Make your own hotly criticised circle graph!!!

Make your own hotly criticized circle graph! The recent critical post on R-bloggers has got me wanting to make my own bubble graphs despite the popular criticism of the graph. After some attempts I have produced something that looks similar but needs some more work.

I would like to say briefly that R-blogger post by PirateGrunt while interesting and reasonable seemed to miss the point. The graphic was not supposed to convey massive amounts of information but simply to make obvious the scope and degree of how the current Ebola epidemic compares with other epidemics.

There are many good reasons to critique this graph from an epidemiological perspective (forthcoming), yet from a graphing perspective I think  this graph format is extremely efficient at conveying memorable information while the graphs assembled by PirateGrunt were more detailed they were also entirely forgettable unfortunately.

Here are a couple of examples. The first is with meaningless simulated data demonstrating how large differences in scale may appear. The second is actually using HIV prevalence data to give a graphical representation of how such rates have changed over the years. You can fine the code as well as the data on github.

Table 1: Demonstrates how scale can be communicated by size of circle

Table 2: Demonstrates the relative stability of AIDS in terms of people ill. Top number is  year while lower number is millions of cases.

Ebola in Liberia could be different than Ebola in New Jersey

The Ebola outbreak in Western Africa has initiated panic throughout the world. Thirty-seven countries so far have implemented policies to restrict the international spread of Ebola. In the United States, governor Chris Christie has initiated additional travel restrictions implementing a mandatory quarantine of health workers returning from Liberia, Sierra Leone, and Guinea even when no symptoms are present.

Should Chris Christie be concerned about Ebola in New Jersey? In some ways New Jersey is not so different from Liberia which has already suffered 4,665 deaths according to the CDC.

New Jersey has a population of around 8.9 million people while Liberia has a population a little less than half that of 4.1 million. Both countries were established by foreigners. New Jersey was is first established by Dutch people in America while Liberia was established by Americans in Africa. Both states gained their current affiliation within 60 years ago with New Jersey entering the Union in 1787 and Liberia gaining independence in 1847. The official language of both New Jersey and Liberia is English and both governments use the dollar though one is the US Dollar and the other is the Liberian Dollar. In both states they drive on the right side of the road and the largest religious affiliation in both regions is Christianity.

I know, from this description thus far most people would find it very difficult to tell if they were in Liberia or New Jersey. Under this reasoning it seems very important for New Jersey to implement stringent rules to keep out any chance of Ebola entering its borders. However, there are some minor differences between New Jersey and Liberia that may bear mentioning.

Overall the United States is listed top 4 on the United Nation's Human Development Index while Liberia is ranked the 5th lowest in 2011 out 187 countries. Within the United States New Jersey is ranked the third most developed states. But what does this really mean?

The per capita earnings in Liberia was reported at 436 USD while in New Jersey the per capita was 54,699 USD. Thus in a year a New Jersey resident could be expected to earn more than 125 Liberians. But does this somewhat noticeable difference in earnings translate into differences in medical services?

According to the US census there was about 311 doctors per 100,000 residents of New Jersey in 2006 translating to about 27,680 medical doctors. Liberia on the other hand reported having 51 doctors in the entire country in 2006. If doctors are distributed evenly throughout both states then you are likely to find one medical doctor within every 0.31 square miles in New Jersey while in Liberia you are likely to find only one medical doctor every 811 square miles.

New Jersey residents can expect to live 80.3 years on average and struggle with different health related issues than those faced by Liberians who can only expect to live 57.4 years. In New Jersey where over-consumption presents a major concern with 60.7% of the population is overweight or obese and the chief causes of death is related to over-consumption including heart disease and diabetes. In contrast 38.5% of Liberians suffer from malnutrition with deaths caused from easily treatable diseases such as malaria, pneumonia, and diarrhea which are collectively responsible for 50% of the deaths in the country.

Overall, New Jersey is vastly more wealthy in terms of both income as well as medial expertise, New Jersey is vastly better prepared for Ebola than Liberia. Within Africa there have been over twenty outbreaks of Ebola, the majority of which have been rapidly contained despite most countries not being significantly better off than Liberia. Ebola is a reasonably well understood disease which can be rapidly controlled when appropriate steps are taken using infrastructure much worse off than that faced in New Jersey.

Despite the numerous similarities between New Jersey and Liberia, we should expect any outbreak of Ebola in New Jersey to be rapidly contained. It is unlikely that this outbreak of Ebola in Liberia will result in any significant outbreak in New Jersey or anywhere in the United States in which even our poorest areas are much better equipped than any of the countries now suffering from the outbreak.

