**12. Evolution of mathematical modeling**

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 statistics and models for estimation of parameters from epidemiological data.

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 influenza or H1N1 require careful parameterization to match available data and it should reflect both statistical uncertainty and uncertainty in data itself [13].

does not affect severely the transmission of the disease [11], [27]. For better understanding of

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Assumptions that have been discussed in a previous section hold true for developing countries where all individuals are exposed to similar death pressure, irrespective of age, due to the environmental conditions and lack of medical facilities. But the situation differs in developed countries where square shape age distribution is manifested. This can be attributed to availability of proper medical care. Modeling becomes somewhat complicated in analysis of macroparasitic infections owing to their inherent differences from the infections caused due to microparasites. In case of microparasitic infections, the internal dynamics of the pathogen within the host is not as crucial as the host's infection status. But, the complex life cycle of the macroparasite within the host necessitates inclusion of this parameter in modeling of macro‐ parasitic infections. Moreover, disease transmission and pathogenicity are highly related to

Furthermore, complex dynamics may be followed by the infections, where the causative agent can undergo mutation and can create a class of population, susceptible to the new strains. Previous infections in those cases confer only partial cross-immunity and thus, history of infection is important. Number of parameters in the model therefore, increases exponentially with the number of strains. In these circumstances, either reduced transmission approach or polarized immunity concept is applied [11]. Emergence of drug resistant strains is an issue of great concern which should be included in modeling disease dynamics. Sexually transmitted diseases unfold a different picture because of high risk of exposure of sexually active individ‐

Complexities may also arise in modeling of diseases in presence of an optimal control measure where there is a delay element between implementation of isolation and quaran‐ tining. Inclusion of delay increases the proportion of transmission occurring with an asymptomatic or pre-symptomatic infector and hence, necessitating adoption of stringent measure including contact tracing. Oversimplification may fail to estimate the efficacy of contact tracing in reducing the transmission. However, in this delay period, the patient may practice self-isolation depending on the nature and severity of the symptoms and the time

In a modified SIR model with "skipping" dynamics for diseases like influenza, it has been shown that the immunity of hosts depends on previous exposure to the disease and immune memory. Due to mutational changes in influenza virus or antigenic evolution, hosts may be re-infected with the disease every few years, with years of 'skips' or gaps in between [31].

The simple SIR model has been extended to include the effects of saturation where the incidence rate is not bilinear in S and I but a general function *f*(S,I). It is assumed that in presence of large number of infectives in the population, the number of contacts per time diminishes [3]. The simplistic SIR model has evolved into biphasic SIR model (B-SIR) to explain the key features of multi-ennial epidemic cycle. In this model, the SIR dynamics alternates between a relatively fast epidemic phase in which there is a significant increase in the number of infected as well as a rapid decline in the count of the susceptible and a slow build-up phase

the model with periodic perturbations, bifurcation diagram proves beneficial [7].

the load of the causative organisms within the host system [6], [7].

uals, belonging to a particular age group.

scales involved [22].

A new rule, known as 20-80 rule has been proposed by researchers for certain infectious diseases which show unique features. According to the rule, 20% of the individuals are responsible for 80% of onward transmission and they are known as super-spreaders. This gives rise to variation in the number of secondary infections per infected individual which is denoted by Z. It is described by two parameters-mean R0 among infections and dispersion parameter, K. A small K (<0.1) indicates that a small proportion of infected individuals actively transmit the pathogens whereas, a large K (>4) indicates that all infected individuals are equally responsible for onward transmission [19].

In multiple-host diseases, different hosts or reservoirs are involved. Dynamics of the disease itself may vary in the different hosts and the mode of transmission from one host to another is quite complex. All these factors should be borne in mind during study of these diseases [6].

