**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 covariances that can influence the deterministic behavior. Stochastic simulations are computa‐ tionally intensive.

have T values in the order of months or years [19]. But, determination of generation time becomes complicated if the disease possesses asymptomatic periods of infection of variable or unknown duration. Another quantity that is estimated is the serial interval. It is the time between clinical onset of symptoms in the index case and the clinical onset of symptoms in the

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In a system of differential equations, an equilibrium point is a point at which all the equations equal zero. This indicates that the state of the system is not changing. In a mathematical model, a disease-free equilibrium (DFE) and one or more endemic equilibria are present. The DFE indicates that the entire population is susceptible since not a single infective exists in the population. Endemic equilibrium is steady-state equilibrium produced by spread of infection. If the solutions of the equations near the equilibrium points tend toward the points with time, they are said to be locally stable. An equilibrium point is referred to as globally asymptotically stable if the behavior of the system at any point tends toward the equilibrium point as time

The stability of DFE and existence of other nontrivial equilibria can be determined from a ratio, known as basic reproduction ratio [20]. A very brief description of the essential conceptual

**Parameters Description** Disease generation time Time from the moment one person becomes infected until

Equilibrium point Can be categorized as a. disease-free equilibrium point

Basic reproduction number Number of secondary cases caused by one primary case

A key concept or parameter in epidemiology, the basic reproduction ratio, R0 is being exten‐ sively studied during deterministic, non-spatial, unstructured modeling of in-host population dynamics of microparasitic infectious diseases, once they have been established. It is defined as the expected number of secondary individuals infected by an individual during his or her entire tenure of infectious period. Its derivation is applicable even when non-constant transmission probabilities between classes (i.e., non-exponential lifetime distributions) are assumed. If its value is greater than 1, the infection spreads across a non-zero fraction of

**Table 1.** Glossary of important parameters associated with mathematical modeling of infectious diseases

state solutions.

that person infects another person

when there is no infection in either the host or the vector or there is no pathogen and b. endemic equilibrium where the disease persists in the population. Both are steady

introduced into a population that is wholly susceptible

results from mathematical modeling has been given in Table 1.

average secondary case [2].

tends toward infinity.

**8. Basic reproduction ratio**

In stochastic modeling of HIV dynamics, it is assumed that the viral population is governed by the availability of target cells that can be infected and does not take into account the contribution of the immune responses in the control of virus load.

Stochastic extinction or disease eradication by chance occurs when an infected individual fails to reproduce and transmit the infection and ultimately the pathogen dies out.

An interesting thing to note at this juncture is that deterministic model can be transformed to the corresponding stochastic model by conversion of deterministic rates into the probabilistic ones on the basis of a fixed reference volume of the model. This has been done in the analysis of dynamics of HIV and the opportunistic co-infection TB by incorporation of the response of cytotoxic T-lymphocytes in absence and presence of HAART therapy by altering the model's parameters. Evolution of drug resistant strains can also be assumed to exhibit a stochastic pattern.

A combination of deterministic and stochastic approaches will be the most effective one because of their complementary features, although it may be time-consuming and may exploit more resources [1], [7], 12, [16-18].

As no element of chance or uncertainty is involved in the development of deterministic models, they account for the mean trend of a process only. However in addition to the above feature, stochastic model also accounts for the variance component around the process. Initial epidemic growth of an infection cannot be properly approximated by deterministic SIR model. This occurs because at this stage only a seed of infection is introduced in contrast to the large population at later stages. Probability that an infection will occur is governed by demographic stochasticity. In case of stochastic models, some parameters are characterized by a probability distribution, instead of a fixed constant value, as observed with the deterministic models [6]. Given a stochastic transmission model, most inferential methods rely on likelihood. Given a likelihood, inference can proceed along conventional lines, using tools such as maximum likelihood estimation, expectation maximization algorithm, rejection sampling and Markov chain Monte Carlo methods.