This post thus far has been meant as a jab at the hysteria and political maneuvering that has surrounded Ebola. However, the fear of this disease is entirely appropriate if not well placed. This disease with its high mortality rate (~70%) and rapid transmission within these impoverished nations (Sierra Leone, Liberia, and Guinea) has the potential if not stopped to cause as much loss of life and suffering as the worse currently communicable diseases which plague humanity such as malaria (627,000 annual deaths) or tuberculosis (1,460,000 annual deaths).

I believe it is still possible to turn the tide of Ebola around if sufficient international aid is brought to bear in these afflicted nations. Becoming distracted with enacting useless policies which harm the ability of health care workers to travel between wealthy developed nations and West African nations not only misses the point, but actively undermines the ability of philanthropic health care workers to control this disease.

Ebola: Beds, Labs, and Warnings? Can they help? (Shiny App)

A month ago when the WHO was projecting estimates of the effect of current outbreak of Ebola being as deadly as affecting 20,000 people, I ran some elementary modelling and found that these estimates are far too small given the current trend.  The motivation for the post was to raise awareness that situation could get far worse than anybody was talking about at the time. Since then, most of my 'back of the envelope' estimates have ended up being disturbingly close to reports the World Health Organization has been releasing.



Which frankly is extremely scary. Currently I am living in Mozambique in Southern Africa and though Mozambique is slightly more developed than Liberia, I have no reason to believe that things would be any different here than in Western Africa if an outbreak went undetected for a month as it did in Western Africa.

This among other things has made me wonder how inevitable such an outcome is. Should everybody who can leave Africa and find a nice little bunker to hide in until this whole thing passes? Well probably not but only if Ebola can be stopped.

Currently the world seems to be responding to the crisis in the countries affect in three major ways: 1. provide beds, 2. provide laboratory capabilities to diagnose Ebola, and 3. provide advertisements to increase awareness. But how can we know how effective these measures can be against such a seemly unstoppable force?

Time to break out our models!

Unfortunately there is no really easy way to model this. However, modifying the standard epidemiological SIR (susceptible, infected, recovered) model I am able create a model which looks to be functioning the way we would like it to by including some additional parameters. To see details of the model's construction, see the technical appendix.

Beds

The primary new parameters of consideration are 'beds' which represent the number of beds available as well as the food and supplies necessary in order to feed people who are residents of these beds. Infected individuals once detected are transferred to quarantine if beds are available. If they are not available then infected individuals remain contagious until they recover or die.

Social adoption
From a paper by Fisman, Khoo, and Tuite I have incorporated the idea of social adaption to the epidemic. This captures the concept that the infection could be naturally controlled to some extent by changes in the behavior of the susceptible population and that of the contagious population.

The Model
It becomes immediately clear that the model is extremely sensitive to just about every parameter included. If the infection rate is too high then everybody gets sick. If the rate is too low then the epidemic is quickly contained. However, for this exercise let us assume we cannot directly control in any way infection rates but we can choose how many beds, how effective we are at detecting new cases, and we have some influence on how people respond over time to Ebola by taking safety precautions such as not touching the sick or dying.

Figure 1:Base Model After 9 Months

Each of these interventions can have a significant effect on the outbreak. These interventions when looked at carefully turn into two different strategies: 1. Quarantine infected by providing beds and provisions and 2. increasing public awareness to reduce probability of spread over time.

Providing Beds
The effect of a significant investment in beds (500 new beds) after seven months can abruptly turn around the spread of Ebola as the contagious population is rapidly shifted from free and dangerous to safely quarantined (assuming an effective mechanism exists for detecting those who are ill).

Figure 2: Base model after seven month beds intervention.

Changing Behavior
I have not including behavior curbing into the model quite as dramatically. Instead I have specified social behavior changes as a cumulative effect over time. In the base model individuals adapt to the disease by being .03% less likely each day to contract the disease. This is not much though it does accumulate significantly over time. After six months of the epidemic individuals would be about 5% less likely to get Ebola when exposed to an individual with Ebola.

If we are able to increase awareness about prevention of contraction of the disease to say .06% increase per day then individuals are about 10% less likely to contract Ebola after six months. Though these numbers are not large the effect can be profound on our model.

Figure 3: Behavioral adaption can dramatically reduce the lifespan of the outbreak.

However, the significant problem with including social adaption in this way is that this is based on accumulated actions over time. If this is the case then Ebola should already but or its way out.