In viral disease like Hepatitis B, there is a carrier state where the individual is not fully recovered and is able to transmit low level of infection throughout his life. Some diseases like chlamydial infection is characterized by infected individual who is asymptomatic but is able to transmit disease. Again, meningitis may remain benign for long within population but may exhibit sporadic symptomatic outbreaks [7]. Infectious diseases which are characterized by multi-factorial pathologies or where concurrent infections prevail in immune-compromised hosts usher in complexities in mathematical modeling [1]. A common feature of certain childhood infections is the periodical occurrence of high levels of infection of school-going children which necessitates the inclusion of a time-varying contact rate between susceptible and infected classes of individuals in the model, which is then known as seasonally-forced model and is recognized by a sinusoidal function. The concept of seasonality in the coefficient of transmission and temporal heterogeneity was introduced by Soper to account for the high amplitude outbreaks of measles in Glasgow. These types of diseases are characterized by strong annual, biennial and sometimes irregular oscillations. In India, polio has been found to occur annually. Measles is known to exhibit biennial occurrence for extended periods in London. It has been noticed that the same disease can have different temporal patterns during different epochs which may be attributed to changes in epidemiological factors, e.g. population birth rate, magnitude of disease transmission, and strength of seasonality [14]. Modeling of measles dynamics is best achieved by incorporation of an epidemic oscillator, which takes into account the birth rate variations and can analyze chaotic behavior of epidemic outbreaks. It has been observed that high birth rates drive measles dynamics to annual cycles and dimin‐ ishing birth rates result in biennial patterns. Vaccination causes irregular cycles. Complex oscillations arise due to interactions between the externally imposed annual seasonality and intrinsic oscillatory dynamics of the infection itself [26]. The approach of "term-time forcing" should also be employed for temporally forced models of another childhood bacterial infection, pertussis or whooping cough. Though both measles and pertussis possess identical values for basic reproduction ratio, yet they exhibit different dynamics because the infectious period in whooping cough is longer than that of measles. School holidays during Christmas does not affect severely the transmission of the disease [11], [27]. For better understanding of the model with periodic perturbations, bifurcation diagram proves beneficial [7].

influenza or H1N1 require careful parameterization to match available data and it should

A new rule, known as 20-80 rule has been proposed by researchers for certain infectious diseases which show unique features. According to the rule, 20% of the individuals are responsible for 80% of onward transmission and they are known as super-spreaders. This gives rise to variation in the number of secondary infections per infected individual which is denoted by Z. It is described by two parameters-mean R0 among infections and dispersion parameter, K. A small K (<0.1) indicates that a small proportion of infected individuals actively transmit the pathogens whereas, a large K (>4) indicates that all infected individuals are equally

In multiple-host diseases, different hosts or reservoirs are involved. Dynamics of the disease itself may vary in the different hosts and the mode of transmission from one host to another is quite complex. All these factors should be borne in mind during study of these diseases [6].

In viral disease like Hepatitis B, there is a carrier state where the individual is not fully recovered and is able to transmit low level of infection throughout his life. Some diseases like chlamydial infection is characterized by infected individual who is asymptomatic but is able to transmit disease. Again, meningitis may remain benign for long within population but may exhibit sporadic symptomatic outbreaks [7]. Infectious diseases which are characterized by multi-factorial pathologies or where concurrent infections prevail in immune-compromised hosts usher in complexities in mathematical modeling [1]. A common feature of certain childhood infections is the periodical occurrence of high levels of infection of school-going children which necessitates the inclusion of a time-varying contact rate between susceptible and infected classes of individuals in the model, which is then known as seasonally-forced model and is recognized by a sinusoidal function. The concept of seasonality in the coefficient of transmission and temporal heterogeneity was introduced by Soper to account for the high amplitude outbreaks of measles in Glasgow. These types of diseases are characterized by strong annual, biennial and sometimes irregular oscillations. In India, polio has been found to occur annually. Measles is known to exhibit biennial occurrence for extended periods in London. It has been noticed that the same disease can have different temporal patterns during different epochs which may be attributed to changes in epidemiological factors, e.g. population birth rate, magnitude of disease transmission, and strength of seasonality [14]. Modeling of measles dynamics is best achieved by incorporation of an epidemic oscillator, which takes into account the birth rate variations and can analyze chaotic behavior of epidemic outbreaks. It has been observed that high birth rates drive measles dynamics to annual cycles and dimin‐ ishing birth rates result in biennial patterns. Vaccination causes irregular cycles. Complex oscillations arise due to interactions between the externally imposed annual seasonality and intrinsic oscillatory dynamics of the infection itself [26]. The approach of "term-time forcing" should also be employed for temporally forced models of another childhood bacterial infection, pertussis or whooping cough. Though both measles and pertussis possess identical values for basic reproduction ratio, yet they exhibit different dynamics because the infectious period in whooping cough is longer than that of measles. School holidays during Christmas

