**4. Mathematical model**

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

experimental data to obtain estimates for the kinetic parameters associated with the system,

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Qualitative fitting of the data enables deeper understanding of the disease dynamics and quantitative fitting helps in designing and implementing control measures [7]. Apart from constructing a set of equations, data fitting and estimation of parameters, mathematical model also uses some graphical tools for characterization of dynamical systems. Phase plane techni‐ que like, linearization approximation is a graphical tool used to analyze the system dynamics of one state variable as a function of another state variable. In such plot, time dimension is not present but the trajectory of the dynamics is shown by arrows. The second technique is bifurcation diagram reserved for visualization of relatively complex dynamics. Bifurcation diagram is a summary of the asymptomatic dynamics of a dynamical system as a function of

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,

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

the progress of the disease is defined in terms of level of pathogen within the host.

which are otherwise difficult to obtain experimentally.

**5. Compartmental mathematical models**

a bifurcation parameter [6].

**5.1. SIR model**

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 control measure [8].

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

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 experimental data to obtain estimates for the kinetic parameters associated with the system, which are otherwise difficult to obtain experimentally.

Qualitative fitting of the data enables deeper understanding of the disease dynamics and quantitative fitting helps in designing and implementing control measures [7]. Apart from constructing a set of equations, data fitting and estimation of parameters, mathematical model also uses some graphical tools for characterization of dynamical systems. Phase plane techni‐ que like, linearization approximation is a graphical tool used to analyze the system dynamics of one state variable as a function of another state variable. In such plot, time dimension is not present but the trajectory of the dynamics is shown by arrows. The second technique is bifurcation diagram reserved for visualization of relatively complex dynamics. Bifurcation diagram is a summary of the asymptomatic dynamics of a dynamical system as a function of a bifurcation parameter [6].
