Papers in Physics, vol. 5, art. 050002 (2013) Received: 9 October 2012, Accepted: 1 March 2013 Edited by: G. Mindlin Licence: Creative Commons Attribution 3.0 DOI: http://dx.doi.org/10.4279/PIP.050002 www.papersinphysics.org ISSN 1852-4249 A mathematically assisted reconstruction of the initial focus of the yellow fever outbreak in Buenos Aires (1871) M L Fernández,1 M Otero,2 N Schweigmann,3 H G Solari2∗ We discuss the historic mortality record corresponding to the initial focus of the yellow fever epidemic outbreak registered in Buenos Aires during the year 1871 as compared to simulations of a stochastic population dynamics model. This model incorporates the biology of the urban vector of yellow fever, the mosquito Aedes aegypti, the stages of the disease in the human being as well as the spatial extension of the epidemic outbreak. After introducing the historical context and the restrictions it puts on initial conditions and ecological parameters, we discuss the general features of the simulation and the dependence on initial conditions and available sites for breeding the vector. We discuss the sensitivity, to the free parameters, of statistical estimators such as: �nal death toll, day of the year when the outbreak reached half the total mortality and the normalized daily mortality, showing some striking regularities. The model is precise and accurate enough to discuss the truthfulness of the presently accepted historic discussions of the epidemic causes, showing that there are more likely scenarios for the historic facts. I. Introduction Yellow fever (YF) is a disease produced by an arthropod borne virus (arbovirus) of the family �aviviridae and genus Flavivirus. The arthropod vector can be one of several mosquitoes and the usual hosts are monkeys and/or people. Wild mosquitoes of genus Haemagogus, Sabetes and Aedes are responsible for the transmission of the ∗E-mail: solari@df.uba.ar 1 Departamento de Computación, Facultad de Ciencias Exactas y Naturales (FCEN), Universidad de Buenos Aires (UBA) and CONICET. Intendente Güiraldes 2160, Ciudad Universitaria, C1428EGA Buenos Aires, Ar- gentina. 2 Departamento de Física, FCEN�UBA and IFIBA� CONICET. 1428 Buenos Aires, Argentina. 3 Departamento de Ecología, Genética y Evolución, FCEN�UBA and IEGEBA�CONICET. 1428 Buenos Aires, Argentina. virus among wild monkeys, such as the Brown Howler Monkey (Alouatta guariba) associated to recent outbreaks of YF in Brazil, Paraguay and Ar- gentina [1]). In contrast, urban YF is transmitted by a domestic and anthropophilic mosquito, Aedes aegypti, human beings being the host [2]. Aedes ae- gypti is a tree hole mosquito, with origins in Africa, that has been dispersed through the world thanks to its association with people. During the end of the XVIII and the XIX cen- turies, YF caused large urban outbreaks in the Americas from Boston (1798), New York (1798) and Philadelphia (1793, 1797, 1798, 1799) in the North [3] to Montevideo (1857) and Buenos Aires (1858, 1870, 1871) [4] in the South. These historical episodes arise as ideal cases for testing the capabili- ties of YF models in urban settings. Is it possible to reconstruct the evolution of one of these epidemic outbreaks? Can enough information be recovered to produce a thorough test on the models? This 050002-1 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. is seldom the case, for example, for the study of the Memphis (1878) epidemic, with over 10000 ca- sualties, only 1965 were considered potentially us- able [5]. In contrast, the records of the outbreak in Buenos Aires 1871, unearthed and digitized for this work, left us with an amount of 1274 death cases located in time and space for the initial focus in the quarter of San Telmo, about 78% of the total mortality in the quarter [6]. According to the 1869 national census [7] Buenos Aires had 177787 inhabi- tants, 12329 of them living in San Telmo, about half of them just immigrated into the country mostly from Europe. In this work, we will compare the initial devel- opment of the epidemic outbreak (Buenos Aires, 1871) with the simulations resulting from an eco- epidemiological model developed in Refs. [8�10], testing the worth of the predictive model. The simulations were performed under a number of assumptions, most of them essentially forced by the lack of better information. We will assume that: 1. Now and before, YF is the same illness, i.e., we can use current information on YF devel- opment such as: the average extent of the in- cubation, infection, recovery, and toxic peri- ods, as well as the mortality level in 1871. In other words, the virus presents no substantial changes since 1871 to present days. We do not expect this hypothesis to be completely correct: the YF virus is an RNA-virus as op- posed to the stable DNA-viruses, as such, mu- tations in about 140 years of continuous repli- cations in mosquitoes and primates can hardly be ruled out. Furthermore, present-day YF has been subject to di�erent evolutionary pres- sures than the YF in the XIX century. While in the XIX century yellow fever circulated con- tinuously in human populations, today the wild part of the cycle involving wild popula- tions of monkeys plays a substantial role. 2. The epidemic was transmitted by Aedes ae- gypti. There is no evidence of this fact since the scienti�c society and medical doctors in general were not aware of the role played by the mosquito until the con�rmation given by Reed [3] of Finlay's ideas [11]. 1 1According to other sources, it was Beauperthuy [12] the �rst one to accurately describe the transmission of YF We assume that Aedes aegypti has not changed since then, and/or there are no substantial changes in the life cycle, vector capabilities and adaptation between the (assumed) pop- ulation in 1871 and present-day populations in Buenos Aires city. After the eradication campaign (1958�1965) [14], Aedes aegypti was eradicated from Buenos Aires [15]. Hence, the present populations result from a re-infestation and they are not the direct descendants of the mosquitoes population of 1871. 3. Lacking time statistics for the duration of the di�erent stages in the development of the ill- ness, reproduction of the virus and life cycle of the mosquito, we use, as distribution for such events, a maximum likelihood distribu- tion subject to the constrain of the average value for the cycle. In short, we use exponen- tially distributed times for the next event for all type of events. 4. Finally, and most importantly, we assume that the human population mobility is not a factor in the local spread of the disease. We antic- ipate one of our conclusions: this assumption is likely to be false for the full development of the epidemic outbreak but seems reasonable for the early (silent) development. The study of the secondary foci of the epidemic outbreak merits a detailed analysis of the social and po- litical circumstances related to human mobility and it is beyond the possibilities of this study. Since we want the test to be as demanding as possible, more information is needed to simulate the outbreak eliminating sources of ambiguity and parameters to be �tted using the same test data. We recovered the following information: 1. Estimations of daily temperatures. They are relevant since the temperature regulates the developmental rates of the mosquitoes. 2. A very rough, anecdotal, estimation of the availability of breeding sites (BS) that, ulti- mately, control the carrying capacity of the by mosquitoes, as observed in the epidemic outbreak at Cumaná, Venezuela (1853), as well as the e�cient measures of protection taken by Native Americans, the use of nets to prevent the spread of the epidemic [13]. 