reflect both statistical uncertainty and uncertainty in data itself [13].

responsible for onward transmission [19].

198 Trends in Infectious Diseases

Assumptions that have been discussed in a previous section hold true for developing countries where all individuals are exposed to similar death pressure, irrespective of age, due to the environmental conditions and lack of medical facilities. But the situation differs in developed countries where square shape age distribution is manifested. This can be attributed to availability of proper medical care. Modeling becomes somewhat complicated in analysis of macroparasitic infections owing to their inherent differences from the infections caused due to microparasites. In case of microparasitic infections, the internal dynamics of the pathogen within the host is not as crucial as the host's infection status. But, the complex life cycle of the macroparasite within the host necessitates inclusion of this parameter in modeling of macro‐ parasitic infections. Moreover, disease transmission and pathogenicity are highly related to the load of the causative organisms within the host system [6], [7].

Furthermore, complex dynamics may be followed by the infections, where the causative agent can undergo mutation and can create a class of population, susceptible to the new strains. Previous infections in those cases confer only partial cross-immunity and thus, history of infection is important. Number of parameters in the model therefore, increases exponentially with the number of strains. In these circumstances, either reduced transmission approach or polarized immunity concept is applied [11]. Emergence of drug resistant strains is an issue of great concern which should be included in modeling disease dynamics. Sexually transmitted diseases unfold a different picture because of high risk of exposure of sexually active individ‐ uals, belonging to a particular age group.

Complexities may also arise in modeling of diseases in presence of an optimal control measure where there is a delay element between implementation of isolation and quaran‐ tining. Inclusion of delay increases the proportion of transmission occurring with an asymptomatic or pre-symptomatic infector and hence, necessitating adoption of stringent measure including contact tracing. Oversimplification may fail to estimate the efficacy of contact tracing in reducing the transmission. However, in this delay period, the patient may practice self-isolation depending on the nature and severity of the symptoms and the time scales involved [22].

In a modified SIR model with "skipping" dynamics for diseases like influenza, it has been shown that the immunity of hosts depends on previous exposure to the disease and immune memory. Due to mutational changes in influenza virus or antigenic evolution, hosts may be re-infected with the disease every few years, with years of 'skips' or gaps in between [31].

The simple SIR model has been extended to include the effects of saturation where the incidence rate is not bilinear in S and I but a general function *f*(S,I). It is assumed that in presence of large number of infectives in the population, the number of contacts per time diminishes [3].

The simplistic SIR model has evolved into biphasic SIR model (B-SIR) to explain the key features of multi-ennial epidemic cycle. In this model, the SIR dynamics alternates between a relatively fast epidemic phase in which there is a significant increase in the number of infected as well as a rapid decline in the count of the susceptible and a slow build-up phase characterized by continuous replenishment of susceptible. This enables estimation of threshold(bifurcation) values at which there will be a switching from biennial to annual epidemic dynamics [14].

dynamic approach can be successfully utilized in mathematical modeling of such viruses like dengue, human respiratory synctial virus, Hepatitis C, Toxoplasma etc. [19], [33], [34].

