**10. Vaccination**

**9. Strategies for containment of an infectious disease**

194 Trends in Infectious Diseases

practice, attempts are made to prevent spreading of the infection.

little respite [10], [22], [24], [25].

Barlett postulated that any infectious disease cannot be maintained if the population size is below the critical community size unless there is supply of susceptible or regular migration of the infected class from the adjoining places. This will ultimately lead to disease extinction. Disease is said to fade-out if the duration of disease extinction is more than the disease generation length [6]. Although, theoretically, it may be possible to estimate and predict the time for disease eradication from the population, practically it is not feasible. Therefore, in

Once the epidemiological data have been modeled into a reliable mathematical model, they can be used to identify population subgroups at high risk of disease and develop preventive interventions or measures according to time, place and person. These measures include education, immunization, quarantine regulations or social distancing to restrict interaction with others and treatment options. Mathematical modeling of intervention strategies can be done in two ways. In the first method, the goal is to assess the effect on the disease dynamics by changing the value of a constant parameter associated with the disease. This indicates the best parameter value for a given performance measure. In the second method, intervention measures are varied as a function of time and the objective is to determine the best parameter value for a given performance measure. Pontryagin's Maximum Principle (PMP) is applied for comparison of a wide range of time varying functions. The best strategy for mitigation of spreading of a contagious disease would be the one which evolves with time during different phases of infection and thus focuses on progressively changing classes of populations. Optimal control theory suggests the most effective mitigation strategy to minimize the risk of individ‐ uals being infected by applying and balancing vaccination and administering drug in a costeffective manner. Based on this, vaccination has been found to be the most common strategy in reversing the epidemic growth of an infection within the population in the initial stages of an outbreak. But in absence of the strategy, due to some reason or other, medical practitioners either isolate symptomatic individuals or trace and quarantine contacts of symptomatic cases. These strategies require proper diagnosis of the symptoms of the disease in each and every individual. Efficacy of any implementation strategy is decided by the efficacy at which the infected person is isolated and the efficacy at which the persons with whom the infectious person came in contact can be quarantined. An infection can be sufficiently controlled if the values of basic reproduction ratio and the proportion of transmission occurring asymptotically can be reduced below a critical line. Since SARS is characterized by low R0 and low infectious‐ ness prior to clinical symptoms, effective isolation of symptomatic patients can sufficiently control an epidemic outbreak. For those diseases where the proportion of asymptomatic transmission is more than 1/ R0, contact tracing should be added to the set of control measures used. This proportion can be determined from the longitudinal data on clinical symptoms and pathogen load within the patient. Mathematically, it has been deduced that influenza is more difficult to control than small pox. Moreover, quarantining and contact tracing would not provide any extra benefit in case of influenza because of very short incubation and infectious periods. AIDS has taken an alarming proportion because of high risk of pre-symptomatic transmission and under such circumstance, self-isolation and contact tracing would provide

Implementation of proper and effective vaccination protocol is of primary concern to the epidemiologists and public health decision makers. Vaccination program is usually a govern‐ ment initiative applied on large spatial and temporal scales to reduce the level of complexity of disease. The ultimate goal of any vaccination program is to keep the value of basic repro‐ duction ratio below unity by altering the various control parameters. Mathematical modeling of vaccination recognizes that linear transfer occurs between the susceptible and the removed compartments. For modeling purpose, important variables related to vaccination that should be included are vaccination rate, i.e. the rate at which the susceptible individuals are vaccinated and efficacy or the proportion of susceptible left unprotected even, after immunization [26].

Two schemes are being widely employed worldwide for prevention and eradication of vaccine-preventable infections-mass vaccination and pulse vaccination. Mass vaccination is usually carried out in infants before the mean age at infection. But, it may not always be practically the most effective approach for global disease eradication. Pulse vaccination, on the other hand, is a cheaper and a better alternative where periodic vaccination of a certain proportion of the population renders sufficient and enough protection against further spread of the disease as the percentage of susceptible is always maintained below the threshold level required for an epidemic to start. But, pulse vaccination suffers from the side effect of resonance. Impact of single- and multiple-dose vaccination on rubella eradication has been studied extensively in an age-structured model with constant transmission rate using Floquet analysis. It has been established that eradication likelihood is governed by the effective duration of immunity. Booster vaccination has also been studied, but, in less details. Permanent immunity is conferred by booster dose in contrast to partial immunity imparted by primary vaccines. A new threshold quantity, known as re-infection thresh‐ old in case of vaccination-induced partial immunity, indicates the condition when vaccina‐ tion will not succeed thereby producing high levels of infection. Prevention or reduction in epidemicity is more readily achievable by booster vaccination, depending on the level of primary vaccination program, because it has the capacity to increase the vaccination coverage and herd immunity of the population as a whole. Outcome of a booster dose depends highly on precise timing of the additional vaccine doses as well as the propor‐ tion of individuals receiving the second-dose. During vaccination, decision of the individu‐ al or mass to undergo program greatly affects the efficacy of the program. Therefore, human behavior plays a significant role in ensuring success of the strategy [3], [6], [27].

