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# Mathematical Models for Fighting Zika Virus

While the role of mathematics in comprehending and predicting certain sciences  is obvious to most people, the usefulness of mathematical models in the field of epidemiology is often not so clear. Scientists (and laypeople) tend to greatly under or overestimate the state of the art and helpfulness of mathematics in understanding infectious disease spread. Among infectious diseases, mosquito-borne diseases are both ubiquitous and notoriously difficult to control.

When a little-known mosquito-borne pathogen such as Zika virus causes a sudden outbreak accompanied by potentially deadly or disabling side effects, like Guillain-Barré syndrome or microcephaly, public health workers and policy makers need a few basic questions answered: How many people will the outbreak potentially infect? How far and how quickly will the disease spread? What areas and people are at highest risk, and when are they most at risk? How can we best make use of limited resources? How can we best slow or prevent the outbreak and protect vulnerable populations?

Mathematical models that simulate the spread of mosquito-borne disease can provide important guidance and insight to all of these questions. They can help identify trends of where and how the infection will spread. The models can also determine the sensitivity of important epidemic quantities, such as total number of people infected, to input control parameters and thus inform best practices for control. Any known and broadly-held statistical correlations between disease incidence and demographic or weather data from past epidemics can often improve model-based estimates.

Forecasting even the most predictable diseases, such as seasonal influenza, is difficult and often met with little success. Accurate real-time prediction of mosquito-borne epidemics with mechanistic transmission models, particularly when outbreaks are sporadic, is currently beyond our capability. Even though present models forecast the number of infections with available data adequately at best, they can quantify the resulting “what-if” scenarios that may offer insight into disease control.

The first basic mathematical model addressing the spread of mosquito-borne disease was developed by Ronald Ross to model malaria [4]. He realized that disease spread depended on a few important factors, including the rate of contact between mosquitoes and humans, the number of times a female mosquito bites in her lifetime, the number of available susceptible humans, and the length of the infectious period in humans. An interesting and important tidbit: only female mosquitoes bite, as they need the protein in blood to produce eggs. Thus, the mosquito biting rate is intrinsically related to the egg-laying rate (gonotrophic cycle).

Figure 1. The mosquito-human transmission cycle. An infectious mosquito bites a susceptible, uninfected human and transmits the virus via saliva. Once the human become infectious (usually accompanied by symptoms), the human can transmit the pathogen to an uninfected mosquito via the blood the mosquito ingests.
The Ross-MacDonald ordinary differential equation (ODE) model provided insight into how different aspects of the disease transmission cycle interact to cause outbreaks  (see Figure 1 for a schematic of the mosquito-human transmission cycle). The model can be used to compute the basic reproduction number, $$R_0$$, that estimates the expected number of secondary cases resulting from a single infected person in a fully susceptible population. From a mathematical perspective, the stability or instability of the disease-free boundary equilibrium of the system is determined by the basic reproduction number. If $$R_0 < 1,$$ then the disease-free equilibrium is locally asymptotically stable. If $$R_0 > 1,$$ the disease-free equilibrium is unstable and introduction of an infected individual will result in an outbreak. In the early stages of an epidemic, $$R_0$$ is the key quantity of interest, and the goal is to identify mitigation strategies to reduce it below the threshold $$R_0=1.$$ For example, most mosquito-borne disease models can predict how decreasing the bites on humans or reducing the mosquito population below a certain level would impact the spread of Zika.

Local sensitivity analysis is another mathematical tool to quantify which parameters will be most able to reduce the severity of an outbreak. Sensitivity analysis uses the derivative of $$R_0$$ —with respect to the parameters—to determine that in the many mosquito-borne epidemics, $$R_0$$ is most sensitive to the average number of bites of an infectious mosquito before it dies [1]. These bites depend on the mosquito biting rate, the extrinsic incubation period (EIP)—the time for an infected mosquito to become infectious—and the mosquito lifespan. Sensitivity analysis quantifies the way in which reducing the biting rate, say by using pesticides, can slow the epidemic. It explains how rising global temperatures, which could lower the EIP and increase the range of mosquito species, would increase $$R_0,$$ allowing the disease to spread into new areas. The analysis helps predict the effectiveness of current mitigation efforts focused on reducing mosquito lifespan.