050002-2 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. environment, the number of vectors and the infection rate. 3. Human populations discriminated by block in the city. 4. Estimations of the date of arrival of the virus to the city, putting bounds to the reasonable initial conditions for the simulation. This climatological, social and historical informa- tion represents a determining part of the recon- struction as it is integrated into the model jointly with the entomological and medical information to produce stochastic simulations of possible out- breaks to be compared with the historic records of casualties. We will show that the model predicts large prob- abilities for the occurrence of YF in the given his- torical circumstances and it is also able to answer why a minor outbreak in 1870 did not progress to- wards a large epidemic. The total number of deaths and the time-evolution of the death record will be shown to agree between the historical record and the simulated episodes as well, within the original focus. The rest of the manuscript will be organized as follows: we will begin with the description of YF in section II, including the eco-epidemiological model. In section III, we will address the relevant climato- logical, social and historical aspects. In section IV, we will explore the sensitivity of the model to ini- tial conditions and the number of available breed- ing sites, discussing the statistics more clearly in�u- enced by vector abundance. The historic mortality records and the simulated records are compared in section V.. We will �nally discuss the performance of the model in section VI. II. The disease We will simply quote the fact sheet provided by the World Health Organization [16] as the standardized description: �YF is a viral disease, found in tropi- cal regions of Africa and the Americas. It principally a�ects humans and mon- keys, and is transmitted via the bite of Aedes mosquitoes. It can produce devas- tating outbreaks, which can be prevented and controlled by mass vaccination cam- paigns. The �rst symptoms of the disease usually appear 3�6 days after infection. The �rst, or acute, phase is characterized by fever, muscle pain, headache, shivers, loss of ap- petite, nausea and vomiting. After 3�4 days, most patients improve and symp- toms disappear. However, in a few cases, the disease enters a toxicphase: fever reappears, and the patient develops jaun- dice and sometimes bleeding, with blood appearing in the vomit (the typical vomito negro). About 50% of patients who enter the toxic phase die within 10�14 days�. We add that the remission period lasts between 2 and 48 hours [17], and as it was mentioned in the introduction, not only Aedes mosquitoes transmit the disease. i. The model The yellow fever model is rather similar to the al- ready presented dengue model [10], the similarity corresponds to the fact that dengue is produced by a Flavivirus as well, it is transmitted by the same vector and follows the same clinical sequence in the human being, although with substantially lesser mortality. The model describes the life cycle of the mosquito [8] and its dispersalafter a blood meal, seeking oviposition sites [9]. The mosquito goes through several stages: egg, larva, pupa, adult (non parous), �yer (i.e., the mosquito dispersing) and adult (parous). In each stage, the mosquito can die or continue the cycle with a transition rate between the subpopulations that depends on the tempera- ture. The mortality in the larva stage is nonlinear and it regulates the population as a function of the availability of breeding sites. Thus, the transitions from adult to �yer are associated with blood meals, the event that can transmit the virus from human to mosquito and vice-versa. From the epidemiolog- ical point of view, the mosquito follows a SEI se- quence (Susceptible, Exposed �extrinsic period�, Infective). Correspondingly, the adult populations are subdivided according to their status with re- spect to the virus. We assume that there is no vertical transmission of the virus and eggs, larvae, 050002-3 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. pupae and non parous adults are always suscepti- ble. The humans are subdivided in subpopulations according to their status with respect to the ill- ness as: susceptible (S), exposed (E), infective (I), in remission (r), toxic (T) and recovered (R). The temporary remission period is followed by recovery with a probability between 0.75 and 0.85 or a toxic period (probability 0.25 to 0.15) which ends half of the times in death and half of the times in recovery. The yellow fever model di�ers in the structure from the dengue model in Ref. [10], as the human part of the dengue model is SEIR and the yellow fever model is SEIrRTD. However, the additional stages do not alter the evolution of the epidemic since the �in remission�, toxic and dead stages do not par- ticipate in the transmission of the virus. The YF parameters are presented in Table 1. Period value range Intrinsic Incubation (IIP) 4 days 3�6 days Extrinsic Incubation (EIP) 10 days 9�12 days Human Viremic (VP) 4 days 3�4 days Remission (rP) 1 days 0�2 days Toxic (tP) 8 days 7�10 days Probability value range Recovery after remission (rar) 0.75 0.75�0.85 Mortality for toxic patients (mt) 0.5 Transmission host to vector (ahv) 0.75 Transmission vector to host (avh) 0.75 Table 1: Parameters (mean value of state) adopted for YF. The range indicated is taken from PAHO [17]. The model is compartmental, all populations are counted as non-negative integers numbers and evolve by a stochastic process in which the time of the next event is exponentially distributed and the events compete with probabilities proportional to their rates in a process known as density- dependent-Poisson-process [18]. The model can be understood qualitatively with the scheme of the Fig. 1. The model equations are summarized in Appendix A. The city is divided in blocks, roughly follow- ing the actual division (see Fig. 2). The human populations are constrained to the block while the mosquitoes can disperse from block to block. Figure 1: Scheme of the yellow fever model. On the left side, the evolution of the mosquito and, on the right side, the evolution of human subpopulations. Hollow arrows indicate the progression through the life cycle of the mosquito following the sequence: egg, larva, pupa, adult (non-parous), �yer, adult (parous) and the repetition of the two last steps. The mortality events are not shown to lighten the scheme. Eggs are laid in the transition from �yer to adult. The adult mosquito populations are subdi- vided according to their status with respect to the virus as: susceptible (S), exposed (E) and infective (I). The virus is transmitted from mosquitoes to hu- mans and vice-versa in the transition from adult to �yer (blood meal) when either the mosquito or the human is infective and the other susceptible (red arrows). The red bold arrows indicate the progres- sion of the disease, from exposed to infective in the mosquito and, in humans, following the sequence: exposed (E), infective (I), in remission (r), toxic, recovered (R) or dead. III. Historical, social and climato- logical information i. When and how the epidemic started The YF outbreak in Buenos Aires (1871) was one of a series of large epidemic outbreaks associated to the end of the War of the Triple Alliance or Paraguayan war. The war confronted Argentina, Brazil and Uruguay (the three allies) on one side and Paraguay on the other side, and ended by March, 1870. By the end of 1870, Asunción, Paraguay's capital city, was under the rule of the Triple Alliance. The return of Paraguay's war pris- 050002-4 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. oners from Brazil (where YF was almost endemic at that time) to Asunción triggered a large epi- demic outbreak [4]. The allied troops received their main logistic support from Corrientes (Argentina), a city with 11218 inhabitants according to the 1869 census [7], located about 300 km south of Asun- ción (following the waterway) and 1000 km north of Buenos Aires along the Paraná river (see Fig. 3). On December 14, 1870, the �rst case of YF was diagnosed in Corrientes [19], and a focus de- veloped around this case imported from Asunción. According to some sources, the epidemic produced panic, resulting in about half the population