Sensitivity of the Model - the shiny app
As mentioned previously this model is extremely sensitive to parameter choices. It is therefore more of an illustrative tool than actually meant to exactly represent the situation in Western Africa. As a tool we can see under the right circumstances that beds and public information can have a dramatic effect on the spread of Ebola. However, don't take my word for it! Check out the app below and play around with the model yourself.

Technical Appendix

Parameters:

alpha is detection rate.

delta is transition rate to recovery or death.

mu is mortality rate.

State equations:

Change in susceptible population:

$$\dot S = -\frac{\gamma S_R S_t C_t}{S_t C_t}$$

Change in contagious population:

$$\dot C = -\dot S-\min[\alpha C_t , \max(beds-Q_t(1-\delta),0)]-\delta C_t$$

Change in the quarantined population:

$$\dot Q = \min[\alpha C_t , \max(beds-Q_t(1-\delta),0)] - \delta Q_t$$

Change in the recovered population:

$$\dot R = (1-\mu) \delta (Q_t+C_t)$$

Change in the decease population:

$$\dot D = \mu \delta (Q_t+C_t)$$

R Code

The R code used to produce this app can be found on Github. If you prefer running the app from your computer, you can download server.R and ui.R and run the package from your own

1.2 Million Deaths by Ebola projected within Six Months?

The World Health Organization, Samaratins Purse, Doctors Without Borders, and other international medical emergency relief programs are desperately calling for additional resources in the international fight against Ebola that has already killed thousands and is likely to kills thousands more even if a full arsenal of aid was made available.

Figure 1: The lines are projected values while the points are data points.

The World Health Organization released a statement on August 28th that the epidemic could afflict more than 20,000 people before it could be brought under control. This however, assumes full international backing for an intervention to control the deadly outbreak. Failure to fully support the WHO's plan presumably would cause the disease to continue to spread in a similar manner as it already has.

At first a figure as high as 20,000 seems exaggerated especially when looking just at the number of 3,000 cases reported the same day as the announcement. However, I believe that this estimate is vastly too small and is entirely based on an effective and well funded international relief mission. Using a projection from all of the WHO reports to date I calculate that if the disease continues to spread at the rate it currently is then we will have more than 20,000 cases by October 24.

The report also states that it will likely take six to nine months in order to stop the epidemic. However if nothing changes and the epidemic continues to rage as it currently does then my projections estimate that as many as 4.7 million people will have been infected and 1.2 million will have already died.

These are extremely dire predictions and I hope that they are purely extrapolations based on what will later be seen as the scariest month of this epidemic. However, the exponential growth model fits the data very well and at least in the short term should be expected to be fairly accurate.

All of this analysis is done in R and the code can be found on Github.

From 41 CDC Ebola reports I have assembled a small database of cases by country listing the number of 'Suspected or Confirmed Cases', the number of 'Deaths' suspected to be associated with Ebola, and the number of 'Laboratory Confirmed' cases of Ebola. You can find and add to this database as a google spreadsheet here. If running the code for yourself it will import the spreadsheet data directly.

Mapping this data by country and fitting a local polynomial regression to give a fitted line for each country gives us some signs of a very disturbing trend. The country in which the current outbreak originated is Guinea and though the disease continues to claim new victims it is much less worrisome compared with Sierra Leone and Liberia where rates of suspected cases and numbers of deaths are exponentially growing.

Figure 2: The increase of deaths in Liberia is much steeper than the other two heavily afflicted countries of Guinea and Sierra Leone.

Figure 3: The increase of laboratory confirmed cases in Liberia is much less steep than the increase is deaths indicating that the poor medical infrastructure is not able to keep up with the number of diagnoses demanded.

Figure 4: The increase of deaths in Liberia is much steeper than the other two heavily afflicted countries of Guinea and Sierra Leone.

By exponential growth, we mean that whatever the current number of infected people are, we can expect them to infect some additional number of people proportion to the transmission rate. The problem with exponential growth is that while the inclusion of new victims can initially start out small the more victims there are the more are likely to be added to the victim pool each day.

Figure 5: The total number of cases is rising extremely quickly.

When we look at the total numbers of each case summed across country we arrive at the above graph.
From this graph it is clear that a direct linear model cannot fit well at all. Suspecting that the change over time might fit an exponential growth function, I take the natural log of the values mapped above.

Figure 6: A log transformation of the total number of cases creates a relatively linear relationship between time and number of cases reported.

