meaningful data collection and data interpreta‐ tion are the essential components of epidemiology. These data can be exploited to identify trends, make general predictions and assess shortcomings of those predictions. Such predic‐ tion can be highly erroneous unless derived mathematically and here lies the utility of mathematical modeling. Moreover, age, clinical status and socio-economic status of the patient, environmental condition, demographic data, meta-population structure, geographical location etc are of great relevance to the disease occurrence, prevalence, persistence and ultimately eradication from a heterogeneous population.

For an in-depth and complete understanding of the unpredictable behavior and pattern of transmission of infectious diseases, both in time and space, epidemiological modeling proves to be a very powerful tool. It enables the epidemiologist to think rigorously and frame policies for protection and treatment of the unaffected and affected population respectively from the invasion by the pathogen. The use of mathematical model in the study of infectious diseases is being envisaged as an insightful alternative to ethically challenging, expensive and at times, practically impossible in vivo and in vitro detailed experimentation and subsequent compli‐ cated interpretation. There are instances when the models and the experimental data exist in

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a symbiotic relationship and improve our quantitative understanding of the infection dynam‐ ics. These models are used to hypothetically disrupt or neutralize genes and simulate infections within a few seconds. Latent infections that can be maintained for decades in a host can be mathematically reactivated to determine the effect on an outbreak. Sometimes, the concept of modeling has been enriched by introduction of an idea from a related field of science as in the case of HIV quasi-species model inspired by molecular quasi-species model in chemistry [1].

genicity or illness rate, which is expressed as the ratio of number with symptoms and number infected and lastly, (c) virulence which may be written as the ratio of number of severe/fatal cases and total number of cases. For example, chicken pox and measles are characterized by high infectivity, high pathogenicity and low virulence whereas, smallpox is recognized by high levels of all the three variables and tuberculosis is found to possess low infectivity, low

Unearthing the Complexities of Mathematical Modeling of Infectious Disease Transmission Dynamics

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

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The etiological agents of micro-parasitic infections are viruses, bacteria, protozoa or prions, all of which are usually unicellular and of microscopic size and can reproduce very fast. High degree of prevalence of this class of infections among the children worldwide is a matter of great concern to the parents, school authorities and the government. They include measles, rubella, chickenpox, mumps, whooping cough etc. Usual route of transmission is by direct contact through air-borne droplets. These diseases are characterized by high infectivity, short disease generation length and low mean age at infection, on one hand and lifelong acquired

Macroparasitic infections, caused by parasites, visible to the naked eye (e.g. helminthes, arthropods) are characterized by short duration of immunity following recovery. The number of parasites per host is a critical factor in epidemiology of this category of infections [6].

Microparasitic or macroparasitic infections can occur by either direct or indirect transmission. Usually microparasitic infections are transmitted by direct contact between two individuals as in the case of influenza, HIV, measles etc. The pathogen cannot survive outside the host body. Macroparasites, on the other hand, are indirectly transmitted and they spend a part of their lifecycle outside the host system, freely in the environment. There is a third category of diseases or vector-borne diseases where the causative organism is passed from primary host to the vector and from vector to another primary host as found in case of malaria, filariasis,

Variation in the characteristics of the infectious agent is manifested as difference in traits and

Mathematical models help in generating and clarifying hypotheses, assessing quantitative conjectures, finding answers to specific questions, determining sensitivities to changes in parameter values and estimating parameters from data in absence and presence of preventive and therapeutic interventions. Therefore, according to input-output approach, mathematical model may be viewed as a system where the facts about the disease serve as the inputs and prediction about the number of infected and uninfected people over time is regarded as the

pathogenicity and high degree of virulence.

**3. Categories of infectious diseases**

immunity following recovery on the other hand.

sleeping sickness etc. [7].

**4. Mathematical model**

thus arises the need of different dynamics.

Interpretation of variables associated with mathematical modeling helps in estimation of parameters of biological significance and deduction of concepts not directly perceptible from the data. These concepts prove invaluable in giving rise to the observed patterns as also unearthing the complexities underlying the infection. Concept-building, as derived from the framework of mathematical model, gives lead-time to the medical fraternity and the govern‐ ment policy-makers in designing and implementing timely intervention measures for preven‐ tion and control of spread of a communicable disease.

Daniel Bernoulli is regarded as the father of epidemiological modeling since he investigated the influence of variolation on life expectancy as early as 1760. One of the fundamental principles of epidemiology, the threshold theorem was established by Kermack and McKen‐ drick in 1927 to establish the dynamics of bubonic plague. Disease persistence in large host population was determined by Barlett.