An interesting thing to note while conducting mass scale immunization is that it is possible to protect the whole population from an outbreak even if there are some susceptibles in the population, at less than 100% immunization. This effect is called herd immunity. When R0 < 1, the DFE is globally asymptotically stable resulting in disease eradication as observed in case of measles. Vaccination reduces the force of infection and increases the mean age at infection when the infection may be acquired. Vaccination desynchronizes local dynamics which will prevent migration of susceptible population from neighboring places and thus can facilitate extinction of the disease. However, vaccination of each and every individual in the susceptible class is practically not feasible and more so, in economically backward countries. Therefore, the critical fraction or proportion of population that needs to be vaccinated is to be determined. Moreover, desired degree of success may not be always obtained due to vaccination because of less than optimum coverage, irregularities in the supply of vaccines, use of low-efficiency vaccines or waning rate of vaccines. All these may lead to re-emergence of disease outbreak, sometimes with increased intensity, owing to the resurgence of the susceptible class finally resulting in serious side effects.

**11. Quarantine**

Infectious class into the Quarantine class [28].

**12. Evolution of mathematical modeling**

Quarantine refers to intentional or forceful isolation of individuals suffering from diseas‐ es like leprosy, plague, cholera, typhus, yellow fever, smallpox, diphtheria, tuberculosis etc. The ultimate objective of the process is to reduce the average infectious period by isolat‐ ing some infectives, so that they do not transmit the infection. To study the effect of quarantine, a new class Q of quarantined individuals has been included in standard SIS and SIR endemic models. They include those who have been removed and isolated either voluntarily or coercively from the Infectious class. The quarantine reproduction number, R0 depends on the quarantine rate constant which governs the transfer rate out of the

Unearthing the Complexities of Mathematical Modeling of Infectious Disease Transmission Dynamics

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

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Epidemiological modeling has undergone numerous revisions and improvements to cope up with emerging new infections and discovery of new concepts and basis of existing infections. It is well-known that increase in model complexity by including more relevant biological details improves the accuracy. But it is practically not feasible to construct a fully accurate model. There will always be some factors related to host, pathogen, environment or population which cannot be estimated or predicted. Processes of random occurrence affect the degree of accuracy of model. Predictability of the model depends highly on a strong interplay between

The key assumption of deterministic mathematical model is the existence of homogeneous and constant population, where it is assumed that, the death of a susceptible, infectious or recovered individual is immediately compensated by the birth of a new susceptible. In actual practice, heterogeneity is observed where the population can be divided into several homo‐ geneous subpopulations or groups on the basis of mode of transmission, contact patterns, latent period, infectious period, genetic susceptibility or resistance as well as socio-economic, cultural, demographic and geographic factors. All these complexities can be suitably incorpo‐ rated in a multi-group model and can prove effective in explaining the dynamics of sexually transmitted diseases such as gonorrhea or AIDS [29]. Population-level heterogeneities that, if included can improve the model's accuracy and predictive ability, include age, gender, behavior, genetic susceptibility [7]. Influenza infection is well-studied and well-modeled. But, there are still several aspects of the infection which have not been included in model construc‐ tion like, the contribution of strain-specific cell tropism, pre-existing immunity, effect of host genetic factors on virulence and transmissibility of a particular strain. The model may be modified to determine the severity, duration and outcome of infection progression within an individual [30]. Moreover, drug resistance may develop. Therefore, a model with an immune response can be generated for better insight into the disease dynamics and the predictions from the model are different from those of a model formulated without an immune response. Twophase solution can be used to study different viral infections. Study of models, specific for

statistics and models for estimation of parameters from epidemiological data.

Vaccination coverage depends on the characteristics of the endemic equilibrium. At equilibri‐ um, the replacement number, R is equal to the product of basic reproduction ratio and the proportion of susceptibles i.e. R=R0*s*\*. The replacement number is defined as the average number of secondary infections produced by a typical infective during its entire course of infectiousness. Vaccination coverage of *p* reduces the proportion of susceptible to 1-*p*. Higher the magnitude of R0, higher is the vaccination coverage and is complicated by parameter such as vaccine efficacy. Therefore, for disease eradication to occur, R=R0*s*\*(1-*p*) < 1 or *p* > 1-1/ R0. The critical vaccination coverage is expressed by the formula, *pc=*1-1/ R0 and the value has been found to be 0.94 and 0.86 respectively for measles and rubella. An increase in the value results in a decrease of the spatial synchrony of disease dynamics. Smallpox has been successfully removed from the face of the earth because it possesses the least critical vaccination coverage. Vaccine efficacy of 0.97 represents that 3% of those vaccinated do not become immune.

**Parameters Description** Herd immunity Immunity and protection of the entire community achievable by vaccinating a proportion of the population and creating immune individuals Basic reproduction number under vaccination Number of secondary cases caused by one primary case introduced into a population in which a proportion has been vaccinated Critical vaccination proportion Proportion of population to be vaccinated to achieve eradication by maintaining Basic reproduction number under vaccination equal to one Vaccine efficacy Effectiveness of the vaccine to induce immunity Eradication likelihood Determined by effective period of immunity Re-infection threshold Occurs during transmission induced by partial immunity. It is a threshold quantity above which levels of infection will be high and vaccination fails

Optimal vaccination coverage and frequency of pulse vaccination has been studied with the help of Pythagore theorem [6]. The important terminologies associated with vaccination has been presented in Table 2.

**Table 2.** Glossary of important parameters related to vaccination
