**8. Basic reproduction ratio**

variances that can influence the deterministic behavior. Stochastic simulations are computa‐

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

Stochastic extinction or disease eradication by chance occurs when an infected individual fails

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

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

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

**7. Insights into conceptual results from mathematical model**

Two approaches can be used to determine the time-scale of disease transmission by utilizing data on individual-to-individual chains of transmission. Estimation of disease generation time (T) is one approach. It may be defined as the expected length of time between infection of an index case and infection of his or her secondary cases. It is the duration of latency plus infectiousness. The generation time of measles is approximately 14 days. A short generation time indicates rapid transmission whereas a longer T suggests slower spread but longer carriage. The duration of carriage of pathogens represents an upper limit on T and it can be concluded that directly transmitted acute infections have T < 1 month and chronic infections

contribution of the immune responses in the control of virus load.

to reproduce and transmit the infection and ultimately the pathogen dies out.

tionally intensive.

190 Trends in Infectious Diseases

pattern.

more resources [1], [7], 12, [16-18].

chain Monte Carlo methods.

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 susceptible population. If the spreading rate is too low and R0 cannot cross beyond the threshold level, it is not feasible to affect a finite proportion of population and disease dies out in a finite time. In a contact network model, the disease cannot replenish itself and ultimately dies out after a finite number of waves, if R0< 1 with probability 1. Disease persists with positive probability, at least by infecting one person in each wave, if R0 > 1. Keeping and maintaining the value of R0 < 1 reflects the stability of the disease-free equilibrium and creates a condition for clearing of pathogen from the population and thus it is the goal of any public health initiative designed for containment or control of infection. Moreover, the estimation of R0 plays a crucial role in understanding the outbreak and potential danger from emerging infectious disease. The concept of R0 has also been developed for complex models like stochastic and finite systems, models with spatial structure and also macroparasite infections. Comparison of R0 values, based either on their numerical values or area under the infectiousness curve, helps in estimation of relative intrinsic transmissibility of the pathogens [9], [10], [21], [22].

homosexual population in the United Kingdom, R0 is close to 4 and approximately 11 for

Unearthing the Complexities of Mathematical Modeling of Infectious Disease Transmission Dynamics

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

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The magnitude of R0, along with the disease generation time, help in assessing the time scale of infection, implementing sustainable control measures at the most appropriate time and justifying implementation of costly approaches in management of infectious diseases. Low value of R0 for any infection suggests that the epidemic can be controlled, either by adopting

Condition for endemicity can be deduced from an idea of R0. Disease is said to exist in an endemic state when it persists in the population at a low and constant level of prevalence, for which there should be a continuous supply of susceptibles. This happens if deaths and births occur at equal constant rates keeping the population turnover rate at a fixed value. An outcome of this assumption is the negative exponential distribution of age. At endemic equilibrium, the relationship between the proportion of susceptible in the population and the basic reproduc‐

Though R0 is the widely accepted and used indicator of control measure, it has been observed that different control approaches may produce same degree of reduction of R0 but not same effect on the growth rate. Factors such as timing of secondary infections, negative impact of control measures on the population are not considered while implementing public health

R0 is a highly pathogen-centered parameter and recently, a host-centered parameter has been evolved, the basic depression ratio, D0. An alternative parameter for heterogeneous population has been developed, type reproduction number, T which indicates the efficacy of the control measure against a particular subtype of host population, from which if infection is eradicated,

In case of a population which is susceptible after acquiring immunity from a previous epidemic or due to vaccination, instead of R0, Reff or effective value of R0 is used. Reff is a time-dependent quantity that accounts for the population's reduced susceptibility. If Reff is greater than one, the number of infected individuals grows and decreases if Reff is less than one. Therefore, the critical proportion of susceptible is given by Reff equal to one [11]. The parameter can be determined by fitting deterministic epidemiological model employing a generalized least

In case of seasonally driven epidemics as with different childhood microparasitic infections, it is necessary to determine the number of susceptible left after a major epidemic (S0). If S0 goes above a critical threshold value, epidemic outbreak may recur in the next year or there will be

female prostitutes in Kenya [7], [13], [22].

**8.1. Significance of basic reproduction ratio**

tion ratio is inverse [6].

**8.2. Alternatives to R0**

the disease will not sustain at all [21].

squares estimation scheme [23].

single or combined putative containment procedures [21], [22].

initiatives based on the basic reproduction ratio [21].

a skip i.e., a year when epidemic fails to initiate [14].

There are also alternative approaches of estimating R0 from available incidence or epidemio‐ logical data which require simplifying assumptions for numerical estimation of some un‐ known parameters. It is assumed that the host population is homogeneous, mixes uniformly and is of constant size in a constant state. The number of contacts per infective is independent of the number of infectives. Infectivity and mortality do not depend on age, genetic make-up, geography. Moreover, it is assumed that all individuals are born susceptible and as soon as disease is acquired, they are no longer considered susceptible. Spreading pattern of the epidemics is controlled by the generation time-scale. All these assumptions may never be fully realized in a practical clinical setting. R0 can be expressed as the ratio of the life expectancy and mean age of acquiring the infection. Thus, higher the R0, lower is the mean age at infection. Another important relationship that can be derived is that the mean age at infection is the reciprocal of the force of infection. A key parameter, the coefficient of transmission for airborne diseases, can be determined from the value of the force of infection. Alternatively, R0 can be estimated from the intrinsic growth rate of the infected class, which is highly dependent on collecting accurate data. Stochastic fluctuations can affect the value of growth rate. In case of vector-borne disease like dengue, R0 was calculated from the survival function, assuming spatial compartments of varying vector density. For multiple classes of infectives, R0 can be defined per infection cycle [6], [10], [21].

From sensitivity analysis, it has been found that different infection-and population-related factors may affect R0 like, transmission rate, vector mortality, incubation period of the vector, the relative infectiousness after isolation [21].

For both measles and whooping cough in England and Wales from 1945 to 1965, R0 has been found to be nearly 17. For the H1N1 epidemic in UK, R0 has been estimated to be 1.4. Estimation of R0 for SARS by different groups have mostly given the values in the range of 2-4, although a wide range of 1-7 has also been found in the literature. For smallpox, the R0 was found to be in the range of 4< R0 <10. Though an upper bound has been found for pandemic influenza, no lower bound could be obtained. For AIDS, R0 is always greater than 1, especially in African countries and it depends highly on the sexual behavior. For homosexual population in the United Kingdom, R0 is close to 4 and approximately 11 for female prostitutes in Kenya [7], [13], [22].