leav- ing the city between December 15 and January 15 [20]. However, other historical reasons might have played a relevant role since the city of Corrientes was under the in�uence and ruling of Buenos Aires, while in the farmlands, the General Ricardo López Jordan was commanding a rebel army (a sequel of Argentina civil wars and the war of the Triple Al- liance). The subversion ended with the battle of Ñaembé, about 200 km east of Corrientes, on Jan- uary 26, 1871. Putting things in perspective, we must realize that in those times, YF was recognized only in its toxic stage associated to the black vomit, it is then perfectly plausible that recently infected individu- als would have left Corrientes and Asunción reach- ing Buenos Aires, despite quarantine measures that were late and leaky [4,22].2 The death toll in Corri- entes was of 1289 people in the city (and about 700 more in places around the city) [20], representing a 11,5% of the population (notice that this num- ber is not consistent with current numbers in use by WHO [17] which indicate a 7.5% of mortality in diagnosed cases of YF but is well in line with historical reports [23] of 20% to 70% mortality in diagnosed cases �the statistical basis has changed with the improved knowledge of early, not toxic, YF cases. According to this historical view, the initial ar- rival of infectious people to Buenos Aires happened, more likely, during December 1870 and January 1871. In his study of the YF epidemic, written twenty three years after the epidemic outbreak, José Penna (MD) [4] quotes the issue of the journal 2On December 16, a sanitary o�cial from Buenos Aires was commissioned to Corrientes to organize the quarantine, a measure that was applied to ships coming from Paraguay but not to those with Corrientes as departing port. Figure 2: Police districts from a map of the time and computer representation. The red colored area in district 14 (San Telmo) are the two blocks where the 1871 epidemic started. The green colored area in district 3 is the block where the Hotel Roma was and where the 1870 focus began (see section V.iii.). The red and green lines sourround the region sim- ulated for the 1871 epidemics and the 1870 focus, respectively. Notice that districts 15 and 13 dis- agree in the maps. The computer representation follows the information in Ref. [21] from where the population information was obtained. �Revista Médico Quirúrgica�, published in Buenos Aires on December 23, 1870 [24], which presents a report regarding the sanitary situation during the last �fteen days, indicating the emergence of a �bil- ious fever� and a general tendency of other fevers to produce icterus or jaundice. In the next issue, dated January 8, 1871, the �Revista� indicates an important increase in the number of bilious fever cases reported [22]. In a separate article, the doc- tors call the attention on how easily and how of- 050002-5 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. ten the quarantine to ships coming from Paraguay is avoided, and calls for strengthening the mea- sures. Penna indicates that the �bilious fever� (not a standard term in medicine) likely corresponded to milder cases of YF. We will term this idea �Penna's conjecture� and will come back to it later. For our initial guess, we considered this informa- tion as evidence that the epidemic outbreak started during December, 1870. Exploring the model, and arbitrarily, we took December 16, 1870, as the time to introduce two infectious people with YF in the simulations, at the blocks where the mortality started. Yet, an educated guess for Penna's con- jecture is to consider the 3�6 days needed from in- fection to clinical manifestation and the 9�12 days of the extrinsic cycle. Hence, since the �rst clini- cal manifestations of transmitted YF happened be- tween December 11�23, we would guess the infected people arriving somewhere between November 21 and December 11. Figure 3: A 1870 map [25] showing Asunción next to the Paraguay river, Corrientes and Rosario next to the Paraná river and Buenos Aires (spelled Buenos Ayres) next to the Rio de la Plata. Yet, we must take into account that Penna's con- jecture contrasts with the conjectures presented by MD Wilde and MD Mallo, members of the San- ity committee in charge during the YF epidemic. Wilde and Mallo advocated for the spontaneous origin of the disease, very much in line with the theories of miasmas in use in those times, theo- ries that guided the sanitary measures taken [19]. Wilde and Mallo also argued that Asunción could not be the origin of the epidemic, because of their belief that the ten or �fteen days quarantine (count- ing since the last port touched) was enough to avoid the propagation of the disease. This belief contrasts with the experience of 1870 (in Buenos Aires) where a ten day quarantine was not enough to prevent a minor epidemic [4]. Nevertheless, the quarantine measures werefully implemented in Corrientes by December 31, 1870. The measures were later lifted because of the epidemic in Corrientes and imple- mented at ports down the Paraná river, being com- pleted nearby Buenos Aires (ports of La Conchas, Tigre, San Fernando and �La Boca� within Buenos Aires city) by mid-February when the epidemic was in full development in Buenos Aires according to the port sanitary authorities, Wilde and Mallo [19]. Being Corrientes the source of infected people can hardly be disregarded. With about 5000 people leaving the city between December 15 and January 15 [19], a city where YF was developing. According to Wilde and Mallo [19], there were (non-fatal?) YF cases in Buenos Aires as early as January 6, 1871, reported by MDs Argerich and Gallarani as well as documented cases of YF death after disembarking in Rosario (200 km North of Buenos Aires along the Paraná river) having boarded in Corrientes. ii. Breeding sites One of the key elements in the reconstruction and simulation of an epidemic transmitted by mosquitoes is to have an estimation of their num- bers which will be re�ected directly in the propa- gation of the epidemic. In the mosquito model [8], this number is regulated by the quality and abun- dance of breeding sites. The production of a single breeding site, normalized to be a �ower pot in a local cemetery, was taken as unit in Ref. [8] and the number of breeding sites measured in this unit roughly corresponds to half a liter of water. The Aedes aegypti population in monitored areas of Buenos Aires, today, is compatible with about 20 to 30 breeding sites per block [9]. Estimating the number of sites available for breeding today is already a di�cult task, the estimation of breeding 050002-6 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. sites available in 1871 is a nearly impossible one. In what remains of this subsection, we will try to get a very rough a-priori estimate. District Population/(100 m)2 BS/(100 m)2 1 339 391.0 2 279 300.0 3 428 522.0 4 353 443.0 5 330 430.0 6 259 365.0 13 160 196.0 14 224 300.0 15 90 157.0 16 165 316.0 18 23 52.0 19 13 26.0 20 30 52.0 Table 2: Population data. Buenos Aires, 1869 [7]. Population density by police district (see Fig. 2) and equivalent breeding sites, BS, originally esti- mated as proportional to the house density in the police district. A very important di�erence between those days and the present corresponds to the supply of fresh water which today is taken from the river, pro- cessed and distributed through pipes; but in those days, it was an expensive commodity taken from the river by the �waterman� and sold to the cus- tomers who, in turn, had to let it rest so that the clay in suspension decanted to the bottom of the vessel (a process that takes at least 3 days). Ad- ditionally, there were some wells available but the water was (is) of low quality (salty). The last, and rather common resource [19,26], was the collection of rain water in cisterns. iii. Temperature reconstruction Aedes aegypti developmental times depend on tem- perature. Although it would seem reasonable to use as substitute of