Data150-Aisling

Assignment 3 - Methods

Aisling Halliden

Professor Frazier

Human Development/Data Science

25 April 2021

SIR/TSIR and Markov Modelling as an Aid to Human Development

The effects of disease burden we see throughout the world and regions of focus in Asia are extreme. There is zero argument that the effects of disease are always negative, rather than positive. Despite this notion, people coming from lower socioeconomic statuses are more greatly burdened than wealthier individuals or communities. The harm is apparent, in that, there is limited access to healthcare for people in poorer communities. Not to mention, on a global scale, children are dying, people are losing working hours because they are sick, and thus are unable to live a life with freedom in which people have the ability to provide for their families. The problem faced on the individual level is somewhat the same as diseases cause death, personal pain, loss of work, but if we were to aggregate them to create ‘significance,’ I would argue that it is an economic problem. The economy will eventually suffer if people are unable to go to work because they are sick. The inherent nature of the problem is related to government and their public policy that surrounds healthcare. However, it is a fact that people are getting sick and we are aiming to discover why this is and why certain populations of poorer individuals are disparagingly burdened. There is a great focus as of now, that focuses on the global issue, regardless of socioeconomic status, but there is an urgent need to focus on the lack of accessibility to healthcare is inhibiting the development (as freedom) of developing regions. This problem is significant as there have been few improvements to disease burden. The solution will come from efforts towards improving healthcare and access, rather than trying to simply eradicate these diseases completely.

Extremely important to the estimation of global disease burden, is the SIR model. The SIR model is one of the most simplified mathematical models used to predict the dynamics of infectious diseases. The interaction between mathematics and epidemiology has been growing. The SIR model and also the basis for several derivatives of it, was created in the early 20th century by mathematicians Ronald Ross, William Hamer, and more. In the simplest terms, it is a system of three coupled nonlinear ordinary differential equations. When considering SIR it is important to note that if the demography is disregarded, an epidemic model transitions into the aggregation of several factors: a rule of individual contacts; a rule of per contact transmission; a rule of the infection’s development at the individuals’ level.

In one specific study they used a time series Susceptible-Infected-Recovered model (TSIR) to estimate the burden of HFMD (Hand, foot, and mouth disease) in Guangdong, China. They did this using varying province, city, and county geographical levels. The TSIR model is an extension of the previously created compartmental model (SIR). The model was developed by including the time series as a covariate in a nonparametric autoregressive modelling approach. This made forecasting performance more efficient which then led to transmission dynamics estimations. The TSIR model is arguably simpler to process because of its basis which is just a simple linear regression. In comparison, the SIR model is continuous in which individuals are born and transition into the class of individuals that are susceptible, then become infected and thus infectious with the disease, recover from the disease and are then removed. The TSIR model is a more “discrete-time version” (Du 3). The TSIR model considers the time lags in transmission of vector pathogens. “I sub t is defined as the number of new local cases at time t and I’ sub t is the effective number of cases, both local and imported, that could have generated a local case at time t” (Oidtman). To ascertain transmission coefficients, they use an equation. The S represents the number of individuals that are susceptible. The ‘I’ represents the number of individuals that are infectious. Finally, the R represents the number of individuals who are either immune or deceased and are therefore considered ‘removed.’

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While these models may be simple, detailed interpretation is key to better understand data trends. “Studying them is crucial in gaining important knowledge of the underlying aspects of the infectious diseases spread out, and to evaluate the potential impact of control programs in reducing morbidity and mortality” (Rodrigues 92). While it is easy to compute, the SIR model leaves room for it to oversimplify complex diseases. The model doesn’t incorporate the latent period between when an individual is exposed to a disease and when that individual becomes infected and contagious. It is usually admitted that the SIR model is derived by using a ‘‘fully mixed’’ population. This means that all individuals have the same probability to contact with any other individuals in the population. Despite a population that is fully mixed, it is not a 100% guarantee that the SIR model will be able to propagate the dynamics of the epidemic. In order to better understand the dynamic properties of any given epidemic, more sophisticated models are required.