Unearthing the Complexities of Mathematical Modeling of Infectious Disease Transmission Dynamics

Several new modeling concepts are being explored for detailed description of diseases like AIDS, tuberculosis. These include differential infectivity (DI) and staged progression (SP) model [29]. For modeling of infectious diseases with long infectious period and where there are multiple alternative disease progression pathways and branching, or where there is considerable difference in virulence or when only a part of the infected population undergoes a treatment whereas the rest remains untreated, staged progression models seem to be the most suitable [35]. Individuals infected with HIV sequentially pass through a series of stages, being highly infectious in first few weeks after their own infection, then having low infectivity for many years, and finally becoming more infectious as their immune system breaks down and they progress to full-blown AIDS. Investigation of the influence of imperfect vaccine on HIV transmission by individuals in AIDS stage with the help of SP model revealed that the imperfect vaccine can eliminate HIV in a given community at vaccination reproduction number less than unity, but the disease will persist otherwise [36], [37]. Since, tuberculosis is characterized by presence of very long latent period and infectious period, time is not considered to be constant in modeling of the disease. The latent period and infectious period are divided into n-stages and stage progression model with bilinear incidence was formulated [39]. Characteristic features of DI and SP models have been given in a tabular form (Table 3).

**DI model SP model**

Healthcare-associated infections (HCAI) or nosocomial infections or hospital acquired infections are source of great concern in developed as well as developing countries. Such infections can be modeled using either deterministic or stochastic approach for evaluation of control policies. This requires sound model parameterization and sensitivity analyses [40].

Suitable for modeling of diseases exhibiting variability of infectiousness with time as in AIDS or where time-scale of

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201

Total host population is partitioned into the following compartments: the susceptible compartment, the infectious compartment , whose members are in the *i*-th stage of the disease progression, and the terminal compartment. It is assumed that there is no recovery from the disease, and thus the only exit from the terminal

Disease-free equilibrium and unique endemic equilibrium

have also been established with these models

disease transmission is too long.

compartment is death.

Suitable for diseases where viral levels differ between individuals as in sexually transmitted diseases and also diseases where infectivity depends on parasite or viral loads in infected hosts or vectors e.g. malaria, dengue

groups according to their infectivities. Total population

Infection-free equilibrium is globally stable and there exists

a unique endemic equilibrium for these models

**Table 3.** Characteristic features of DI and SP model

Infectives are divided into a number of a

size is assumed constant.

fever.

A likelihood-based methodology has been developed which assumes the generation interval to follow Weibull distribution and a specific infection network underlies the observed epidemic curve [23].

An agent-based model in the study of tuberculosis has revealed that recruitment of increased number of resting macrophages to the infection site in tuberculosis increases bacterial load. This suggests that the inflammatory response may be detrimental to the host [32].

Since, application of mathematical modeling approach shows that vaccination decreases the mean age at infection, it deduces that implementation of vaccination programs can actually increase the incidences of absolute number of serious cases, if the probability of disease complications increases with age. This is an interesting paradigm of mathematical modeling. In case of imperfect or ineffective vaccination, a backward transfer between the susceptible and recovered compartments must be considered because the vaccinated individuals may come back to the susceptible category or may become directly infected through nonlinear transmission. This behavior gives rise to bi-stability and backward bifurcation [26].

Exponentially growing population and incidences of mortality due to childhood infections in developing countries are not considered in modeling of infections where the host population is assumed to be of constant size. The component of heterogeneity also exists in cases of sexually transmitted diseases. This is further complicated by the fact that, unlike other infections, recovery from STD does not guarantee development of acquired immunity. The currently existing rationale of mathematical modeling may need to be modified while investigating mother-to-child diseases and diseases transmitted by multiple users of syringe as in AIDS [6].

A hierarchical dynamics has been observed in epidemic outbreaks of complex heterogene‐ ous networks. Propagation of infection occurs via a cascade from higher to lower degree classes [10].