the real data of the average temperature registered since systematic data collec- tion began, records of temperature in those times were kept privately [27] and are available. The data set consists of three daily measurements made from January 1866 until December 1871, at 7AM, 2PM and 9PM. When averaged, the records allow an es- timation of the average temperature of the day bet- ter than the usual procedure of adding maximum and minimum dividing by two. Unfortunately, the 0 250 500 750 1000 1250 1500 1750 2000 2250 Time / days 0 10 20 30 T e m p e ra tu re / º C 1900 2000 2100 2200 Time / days -10 -5 0 5 10 R e si d u a ls January 1st, 1871January 1st, 1866 Figure 4: Average daily temperature and periodic approximation �tted accordng to Eq. (1), t = 0 corresponds to January 1, 1866. The inset shows the di�erence between the measured temperatures and the �t (residuals) during 1871. register has some important missing points during the epidemic outbreak. Because of this problem, the data in Ref. [27] was used to �t an approxima- tion in the form: T =7.22◦C× cos(2πt/(365.25 days) + 5.9484) + 17.21◦C, (1) following Ref. [28] and then extrapolating to the epidemic period. In Fig. 4, the data and the �t are displayed. The residuals of the �t do not present seasonality or sistematic deviations, as we can see in the inset of Fig. 4. It is worth noticing that a similar �t on temper- ature data from the period 1980�1990 presents a mean temperature of 18.0◦C, amplitude of 6.7◦C and a phase shift of 6.058◦C [8] (notice that t = 0 in the reference corresponds to July 1 while in this work it corresponds to January 1). According to the threshold computations in Ref. [8], the cli- matic situation was less favorable for the mosquito in 1866�1871 than in the 1960�1991 period. The reconstruction of temperatures needs to be performed at least from the 1868 winter, since a relatively arbitrary initial condition in the form of eggs for July 1, 1868 is used to initialize the code, and then run over a transitory of two years. Such 050002-7 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. procedure has been found to give reliable results [8]. There are several factors in the biology of Ae. ae. that indicate that the biological response to air-temperature �uctuations is re�ected in attenu- ated �uctuations of biological variables. First, the larvae and pupae develop in water containers, thus, what matters is the water temperature. This fact represents a �rst smoothing of air-temperature �uc- tuations. Second, insects developmental rates for �uctuating temperature environments correspond to averages in time of rates obtained in constant temperature environments [29], an alternative view is that development depends on accumulated heat [30]. Such averages occur over a period of about 6 days at 30◦C and longer times for other tempera- tures and non-optimal food conditions [31]. Third, the biting rate (completion of the gonotrophic cy- cle) depends as well on temperatures averaged over a period of a few days. Last, mosquitoes actively seek the conditions that �t best to them and more often than not, they are found resting inside the houses. iv. Mortality data For this work, the daily mortality data recorded during the 1871 epidemic outbreak [6] is key. This statistical work has received no attention in the past, and no study of the YF outbreak in Buenos Aires made reference to this information. We have cross-checked the information with data in the 1869 national census [7], as well as with data in published works [4], and the details are consistent among these sources. The data set is presented here closing the historic research part. v. Revising the clinical development of yel- low fever. A YF epidemic outbreak happened in Buenos Aires, in 1870, developing about 200 cases [4] (the text is ambiguous on whether the cases are toxic or fatal). The epidemic outbreak was noticed by February 22 (�rst death), a sailor who left Rio de Janeiro (Brazil) on February 7, and presumably landed on February 17 (no cases of YF were re- ported on board of the Poitou �the boat). This well documented case allows us to see the margins of tolerance that have to be exercised in 0 30 60 90 120 150 Time / days 0 10 20 30 40 50 60 70 D a il y m o rt a li ty ( h is to ri c ) January 1st, 1871 Figure 5: Daily mortality cases in the police district 14 (see Fig. 2) corresponding to the quarter San Telmo [6], t = 0 corresponds to January 1, 1871. taking medical information prepared for clinical use as statistical information. Assume, following Penna, that the sailor was exposed to YF before boarding in Rio de Janeiro, according to informa- tion in Table 1 collected from the Pan American Health Organization [17], adding incubation and viremic period, we have a range of 6�10 days, hence the sailor was close to the limit of his infectious pe- riod. He was not toxic, according to the MD on board who signed a certi�cate accepted by the san- itary authority. Yet, �ve days later, he was dying, making the remission plus toxic period of 5 days, shorter than the range of 7�12 days listed in Ta- ble 1 and substantially shorter than the 10�15 days (remission plus toxic) communicated in Ref. [16]. Shall we assume as precise the values reported in Ref. [16, 17] we would have to conclude that the disease was substantially di�erent, at least in its clinical evolution, in 1871 as compared with present days. In clinical studies performed during a YF epi- demic on the Jos Plateau, Nigeria, Jones and Wil- son [32] report a 45.6% overall mortality and a signi�cant di�erence in the duration of the illness for fatal and non-fatal cases with averages of 6.4 and 17.8 days. Série et al. [33] reports for the 1960�1962 epidemic in Ethiopia a mortality rang- ing from 43% (Kouré) to 100% (Boloso) and 50% (Menéra) with a total duration of the clinic phase 050002-8 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. of the illness of 7.14, 2.14 and 4.5 days, respectively (weighted average of 4.6 days in 18 cases). We must conclude that the extension of the toxic period preceding the death presents high variabil- ity. This variability may represent variability in the illness or in medical criteria. For example, Série [33] indicates that the 100% mortality found at the Boloso Hospital is associated to the admission cri- teria giving priority to the most severe cases. In correspondence with this extremely high mortality level, the survival period is the shortest registered. The minimum length of the clinical phase is of 10� 14 days or 13�18 days, depending of the source (adding viremic, remission and toxic periods). We note that not only the toxic period of fatal cases must be shorter than the same period for non-fatal cases, but also the viremic period must be shorter in average, if all the pieces of data are consistent. The time elapsed between the �rst symptoms and death is probably longer today than in 1871, since it, in part, re�ects the evolution of medical knowl- edge. The hospitalization time is also rather arbi- trary and changes with medical practices which do not re�ect changes in the disease. A rudimentary procedure to correct for this dif- ferences is to shift the simulated mortality some �xed time between 5 and 8 days (the di�erence be- tween our 13 days guessed (Table 1) and the 4.5� 6.4 reported for Africa [32,33]). Such a procedure is not conceptually optimal, but it is as much as it can be done within present knowledge. We cer- tainly do not know whether just the toxic period must be shortened or the viremic period must be shortened as well, and in the latter case, how this would a�ect the spreading of the disease. A second source of discrepancies between recorded data and simulations are the inaccuracies in the historic record. Can we consider the daily mortality record as a perfect account? Which was the protocol used to produce it? We can hardly expect it to be perfect, although we will not make any provision for this potential source of error. IV. Simulation results The simulations