The Markov-type model is also significant in the move towards disease burden estimation. It was named after Russian mathematician, Andrei Markov. This model is based on stochasticity, which just means randomness. The randomness is in the change of systems. One of the biggest assumptions of this model is that future states are not dependent on past states. While there are four types of Markov models that are used situationally, the Markov chain, which is used by fully autonomous systems that have complete observable states, is the most commonly used on a continuous-time scale. The other subsets of the Markov model are: the hidden Markov model which is used by autonomous systems with partially obersvable states; the Markov decision processes which is used by systems that are fully controlled and have a full observable state; the partially observable Markov decision processes which is used by controlled systems where there is a partially observable state. These models display any possible states, the transitions between states, the speed of transitions between states as well as the probabilities between them. No single individual can be in more than one state at any given moment in time. This is because these states are mutually exclusive as well as exhaustive. To include some examples of varying states that could be incorporated into a Markov model that was being made in the context of cancer intervention, here is a list: no progression, after progression, and death. As an individuals condition transitions over time into something serious or less worse, they also move between disease states.

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Figure 2 The joint probability function of the driving events (I1(t), …, IK(t)) is calculated in exactly the same way as for a disease with serial classes (refer to §3 and Figure 1). The joint probability function of the driving events (L1(t), …, LK(t)) can also be easily calculated noting that when L(t) individuals leave the susceptible class, then the driving events (L1(t), …, LK(t)) follow a multinomial distribution with L(t) number of trials and probability of success (α1, …, αK)” (Yaesoubi 21).

The formulation of mathematical models serves as a catalyst for the scrupulous analysis of disease burden and the effectiveness of proposed interventions. Through these models, they are able to make quantitave predictions. The epidemiology of infectious diseases is intrinsically convoluted because infection transmission is affected by several different factors. It is affected by biological characteristics of agent and host, contact sequences between hosts, environmental conditions, and finally human intervention through health services. Mathematical models are extremely important as they distinguish the complicated interplay between all of these factors. They help to aggregate data from varying sources. An important note is that when displaying effectiveness, simply observing an incidence decrease after public intervention, is not sufficient. On the other hand, in some cases there is the possibility that even though intervention was effective, there may continue to be an increase in incidence rate due to a rise in prevalence. Interventions are assessed by models through a comparison made against the ‘counterfactual’ scenario which displays what would have occurred had there not been any intervention at all. “Crucially, quantitative analysis can determine if a putative cause for an observed effect would have been strong enough to cause the effect – e.g. a modeling of the Ugandan HIV epidemic found that several modes of behavior change (delaying sexual debut, reducing numbers of sexual partners, increasing condom use) must have occurred to explain the observed decline in prevalence” (White 50). The realized decrease could not be accounted for by any single change in behavior. Another reason models are important is because they can fill in the gaps in knowledge by empirical research that assigns a certain significance to them. They do this by examining how a model reacts to varying parameter values. While it is important to talk about all the ways in which these models are beneficial and effective, it is also crucial to discuss the limits of them, just as we did briefly when talking about the SIR model. Over significantly shorter intervals of time, the Markov models do not work well. This is because, the short time intervals take away the stochasticity and are rather, deterministic.

References

  1. Du, Zhicheng et al. “Estimating the basic reproduction rate of HFMD using the time series SIR model in Guangdong, China.” PloS one vol. 12,7 e0179623. 10 Jul. 2017, doi:10.1371/journal.pone.0179623
  2. Oidtman, Rachel J et al. “Inter-annual variation in seasonal dengue epidemics driven by multiple interacting factors in Guangzhou, China.” Nature communications vol. 10,1 1148. 8 Mar. 2019, doi:10.1038/s41467-019-09035-x
  3. Yaesoubi, Reza, and Ted Cohen. “Generalized Markov Models of Infectious Disease Spread: A Novel Framework for Developing Dynamic Health Policies.” European journal of operational research vol. 215,3 (2011): 679-687. doi:10.1016/j.ejor.2011.07.016
  4. Magal, Pierre, and Shigui Ruan. “Susceptible-infectious-recovered models revisited: from the individual level to the population level.” Mathematical biosciences vol. 250 (2014): 26-40. doi:10.1016/j.mbs.2014.02.001
  5. White, Peter J.. “Mathematical Models in Infectious Disease Epidemiology.” Infectious Diseases (2017): 49–53.e1. doi:10.1016/B978-0-7020-6285-8.00005-8
  6. van Seventer, Jean Maguire. and Natasha S. Hochberg. “Principles of Infectious Diseases: Transmission, Diagnosis, Prevention, and Control.” International Encyclopedia of Public Health (2017): 22–39. doi:10.1016/B978-0-12-803678-5.00516-6