This new transformed graph demonstrates an extremely distributing confirmation that using an exponential growth model would be an appropriate way of modelling the spread of Ebola. In order to estimate the spread of Ebola I define a simple model with a constant and a linear relationship between days since the outbreak was announced and the log of the measure we are concerned with:
$$log(Y)=\alpha+\beta_1 Day$$
And estimate the model using weights to weight the data based on the number of days into the survey so that more recent observations are considered more important. I also discard the observations for the first 21 days because we can expect the preliminary data at that time was less accurate. Using the above model gives:

           Intercept          Day 
Suspected  4.38881946  0.02245505
Deaths     4.00491144  0.02096758
Laboratory 3.86052949  0.02314866

While intercept estimates are generally considered to be less important the coefficients on Day can be directly interpreted as percent changes by day. Thus we can expect from the current data that each day we will have a little over 2% additional suspected cases, deaths, and laboratory confirmations.

In order allow for the model to be a little more flexible in my projections I include a slightly more complex model including a squared term for the days since announcement.
$$log(Y)=\alpha+\beta_1 Day+\beta_2 Day^2$$
I use this model to project suspected cases, deaths, and laboratory results for the next three weeks. The values up until today show the comparison between the expected values estimated from the model (EDeaths, ESusp, and ELab) with that from the data (Death, Susp, and Lab). We can see the model fits the data quite well with all estimates within 100 of the observed while most are much closer. Using this model we can see that the total number of deaths is expected to be around 3,500 by the 24th and 7,200 suspected cases. Things just get worse from there.

           date day   Deaths EDeaths  Susp ESusp     Lab ELab
1   2014-03-25   1       59   89       86  140        0   49
2   2014-03-26   2       60   90       86  141        1   50
3   2014-03-27   3       66   90      103  143        4   51
7   2014-03-31   7       70   94      112  149       24   56
8   2014-04-01   8       80   95      122  151       24   57
9   2014-04-02   9       83   97      127  152       35   59
14  2014-04-07  14       95  102      151  161       52   66
17  2014-04-10  17      101  106      157  167       66   71
24  2014-04-17  24      122  115      197  182      101   83
28  2014-04-21  28      129  122      203  192      109   91
30  2014-04-23  30      136  125      208  197      112   95
37  2014-04-30  37      146  137      221  218      126  112
42  2014-05-05  42      155  148      231  235      127  126
51  2014-05-14  51      157  169      233  270      129  155
60  2014-05-23  60      174  196      258  315      146  190
65  2014-05-28  65      191  213      290  344      170  214
70  2014-06-02  70      199  232      341  377      186  240
73  2014-06-05  73      222  245      425  399      238  257
78  2014-06-10  78      244  269      462  440      253  289
79  2014-06-11  79      261  274      494  449      277  296
86  2014-06-18  86      337  313      528  517      364  348
92  2014-06-24  92      338  353      599  587      441  399
100 2014-07-02 100      467  416      759  700      544  481
105 2014-07-07 105      481  462      779  784      557  540
106 2014-07-08 106      518  472      844  803      626  552
112 2014-07-14 112      539  539      888  925      664  634
114 2014-07-16 114      601  564      964  971      706  665
122 2014-07-24 122      632  677     1048 1183      745  800
126 2014-07-28 126      672  744     1201 1311      814  877
129 2014-07-31 129      728  800     1323 1417      909  941
132 2014-08-03 132      826  860     1439 1533      953 1008
133 2014-08-04 133      887  882     1603 1574     1009 1032
137 2014-08-08 137      961  974     1779 1753     1134 1132
141 2014-08-12 141     1013 1077     1848 1956     1176 1242
142 2014-08-13 142     1069 1105     1975 2011     1251 1271
144 2014-08-15 144     1145 1163     2127 2127     1310 1332
148 2014-08-19 148     1229 1290     2240 2381     1383 1461
150 2014-08-21 150     1350 1360     2473 2522     1460 1530
151 2014-08-22 151     1427 1397     2561 2596     1528 1566
157 2014-08-28 157     1552 1641     3069 3094     1752 1800
158 2014-08-29 158          1686          3188          1842
159 2014-08-30 159          1733          3284          1885
160 2014-08-31 160     1841 1782     3685 3384          1930 Update 9/4
161 2014-09-01 161          1831          3488          1975
162 2014-09-02 162          1883          3595          2021
163 2014-09-03 163          1936          3705          2069
164 2014-09-04 164          1991          3820          2117
165 2014-09-05 165     2097 2047     3944 3939          2167 Updated 9/11
166 2014-09-06 166          2106          4062          2218
167 2014-09-07 167          2166          4189          2270
168 2014-09-08 168          2228          4321          2323
169 2014-09-09 169          2292          4457          2378
170 2014-09-10 170          2359          4599          2433
171 2014-09-11 171          2427          4745          2491
172 2014-09-12 172          2498          4897          2549
173 2014-09-13 173          2572          5055          2609
174 2014-09-14 174     2630 2647     5347 5218     3095 2670 Updated 9/18
175 2014-09-15 175          2725          5386          2733
176 2014-09-16 176          2806          5562          2797
177 2014-09-17 177          2890          5743          2863
178 2014-09-18 178          2976          5931          2930
179 2014-09-19 179          3065          6126          2998
180 2014-09-20 180          3157          6329          3069
181 2014-09-21 181     2917 3253     6263 6539     3487 3141 Updated 9/25
182 2014-09-22 182          3351          6756          3215
183 2014-09-23 183          3453          6982          3290
184 2014-09-24 184          3559          7217          3367
...
    2014-09-30 190     3441          7470          4087     Updated 10/3
... 
    2014-10-03 193     3857          8011          4440     Updated 10/8
... 
    2014-10-08 198     4033          8400          4656     Updated 10/10