were performed using a one-block spatial resolution, with the division in square blocks of the police districts 14 (San Telmo), 16, 2 and 4, corresponding to Concepción, Catedral Sur and Montserrat; and part of the districts 6, 18, 19, 19A and 20 (see Fig. 2), and totalized for each po- lice district to obtain daily mortality comparable to those reported in Ref. [6] and picture in Fig. 5. Numerical mosquitoes were not allowed to �y over the river. At the remaining borders of the simulated region, a Stochastic Newmann Bound- ary Condition was used, meaning that the mosquito population of the next block across the boundary was considered equal to the block inside the region; but the number of mosquito dispersion events as- sociated to the outside block was drawn randomly, independently of the events in the corresponding in- ner block. Larger regions for the simulations were tested producing no visible di�erences. The time step was set to the small value of 30 s, avoiding the introduction of further complica- tions in the program related to fast event rates for tiny populations [34], although an implementation of the method in Ref. [34], not relying on the small- ness of the time step so heavily, is desirable for a production phase of the program. Before we proceed to the comparison between the historic mortality records of the epidemic and the simulated results, we need to gain some un- derstanding regarding the sensitivity of the simula- tions to the parameters guessed and the best forms of presenting these results. We performed a mod- erate set of computations, since the code has not been optimized for speed and it is highly demand- ing for the personal computers where it runs for several days. Here, we illustrate the main lessons learned in our explorations. Poor people, unable to buy large quantities of water, had to rely mostly on the cisterns and other forms of keeping rain wa- ter. Since 1852, when the population of Buenos Aires was about 76000 people, there was an impor- tant immigration �ow, increasing the population to about 178000 people by 1869 [7]. The immi- grants occupied large houses where they rented a room, usually for an entire family, a housing that was known as �conventillo� and was the dominant form of housing in some districts such as San Telmo, where the epidemic started [20]. In some police chronicles of the time, houses with as many as 300 residents are mentioned [35]. Under such di�cult social circumstances, we can only imagine that the number of breeding sites available to mosquitoes has to be counted as orders of magnitude larger than present-day available sites. 050002-9 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. An a-priori and conservative estimation is to con- sider about ten times the number of breeding sites estimated today. Thus, we assume, as a �rst guess, 300 breeding sites per block in San Telmo. We will have to tune this number later as it regulates mosquito populations and the development of the epidemic focus. The number of breeding sites is the only parameter tuned to the results in this work. More precisely, the criteria adopted was to make the number of (normalized) BS proportional to the number of houses per block taken from historic records [21], adjusting the proportionality factor to the observed dynamics. We introduce the notation BSxY to indicate a multiplicative factor of Y. The population of each police district was set to the density values reported in the 1869 census [7,21] and the police districts geography was taken from police records [36] and referenced according to maps of the city at the time [37�40]. Table 2 shows the average population per block, initially estimated number of breeding sites and number of houses for the district of the initial focus, San Telmo #14 and nearby districts (# 16, 4, 2). A sketch of Buenos Aires police districts according to a 1887 map [37] is displayed alongside with the computer representation in Fig. 2. It is a known feature of stochastic epidemic mod- els [41] that the distribution of totals of infected people has two main contributions. One is that the small epidemic outbreaks when none or a few sec- ondary cases are produced and the extinction time of the outbreak comes quickly. The otheris that the large epidemic outbreaks which, if the basic repro- ductive number is large enough, present a Gaussian shape separated by a valley of improbable epidemic sizes from the small outbreaks. While the present model does not fall within the class of models discussed in Ref. [41], the gen- eral considerations applied to stochastic SIR mod- els qualitatively apply to the present study. Yet, simulations started early during the summer season follow the pattern just described in Ref. [41], but simulations started later do not present the proba- bility valley between large and small epidemics. We have found useful to present the results dis- aggregated in the form: epidemic size, daily per- centage of mortality relative to the total mortality and time to achieve half of the �nal mortality. This presentation will let us realize that most of the �uc- tuation is concentrated in the total epidemic size, while the daily evolution is relatively regular, ex- cept, perhaps, in the time taken to develop up to 50% of the mortality (depending on the abundance of vectors and the initial number of infected hu- mans and chance). i. Total mortality (epidemic size) Since historical records include mostly the number of causalities, it appears sensible for the purposes of this study to use the total number of deaths as a proxy statistics for epidemic size. 0 20 40 0 20 40 F re q u e n c y i n 1 0 0 r u n s 0 20 40 0 300 600 900 1200 Total mortality 0 20 40 Figure 6: San Telmo. Total mortality histograms for di�erent number of breeding sites, computed af- ter 100 simulations with the same initial condition corresponding to 2 infectious people located in San Telmo on January 1, 1871, at the same location where the initial death happened in the historical event. From top to bottom, multiplication factors (bin-width : frequency of no-epidemic) BSx1 (126.2 : 0.23), BSx2 (209.8 : 0.09), BSx3 (128.2 : 0.06) and BSx4 (50.6 : 0.02). The y-axis indicates fre- quency in a set of 100 simulations. The total mortality depends strongly on the stochastic nature of the simulations, initial condi- tions and ecological parameters guessed. Qualita- tively, the results agree with the intuition, although this is an a-posteriori statement, i.e., only after see- ing the results we can �nd intuitive interpretations for them. The discussion assumes that the development of the epidemic outbreak was regulated by either the 050002-10 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. availability of vectors (mosquitoes) or the exhaus- tion of susceptible people, the �rst situation repre- sents a striking di�erence with standard SIR mod- els without seasonal dependence of the biological parameters. Actually, in Fig. 6, we compare fre- quencies of epidemics binned in �ve bins by �nal epidemic size for di�erent sets of 100 simulations with di�erent number of breeding sites. The num- ber of breeding sites is varied in the same form all along the city, keeping the proportionality with housing, and it is expressed as multiplicative factor (BSx1=1, BSx2=2, BSx3=3, BSx4=4) presented in Table 2. Notice also that the width of the bins progresses as 126.2, 209.8, 128.2 and 50.6 indicat- ing how the dispersion of �nal epidemic sizes �rst increases with the number of breeding sites but for larger numbers decreases. We can see how for a factor 2 (BSx2) (and larger) the mortality saturates, indicating the epidemic outbreaks are limited by the number of susceptible people. For our original guess, factor 1, the epi- demic is limited by the seasonal presence/absence of vectors. However, for higher factors, there is a substantial increase in large epidemic outbreaks with larger probabilities for larger epidemics. Only for the factor 4 (BSx4) the most likely bin includes the historical value of 1274 deaths. A second feature, already shown in the Dengue model [10], is that outbreaks starting with the ar- rival of infectious people in late spring will have a lesser chance to evolve into a major epidemic. Yet, those that by chance develop are likely to become large epidemic outbreaks since they have more time to evolve. On the contrary, outbreaks started in autumn will have low chances to evolve and not a large number of casualties. The corresponding histograms can be seen in Fig. 7. Figure 7 also shows how the outbreaks that begin on December 16, as well as simulations starting on January 1, present higher probabilities of large epidemics than of small epidemics, but this tendency is reverted in simulations of outbreaks that start by February 16. This transition is, again, the transition between outbreaks regulated by the number of available sus- ceptible humans and those regulated by the pres- ence or absence of vectors. 