In most of the cases, epidemiological processes within the population and immunological processes involved within the individual host are considered separately, with no explicit interaction between the two. Infectious diseases, caused by RNA viruses are characterized by high mutation rate and short generation time of the viruses indicating that evolutionary processes occur rapidly and evolutionary outcomes depend on the fitness of viral mutants at different stages of viral lifecycles and interactions between viral variants. Moreover, ecological and epidemiological factors, such as host contact patterns, transmission routes, host move‐ ments also govern the success of viral transmission in a population. In this approach, three different time scales are considered-a fast time scale for within-host dynamics, an intermediate time scale for epidemiological process and a slow time scale for the environmental contami‐ nation [5]. Link between the evolutionary process at the pathogen level and the ecological processes at the host population level can be established by phylodynamics [11]. The phylo‐ dynamic approach can be successfully utilized in mathematical modeling of such viruses like dengue, human respiratory synctial virus, Hepatitis C, Toxoplasma etc. [19], [33], [34].

Several new modeling concepts are being explored for detailed description of diseases like AIDS, tuberculosis. These include differential infectivity (DI) and staged progression (SP) model [29]. For modeling of infectious diseases with long infectious period and where there are multiple alternative disease progression pathways and branching, or where there is considerable difference in virulence or when only a part of the infected population undergoes a treatment whereas the rest remains untreated, staged progression models seem to be the most suitable [35]. Individuals infected with HIV sequentially pass through a series of stages, being highly infectious in first few weeks after their own infection, then having low infectivity for many years, and finally becoming more infectious as their immune system breaks down and they progress to full-blown AIDS. Investigation of the influence of imperfect vaccine on HIV transmission by individuals in AIDS stage with the help of SP model revealed that the imperfect vaccine can eliminate HIV in a given community at vaccination reproduction number less than unity, but the disease will persist otherwise [36], [37]. Since, tuberculosis is characterized by presence of very long latent period and infectious period, time is not considered to be constant in modeling of the disease. The latent period and infectious period are divided into n-stages and stage progression model with bilinear incidence was formulated [39]. Characteristic features of DI and SP models have been given in a tabular form (Table 3).


#### **Table 3.** Characteristic features of DI and SP model

characterized by continuous replenishment of susceptible. This enables estimation of threshold(bifurcation) values at which there will be a switching from biennial to annual

A likelihood-based methodology has been developed which assumes the generation interval to follow Weibull distribution and a specific infection network underlies the observed

An agent-based model in the study of tuberculosis has revealed that recruitment of increased number of resting macrophages to the infection site in tuberculosis increases bacterial load.

Since, application of mathematical modeling approach shows that vaccination decreases the mean age at infection, it deduces that implementation of vaccination programs can actually increase the incidences of absolute number of serious cases, if the probability of disease complications increases with age. This is an interesting paradigm of mathematical modeling. In case of imperfect or ineffective vaccination, a backward transfer between the susceptible and recovered compartments must be considered because the vaccinated individuals may come back to the susceptible category or may become directly infected through nonlinear

Exponentially growing population and incidences of mortality due to childhood infections in developing countries are not considered in modeling of infections where the host population is assumed to be of constant size. The component of heterogeneity also exists in cases of sexually transmitted diseases. This is further complicated by the fact that, unlike other infections, recovery from STD does not guarantee development of acquired immunity. The currently existing rationale of mathematical modeling may need to be modified while investigating mother-to-child diseases and diseases transmitted by multiple users of syringe

A hierarchical dynamics has been observed in epidemic outbreaks of complex heterogene‐ ous networks. Propagation of infection occurs via a cascade from higher to lower degree

In most of the cases, epidemiological processes within the population and immunological processes involved within the individual host are considered separately, with no explicit interaction between the two. Infectious diseases, caused by RNA viruses are characterized by high mutation rate and short generation time of the viruses indicating that evolutionary processes occur rapidly and evolutionary outcomes depend on the fitness of viral mutants at different stages of viral lifecycles and interactions between viral variants. Moreover, ecological and epidemiological factors, such as host contact patterns, transmission routes, host move‐ ments also govern the success of viral transmission in a population. In this approach, three different time scales are considered-a fast time scale for within-host dynamics, an intermediate time scale for epidemiological process and a slow time scale for the environmental contami‐ nation [5]. Link between the evolutionary process at the pathogen level and the ecological processes at the host population level can be established by phylodynamics [11]. The phylo‐

This suggests that the inflammatory response may be detrimental to the host [32].

transmission. This behavior gives rise to bi-stability and backward bifurcation [26].

epidemic dynamics [14].

epidemic curve [23].