Falseness of my Model
This model by definition cannot be true globally (into the distant future). This is obvious when we use the model to project out to one year. At one year the number of infected cases is estimated as 436 billion. Since the entire population of the Earth is only 8 billion or so we know that this cannot be true.

However, this kind of model can be a good approximation locally (in the near future). If it is a good approximation locally then the next WHO report is going to list around 2100 deaths and 4060 suspected cases as of today.

So, I ask the question, "is 1.2 million deaths a projection which is either local or global?" I cannot answer this, but it certainly is within the realm of feasibility since the nation of Liberia alone has over 4 million people and Guinea 10 million and Sierra Leone 6 million. The real question becomes, "do we think the ability of Liberia and other afflicted nations to control the spread of Ebola will increase, decrease, or remain the same over time?"

Figure 7: Shows the relationship over time between number of laboratory confirmed cases and suspected cases. If a country is able to apply effective diagnostic tools then this ratio should be high. In most countries we are seeing a rise in the ability to diagnose Ebola except in Liberia where there is a steep decline.

From Figure 7 we can see that Liberia is significantly behind other nations in its ability to diagnose Ebola. This and the well known lack of medical facilities suggests to me that as the crisis escalates the ability of Liberia to maintain any sense of order and with it any hope of controlling the spread of the disease is likely to degrade. If this is the case then it is quite possible that even this horrifying projection is an underestimate of the pain and carnage likely to result from this outbreak.

What to Do?
News reports and the governments they are reporting on seem to have been placing a good deal of emphasis on investing in vaccines and treatment options. However, while all of these options are good, they are long term options (6 to 12 months).  In the meantime, every resource available must be used to contain and restrict the spread of this outbreak.

It is extremely foolish to think that any nation is immune to this disease. So far in the entire history of Ebola outbreaks up until the present less than 10 thousand people have been infected. This relatively low infection count coupled with rapid mortality makes it unlikely that the disease will significantly mutate among the human population.

However, if my projections are anywhere close to accurate then the number of infected people are going to be much higher than has ever occurred previously. This will create many more habitats for which the virus can possible mutate new traits which could increase its transmission rate. These mutations could take the form of longer gestation periods which might lead to a greater time between being infectious and being detectable.

Another possible trait might be the ability to go airborne which would significantly increase its ability ability to be transmitted. Some scientists it very unlikely to become airborne because it is too heavy. This may be the case. However, as the possibility of it becoming airborne could result in a global spread of the disease resulting in unprecedented number of deaths world wide it is more than prudent to heavily invest in controlling the number of new patients infected by this disease.
In addition, even if the disease does not mutate from the state that it is in currently to a new one, it has shown itself to be extremely effective at being transmitted with a large number of health workers becoming infected and dying from the disease. These health workers should have known how to control the spread of the disease and prevent infection. Do we really expect that if the disease were to enter any other nation on Earth that the general population is going to be better prepared to protect themselves than the specialists who have already fallen victim to this disease?

Thus, it is imperative that we do everything within our power to control the spread of this terrible disease. Even if my model only has a ten percent chance of being accurate over the next six months, we would be extremely foolish to risk not responding to this outbreak with every resource within reason humanity can muster.