0 30 60 0 30 60 F re q u e n c y i n 1 0 0 r u n s 0 30 60 0 200 400 600 800 1000 1200 1400 Mortality 0 30 60 Figure 7: San Telmo. Total mortality histograms for di�erent date of arrival of infectious people. Computed after 100 simulations with the same ini- tial condition corresponding to 2 infectious people located in San Telmo and a density factor BSx2.5. From top to bottom: December 16, 1870; January 1, January 16 and February 15, 1871. ii. Mortality progression One of the most remarkable facts unveiled by the simulations is that when the time evolution of the mortality is studied as a fraction of the total mor- tality, much of the stochastic �uctuations are elimi- nated and the curves present only small di�erences (see Fig. 8). The similarity of the normalized evo- lution allows us to focus on the time taken to pro- duce half of the mortality (labelled T1/2). We notice, in Fig. 8, that the T1/2 in these runs lay between 69�110, compared to the historic values of T1/2 =73. The latter observation brings the attention to a remarkable fact of the simulations: not only the normalized progression of the outbreaks are rather similar but also, there is a correspondence between early development and large mortality. Drawing T1/2 against total-mortality (Fig. 9), we see that even for di�erent number of breeding sites, all the simulations indicate that the �nal size is a noisy function of the day when the mortality reaches half its �nal value. This function is almost constant for small T1/2 and becomes linear with increasing dis- persion when T1/2 is relatively large. Once again, the two di�erent forms the outbreak is controlled. 050002-11 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. 0 50 100 150 Day (1 is January 1st 1871) 0 0.5 1 A c c u m u la te d m o rt a li ty / (T o ta l m o rt a li ty ) average average - variance average + variance Figure 8: Evolution of the mortality level for all the epidemics with three or more fatal cases in a batch of 100 simulations (96 runs) di�ering only in the pseudo-random series. The breeding sites number has a factor 4, and the initial contagious people were placed on January 1, 1871, in San Telmo. 60 80 100 120 140 Day of 50% mortality 0 500 1000 1500 T o ta l m o rt a li ty BSx1 BSx1.5 BSx2 BSx3 BSx4 Figure 9: San Telmo. Total mortality versus day when the mortality reached half its �nal number for di�erent number of breeding sites. The mortality appears to be mainly a function of T1/2 and roughly independent of the number of breeding sites. V. Real against simulated epidemic We would like to establish the credibility of the statement: the historic mortality record for the San Telmo focus belongs to the statistics generated by the simulations. To achieve this goal, we need to compare the daily mortality in the historical record and the sim- ulations. Since, in the model, the mortality pro- ceeds day after day with independent random in- crements (as a consequence of the Poisson charac- ter of the model), it is reasonable to consider the statistics χ2 = ∑ i ( HM(i) −MM(i) D(i) )2 (2) where i runs over the days of the year, HM(i) is the fraction of the total death toll in the historic record for the day i, MM(i) is the average of the same fraction obtained in the simulations and D(i) is the corresponding standard deviation for the sim- ulations. The sum runs over the number of days in which the variance is not zero, for BSx3 and BSx4 in no case D(i) = 0 and (HM(i) − MM(i) 6= 0). The number of degrees corresponds to the number of days with non-zero mortality in the simulations minus one. The degree discounted accounts for the fact that ∑ i HM(i) = ∑ i MM(i) = 1. i. Tuning of the simulations Before we proceed, we have to �nd an acceptable number of breeding sites, a reasonable day for the arrival of infected individuals (assumed to be 2 in- dividuals arbitrarily) and adjust for the uncertainty in survival time. Actually, moving the day of arrival d days earlier and shortening the survival time by d will have essentially the same e�ect on the sim- ulated mortality (providing d is small), which is to shift the full series by d days. This is, assigning to the day i the simulated mortality SM of the day i + d, (SM(i + d)). Hence, only two of the param- eters will be obtained from this data. As we have previously observed, the total mor- tality presents a large variance in the simulations. Moreover, in medical accounts of modern time [32, 33], the mortality ranges between 46% and 100% while in historical accounts the percentage goes from 20 to 70 [23]. Hence, a simple adjustment of the mortality coe�cient from our arbitrary 50% within such a wide range would su�ce to eliminate the contributions of the total epidemic size. The average simulated epidemic for BSx4 is of ≈ 1248 deaths while the historic record is of 1274 deaths. 050002-12 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. BSxY d χ2 degree Probability 2 8 490.2 177 0. 3 7 191.7 173 0.15 4 3 151.9 168 0.80 4 4 143.8 168 0.91 4 5 145.6 168 0.89 Table 3: χ2 calculations according to (2). We indi- cate the multiplicative factor applied to the breed- ing sites described in Table 2, the shift applied to the statistics produce with the parameters of Table 1, the value of the statistic χ2, the number of de- grees of freedom, and the probability P(x > χ2) for a random variable x distributed as a χ2-distribution with the indicated degrees of freedom. The stan- dard reading of the χ2 test indicates that, for BSx3 and BSx4, the statement �the deviations from the simulations mean of the historic record (deviations assumed to be distributed as χ2 with the indicated degrees) belong to the simulated set� is not found likely to be false by the test in the cases BSx3 and BSx4. Hence, to match the mean with the historic record, it would su�ce to correct the mortality from 50% to 51%. We can disregard the idea that the epidemic started before December 20, 1870 (Penna's con- jecture), since it is not possible to simultaneously obtain an acceptable �nal mortality and an accept- able evolution of the outbreak. Our best attempt corresponds to an epidemic starting by December 14, 1870, which averages ≈ 1212 deaths and with a deviation of the mortality of χ2 = 219.1 with 175 degrees, giving a probability P(x ≥ 219, 1) = 0.01. We focus on arrival dates around January 1, 1871. We can also disregard, for this initial condi- tion, the original guess BSx1 corresponding to 300 breeding sites per block in San Telmo, since it pro- duces too small epidemics. We present results for the epidemics corresponding to BSx2, BSx3 and BSx4 in Table 3 for di�erent numbers of BS and shift d. ii. Comparison The conclusion of the χ2 tests is that the simula- tions performed with BSx3 and beginning between December 24 1870 and January 1, 1871 (with a sur- vival period shortened between 0 and 8 days) are compatible with the historical record. However, the compatibility is larger when BSx4 is considered and the beginning of the epidemic is situated between December 28, 1870 and January 5, 1871 (with a survival period shortened between 0 and 8 days re- spectively). We illustrate this comparison with Fig. 10. 