200 Trends in Infectious Diseases

as in AIDS [6].

classes [10].

Healthcare-associated infections (HCAI) or nosocomial infections or hospital acquired infections are source of great concern in developed as well as developing countries. Such infections can be modeled using either deterministic or stochastic approach for evaluation of control policies. This requires sound model parameterization and sensitivity analyses [40].

Influenza control is a challenge since it is difficult to predict the predominant strain that will be circulating each season. Successful vaccination program can be designed from a model which includes the drift process (via the emergence of new strains), the co-circulation of existing strains and pre-existing immunity in the population. A good model is able to repro‐ duce the herald wave phenomenon of strain persistence from one influenza season to the next in temperate regions [41].

health response to a pandemic. The foremost important contribution of mathematical epi‐ demiology to healthcare professionals is the concept of basic reproduction ratio and epidemic threshold. From a medical or public health perspective, models are instrumental in policymaking, service planning, risk assessment and monitoring performance of infection control programs. Decision of whom, how and when to quarantine, vaccinate and initiate therapy can be planned and implemented successfully from the outcomes of a model of high quality. The source of complexity in models is the occurrence of huge variability in infection profiles, parameter values and time scales. However, complexity ensures model accuracy. Selection of type of model and its degree of complexity depends on the purpose of modeling and questions to be addressed. Failure to understand, forecast and control an epidemic outbreak in a particular locality can indicate that some of the key disease parameters and elements of biological complexity might have been overlooked while developing the model. One under‐ lying reason for this may be traced to a key fundamental feature of mathematical model. Unless the model is tightly fitted to the experimental data, there remains a doubt on the reliability of the parameter estimates obtained from the model. At the end, it can be said that mathematical model helps in integrating several disease-related facts and factors into a cohesive structure, better visualization of a complex system, determining the plausibility of epidemiological explanations, prediction of unexpected interrelationships among empirical observations and

Unearthing the Complexities of Mathematical Modeling of Infectious Disease Transmission Dynamics

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203

prediction of impact of changes in the system.

of of Institutions Kolkata, West Bengal, India

11(doi:10.1186/1471-2105-11-S1-S67).

Research, INFORMS (ISBN: 978-0-9843378-0-4).

and Gopa Roy Biswas

\*Address all correspondence to: sutapabiswas2001@yahoo.co.in

NSHM College Of Pharmaceutical Technology, NSHM Knowledge Campus, Kolkata-Group

[1] A Sorathiya, A Bracciali and P Liò (2010). Formal reasoning on qualitative models of co-infection of HIV and Tuberculosis and HAART therapy*.* BMC Bioinformatics

[2] N B Dimitrov and L A M Dimitrov (2010). Mathematical approaches to infectious disease prediction and control. In Infectious Disease Models Tutorials in Operations

[3] S Funk, M Salathe and V A A Jansen (2010). Modeling the influence of human behav‐ ior on the spread of infectious diseases: a review J. R. Soc. Interface 7, 1247–1256.

**Author details**

**References**

Sutapa Biswas Majee\*

Recently, multi-scale models are being developed which will help in predicting virulence, transmissibility, at the population level, susceptibility or resistance to drugs without conduct‐ ing tough and time-consuming laboratory experiments. For development of these types of models, complete genome sequence of the pathogen should be made available, which will allow mapping from genotype to complex phenotypes.

Numerous modeling approaches contain data estimated from the literature and for such cases extensive sensitivity analysis may prove beneficial. In uni-variate sensitivity analysis, impact of variation of one parameter by a certain percent on the outcome of the model is measured while all other parameters are held constant. Such analysis can be graphically represented on a tornado plot. In multivariate analysis, impact of multiple parameters is studied through Monte Carlo simulations [38]. Other multilevel fitting schemes and Bayesian/Markov chain Monte Carlo frameworks may also be useful [30].
