Force of Infection

I haven’t posted about a paper recently, so here’s a random grab out of my massive folder of “PDFs to Read Someday.”  (I should use differential equations to model how the number of PDFs in that folder changes over time…except that it appears to be exponentially increasing, so maybe I don’t want to know.) 

First, let’s talk about force of infection (FOI).  FOI is the rate at which susceptible individuals become infected per unit time.  It is an important parameter (the most important?) for transmission in epidemiological models.  The problem is that FOI isn’t as easy to measure as you might think.  In order to get good estimates of the rate of infection, you typically need to monitor infection status in many, many individuals.  For instance, even assuming you can identify infected individuals with 100% accuracy, your estimate of FOI is going to be better if you sample 100 individuals (1% resolution) than 20 individuals (5%) resolution. 

Movement rates between classes of the SIR model

Serious lack of FOI-related pictures on the Internet. Diagram from here.

But of course, you can’t identify infected individuals with 100% accuracy.  This is especially apparent when organisms have a period of latent infection, or when infection gets more obvious the longer the individual is infected.  For instance, if a snail has had first intermediate trematode infection for a while, crushing the snail might result in a mini-explosion of rediae/sporocysts all over your dissection surface.  (A brief aside for the curious.)  If it’s a newly developed infection, you might be lucky to spot a single redia/sporocyst.  The same is true when looking for Plasmodium falciparum (malaria parasite) in blood smears; you’re much more likely to identify an infection if parasitemia is very high.

What once were snail gonads are now trematode duplexes. Trematode sprawl. Photo from here.

Finally, if an individual can get infected multiple times, you would underestimate FOI if you could only identify the first infection event.  This is where the Mueller et al. (2012) paper from PNAS comes in.  For P. falciparum in Papua New Guinea, most people have infections with multiple different clones of P. falciparum.  That means that they got multiple clonal infections from one mosquito bite and/or they got subsequent infections with different clones from different mosquito bites. 

To get a super duper accurate FOI measurement with molecular techniques (molFOI), Mueller et al. (2012) took blood samples from ~400 kids aged 0-4 every two months for 69 weeks.  Impressive!  Then they used genotyping of the mps2 gene to determine how many P. falciparum clones each child had at each time point.

Cutting to the chase:  molFOI and the incidence of clinical cases of malaria (i.e., number of reported cases with high parasite loads) had very similar patterns across child age and week/season.  molFOI had a strong relationship with week/season, and this relationship explained ~50% of the relationship between malaria incidence and season.  Furthermore, most of the age*incidence relationship was explained by the age*molFOI relationship.  So, cool new tool for monitoring transmission in malaria control programs!  Neat!

Though they certainly aren’t the first people to demonstrate the efficacy of using insecticide treated bed nets, they also found that ITNs reduced infection risk by ~50%.  Awesome.

Multiple clonal infections are the norm in the P. falciparum system mentioned here.  Do you think that is true in many disease systems? 


Muellera, I., S. Schoepflind, T.A. Smithd, et al. 2012. Force of infection is key to understanding the epidemiology of Plasmodium falciparum malaria in Papua New Guinean children.  PNAS 109:25, 10030-10035.

Do parasites like it hot?

Today, I’m writing about a 2013 Ecology Letters paper entitled “Optimal temperature for malaria transmission is dramatically lower than previously predicted.”

Ro is the number of new disease cases that arise when just one infected individual is added to a susceptible population.  This is an important parameter for disease transmission models.  (Check out the wikipedia page for more information.)

Many factors affect Ro.  For malaria (Plasmodium falciparum), most of these factors are related to mosquito and parasite biology.  These are life history traits like the rate at which mosquitos bite people, the competence of the mosquito vector, and the mortality rate of the adult mosquitoes.  Mordecai et al. (2013) knew that these life history traits often don’t have linear relationships with temperature.  Instead, these life history trait rates tend to increase with temperature until some optimum, and then decline.


As it turns out, many biological responses have these nonlinear relationships with temperature because above some optimal temperature, enzymes/proteins become denatured. Photo Credit.


Using nonlinear estimates of the relationships between temperature and life history traits from the literature, Mordecai et al. (2013) mathematically modeled Ro.   They found that Ro peaked at 25°C, which is 6°C less than the peak Ro predicted if temperature is assumed to linearly affect life history traits and Ro.   Neat!


If you assume that temperature has a linear effect on life history traits, peak Ro occurs at 31°C (dotted line). If you allow for nonlinear relationships, peak Ro occurs at 25°C (solid line). Stylized results from Mordecai et al. (2013). 

You might be wondering why this 6°C difference is important.  First of all, the 6°C change equates to going from 77°F to 87.8°F, which is a big change!  Imagine which of those temperatures you’d prefer to work in on a summer day.

Second, temperature is one component of the global climate that is expected to change with climate change.  When people try to model how the range of malaria might shift with climate change, they need to know how mosquitos and the malaria parasite (and thus Ro) respond to various temperatures.  And the answer is:  that response is nonlinear, and the peak Ro is around 25°C! 


Predicted temperature increases around the world for 2070-2100. Photo Credit.

What do you think?  Modeling predicted range shifts is complicated business, and temperature is just one consideration.  Do you think malaria might end up in your continent/country/state/city?


Mordecai, E.A., K.P. Paaijmans, L.R. Johnson, C. Balzer, T. Ben-Horin, E. de Moor, A. McNally, S. Pawar, S.J. Ryan, T.C. Smith, K.D. Lafferty.  2013. Optimal temperature for malaria transmission is dramatically lower than previously predicted.  Ecology Letters 16(1): 22-30.