0 50 100 150 200 Days of 1871 0 500 1000 1500 M o rt a li ty Single run Average Historic Average + SD Average - SD Figure 10: The historic record of accumulated mor- tality as a function of the day of the year 1871 is presented. The averaged accumulated mortality as well as curves shifted one standard deviation are shown for comparison. An acceptable look alike in- dividual simulation chose by visual inspection from a set of 100 simulations is included as well. Two exposed individuals (not yet contagious) were in- troduced in the same blocks where the historic epi- demic started the day January 1, 1871, with BSx4. The simulated mortality is anticipated in 4 days. Statistical estimates were taken as averages over 96 runs which resulted in secondary mortality, out of a set of 100 runs. iii. The 1870 outbreak The records for the 1870 outbreak are scarce. Of the recognized cases, only 32 entered the Lazareto (hospital) and 19 of them were originated in the same block that the �rst case. Secondary cases are registered at the Lazareto' books starting on March 30 (two cases) and continuing with daily cases. the �nal outcome of these cases and the cases not en- tered at the Lazareto [4] are not clear. 050002-13 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. Except for the precise initial condition, corre- sponding to one viremic (infective) person located at the Hotel Roma (district 4 in Fig. 2), the in- formation is too imprecise to produce a demanding test for the model. We performed a set of 100 simulations introduc- ing one viremic (infective) person at the precise block where the Hotel Roma was, on February 17. The number of breeding sites was kept at the same factor 4 with respect to the values tabulated in Ta- ble 2 that was used for the best results in the study of the 1871 outbreak. Needless to say, this does not need to be true, as the number of breeding sites may change from season to season. The distribution of the �nal mortality is shown in Fig. 11. As we can see, relatively small epi- demics of less than 200 deaths cannot be ruled out, although there are much larger epidemics also likely to happen. 0 500 1000 1500 2000 Total mortality 0 5 10 15 20 25 30 F re q u e n c y i n 1 0 0 r u n s Figure 11: Mortality distribution for the simula- tions beginning on February 17 with the incorpo- ration of one infective (viremic) in the block of the Hotel Roma and BSx4. The histogram is the result of 87 runs which resulted in epidemics (13 runs did not result in epidemics). The width of the bins is 322.2, and the �rst epidemic bin goes from 17 to 339 deaths with a frequency of 15/100. Actually, a slow start of the epidemic outbreak would favor a small �nal mortality, as it can be seen in Fig. 12. Not only there is a relation between a low mortality early during the outbreak (such as April 15) and the �nal mortality, but we also see that the sharp division between small and large epi- demics is not present in this family of epidemic out- breaks di�ering only in the pseudo-random number sequence. 0 50 100 150 200 250 300 Early mortality 0 500 1000 1500 2000 F in a l m o rt a li ty , 1 8 7 0 April 15th March 30th Figure 12: Final mortality against early mortality for two dates: March 30 and April 15. A low early mortality �predicts� a low �nal mortality as the out- break does not have enough time to develop. Sim- ulations correspond to the conditions of the 1870 small epidemic. BSx4, one infected arriving to the Hotel Roma on February 17. The historic informa- tion indicates that secondary cases were recorded by March 30. Hence, corresponding to a slow start. The �nal mortality is not known, but it is believed it has been in the 100�200 range. For the smallest epidemics simulated, the sec- ondary mortality starts after March 30. Hence, the 1870 focus can be understood as a case of relative good luck and a late start within more or less the same conditions than the outbreak of 1871. VI. Conclusions and �nal discussion In this work, we have studied the development of the initial focus in the YF epidemic that devastated Buenos Aires (Argentina) in 1871 using methods that belong to complex systems epistemology [42]. The core of the research performed has been the development of a model (theory) for an epidemic outbreak spread only by the mosquito Aedes ae- gypti represented according to current biological literature such as Christophers [2] and others. The 050002-14 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. translation of the mosquito's biology into a com- puter code has been performed earlier [8,9] and the basis for the spreading of a disease by this vector has been elaborated in the case of Dengue previ- ously [10]. The present model for YF is then an adaptation of the Dengue model to the particular- ities of YF, and the present attempt of validation (failed falsi�cation) re�ects also on the validity of this earlier work. The present study owes its existence to the work of anonymous police o�cers [6] that gathered and recorded epidemics statistics during the epidemic outbreak, in a city that was not only devastated by the epidemic, but where the political authorities left in the middle of the drama as well [20]. We have gathered (and implemented in a model) entomological, ecological and medical information, as well as geographic, climatological and social in- formation. After establishing the historical con- straints restricting our attempts to simulate the historical event, we have adjusted the density of breeding sites to be the equivalent to 1200 half- liter pots as those encountered in today Buenos Aires cemeteries (the number corresponds to San Telmo quarter). Perhaps a better idea of the num- ber of mosquitoes present is given by the maxi- mum of the average number of bites per person per day estimated by the model, which results in 5 bites/(person day) (to be precise: the ratio between the maximum number of bites in a block during a week and the population of the block, divided by seven). The population of the domestic mosquito Aedes aegypti in Buenos Aires 1870�1871 was large enough to almost assure the propagation of YF during the summer season. The only e�ective measures preventing the epidemic were the natu- ral quarantine resulting from the distance to trop- ical cities were YF was endemic (such as Rio de Janeiro) and the relatively small window for large epidemics, since the extinction of the adult form of the mosquito during the winter months prevents the overwintering of YF. In this sense, the relatively small outbreak of 1870 is an example of how a late arrival of the infected individual combined with a touch of luck produced only a minor sanitary catas- trophe. By 1871, as a consequence of the end of the Paraguayan war and the emergence of YF in Asun- ción, the conditions for an almost unavoidable epi- demic in Buenos Aires were given. The interme- diate step taken in Corrientes, with the panic and partial evacuation of the city, adding the lack of quarantine measures, was more than enough to make certain the epidemic in Buenos Aires. On the contrary, Penna's conjecture of an earlier starting during December, 1870 are inconsistent with the biological and medical times as implemented in the model. We can disregard this conjecture as highly improbable. The historic mortality record is consistent with an epidemic starting between December 28 and January 5, being the symptomatic period (viremic plus remission plus toxic) of the illness between 13 and 5 days. Furthermore, the existence of non- fatal cases of YF by January 6 mentioned by some sources [19] would be consistent, provided the cases were imported. In retrospect, the present research began as an attempt to validate/falsi�cate the YF model and, in more general terms, the model for the transmis- sion of viral diseases by Aedes aegypti using the historic data of this large YF epidemic. As the re- search progressed, it became increasingly evident that the model was robust. In successive