Figure 2. Disease transition arrows are in black, the dashed arrows represent contacts between humans and mosquitoes, and population dynamics are in grey. Susceptible human hosts, $$S_h$$, can be infected when they are bitten by infectious mosquitoes. Infected humans become exposed (infected but not infectious), Eh,Eh, then infectious $$I_h.$$ Infectious humans recover with a constant per capita recovery rate to enter the recovered, $$R_h,$$ class. Susceptible mosquito vectors $$S_v,$$ can become infected when they bite infectious humans. The infected mosquitoes then move through the exposed, $$E_v,$$ and infectious, $$I_v,$$ classes. Population births and deaths are shown as well [1].
Figure 2 shows a diagram depicting disease transition and population dynamics used to derive the model’s system of nonlinear coupled ODEs. In general, there is not enough data to parameterize transmission models with any confidence. The number of infections, as a function of time, in previous outbreaks can help estimate a few missing parameters and ranges for $$R_0.$$ However, mathematical modelers must be careful not to over-fit the data by defining multiple free parameters in a simple mathematical model tailored to inaccurate, aggregated data. As John von Neumann famously stated, “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” Because so little is known about so many model parameters, modelers must use uncertainty analysis to determine the identifiability of the model parameters and prevent over-fitting of the data.

Fitting an entire epidemic curve is different from fitting the beginning of a curve and predicting what will occur subsequently – a daunting task, particularly when evaluating a new outbreak. Currently, statistical and expert opinion models are better at predicting mosquito-borne disease outbreaks in real time than the more complex transmission models, as observed in the recent DARPA chikungunya challenge [2]. However, these statistical models cannot directly predict the impact of mitigation efforts such as the effect of mosquito spraying, release of genetically modified and sterile mosquitoes, or other changes in the underlying system, like differences in climate, infrastructure, culture, etc. Because the statistical models do not account for dynamic nonlinear correlations among the factors driving an epidemic, they are less capable of predicting the impact of changes beyond the data from which they have been derived.

Therefore, the best approach combines these methods to both forecast the disease and quantify the effectiveness of different mitigation strategies. Confidence in these predictions acts as a strong function of the quality and quantity of available data. In a world of “big data,” quality data necessary to understand mosquito-borne epidemics is surprisingly sparse and difficult to obtain. Researchers can combine mechanistic and statistical models to estimate missing data and quantify the uncertainty in the predictions as a function of the reliability of the available data. They can then use these methods to derive a defensible and reproducible framework within which to inform policy and predict risk.

Where do mathematicians go from here? There is rich theory for mosquito-borne disease models, but we need greater understanding of nonlinear, non-autonomous and heterogeneous systems to continue making progress [3]. Most importantly, mathematicians and statisticians must interact often and thoroughly with data collectors, laboratory and field biologists, doctors, mosquito biology experts, sociologists, and public health officials in order to move towards better models, more useful and sustainable mitigation and prevention, and prediction capabilities that will save lives and prevent serious illness. While we have a long way to go, we can derive inspiration from the weather modeling community, which has continued to improve and expand—despite the daunting complexity of the system—to provide accurate forecasts and communicate the uncertainty in these forecasts.

References
[1] Manore, C.A., Hickmann, K.S., Xu, S., Wearing, H.J., & Hyman, J.M. (2014, September 7). Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus. Journal of Theoretical Biology, 356, 174-191.

[2] McMahon, B.H., Del Valle, S.Y., Asher, J., Hatchett, R., Cheever, A.E., Fang, D.Z.,…Mukundan, H. (2016). DARPA Chikungunya Challenge: Implications for Infectious Disease Forecasting. Manuscript in preparation.

[3] Reiner, R.C., Perkins, T.A., Barker, C.M., Niu, T., Chaves, L.F., Ellis, A.M.,…Smith, D.L. (2013, February 13). A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010. J. R. Soc. Interface, 10(81), 20120921. doi:10.1098/rsif.2012.0921

[4] Smith, D.L., Battle, K.E., Hay, S.I., Barker, C.M., Scott T.W., & McKenzie, F.E. (2012). Ross, Macdonald, and a Theory for the Dynamics and Control of Mosquito-Transmitted Pathogens. PLOS Pathog, 8(4), e1002588. doi:10.1371/journal.ppat

Carrie Manore has a Ph.D. in mathematics with a minor in ecosystem informatics. She is an NSF SEES Fellow at Tulane University and a visiting scientist at Los Alamos National Laboratory. She works with multidisciplinary teams to model and understand mosquito-borne disease spread. Mac Hyman has developed and analyzed mathematical models for the spread of malaria, dengue fever, chikungunya, and other vector-borne diseases. His current focus is to identify approaches where these models can help public health workers be more effective in mitigating the impact of these emerging diseases. He holds the Evelyn and John Phillips Distinguished Chair in Mathematics at Tulane University and is a past president of SIAM.

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