attempts, every time the model failed to produce a reasonable result, it forced us to revise the epidemiological and historical hypotheses. In these revisions, we ended up realizing that the accepted origin of the epidemic in imported cases from Brazil, actually hides the central role that the epidemics in Corrientes had, and the gruesome failure of not quarantining Cor- rientes once the mortality started by December 16, 1870, about two weeks before the deducted begin- ning of the outbreak in Buenos Aires. The same study of inconsistencies between the data and the reconstruction made us focus on the survival time of those clinically diagnosed with YF that �nally die. The form in which the illness evolves anticipates the �nal result. Jones and Wil- son [32] indicate the symptoms of cases with a bad prognosis including the rapidity and degree of jaun- dice. This information suggests, in terms of model- ing, that death is not one of two possible outcomes at the end of the �toxic period�, as we have �rst thought. Separation of to-recover and to-die sub- populations could (should?) be performed earlier in the development of the illness, each subpopula- tion having its own parameters for the illness. Yet, while in theory this would be desirable, in prac- 050002-15 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. tice it would have, for the time being, no e�ect, since the characteristic periods of the illness have not been measured in these terms. Epidemics transmitted by vectors come to an end either when the susceptible population has been su�ciently exposed so that the replication of the virus is slowed down (the classical consideration in SIR models) or when the vector's population is dec- imated by other (for example, climatic) reasons. The model shows that both situations can be dis- tinguished in terms of the mortality statistics. We have also shown that the total mortality of the epidemic is not di�cult to adjust by changing the death probability of the toxic phase, and as such, it is not a demanding test for a model. The daily mortality, when normalized, shows sensitivity to the mosquito abundance, specially in the evo- lution times involved, since the general qualitative shape appears to be �xed. In particular, the date in which the epidemic reaches half the total mortality is advanced by larger mosquito populations. How- ever, only comparison of the simulated and histor- ical daily mortality put enough constraints to the free data in the model (date of arrival of infected people and mosquito population) to allow for a se- lection of possible combinations of their values. As successful as the model appears to be, it is completely unable to produce the total mortality in the city, or the spatial extension of the full epi- demic. The simulations produce with BSx4 less than 4500 deaths, while in the historic record, the total mortality in the city is above 13000 cases. The historical account, and the recorded data, show that after the initial San Telmo focus has devel- oped, a second focus in the police district 13 (see Fig. 2) developed, shortly several other foci de- veloped that could not be tracked [4]. Unless the spreading of the illness by infected humans is intro- duced (or some other method to make long jumps by the illness), such events cannot be described. It is worth noticing that the mobility patterns in 1871 are expected to be drastically di�erent from present patterns, and as such, the application of models with human mobility [43] is not straightfor- ward and requires a historical study. One of the most important conclusions of this work is that the logical consistency of mathemati- cal modeling puts a limit to ad-hoc hypotheses, so often used in a-posteriori explanations, as it forces to accept not just the desired consequence of the hypotheses, but all other consequences as well. Last, eco-epidemiological models are adjusted to vector populations pre-existing the actual epi- demics and can therefore be used in prevention to determine epidemic risk and monitor eradication campaigns. In the present work, the tuning was performed in epidemic data only because it is ac- tually impossible to know the environmental con- ditions more than one hundred years ago. Yet, our wild initial guess for the density of breeding sites resulted su�ciently close to allow further tuning. Acknowledgments We want to thank Professor Guillermo Marshall who has been very kind allowing MLF to take time o� her duties to complete this work. We acknowl- edge the grant PICTR0087/2002 by the ANPCyT (Argentina) and the grants X308 and X210 by the Universidad de Buenos Aires. Special thanks are given to the librarians and personnel of the Insti- tuto Histórico de la Ciudad de Buenos Aires, Bib- lioteca Nacional del Maestro, Museo Mitre and the library of the School of Medicine UBA. A Appendix i. Populations and events of the stochastic transmission model We consider a two dimensional space as a mesh of squared patches where the dynamics of vec- tors, hosts and the disease take place. Only adult mosquitoes, Flyers, can �y from one patch to a next one according to a di�usion-like process. The coordinates of a patch are given by two indices, i and j, corresponding to the row and column in the mesh. If Xk is a subpopulation in the stage k, then Xk(i,j) is the Xk subpopulation in the patch of coordinates (i,j). Population of both hosts (Humans) and vectors (Aedes aegypti) were divided into subpopulations representing disease status: SEI for the vectors and SEIrRTD for the human population. Ten di�erent subpopulations for the mosquito were taken into account, three immature subpopu- lations: eggs E(i,j), larvae L(i,j) and pupae P(i,j), and seven adult subpopulations: non parous adults A1(i,j), susceptible �yers Fs(i,j), exposed �yers 050002-16 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. Fe(i,j), infectious �yers Fi(i,j) and parous adults in the three disease status: susceptible A2s(i,j), ex- posed A2e(i,j) and infectious A2i(i,j). The A1(i,j) is always susceptible, after a blood meal it becomes a �yer, susceptible Fs(i,j) or ex- posed Fe(i,j), depending on the disease status of the host. If the host is infectious, A1(i,j) becomes an exposed �yer Fe(i,j) but if the host is not infec- tious, then the A1(i,j) becomes a susceptible �yer Fs(i,j). The transmission of the virus depends not only on the contact between vector and host but also on the transmission probability of the virus. In this case, we have two transmission probabili- ties: the transmission probability from host to vec- tor ahv and the transmission probability from vec- tor to host avh. Human population Nh(i,j) was split into seven di�erent subpopulations according to the disease status: susceptible humans Hs(i,j), exposed hu- mans He(i,j), infectious humans Hi(i,j), humans in remission state Hr(i,j), toxic humans Ht(i,j), re- moved humans HR(i,j) and dead humans because of the disease Hd(i,j). The evolution of the seventeen subpopulations is a�ected by events that occur at rates that depend on subpopulation values and some of them also on temperature, which is a function of time since it changes over the course of the year seasonally [8,9]. ii. Events related to immature stages Table 4 summarizes the events and rates related to immature stages of the mosquito during their �rst gonotrophic cycle. The construction of the transi- tion rates and the election of model parameters re- lated to the mosquito biology such as: me mortality of eggs, elr hatching rate, ml mortality of larvae, α density-dependent mortality of larvae, lpr pupation rate, mp: mortality of pupae, par pupae into adults development coe�cient and the ef emergence fac- tor were described in detail previously [8,9]. The natural regulation of Aedes aegypti popula- tions is due to intra-speci�c competition for food and other resources in the larval stage. This regu- lation was incorporated into the model as a density- dependent transition probability which introduces the necessary nonlinearities that prevent a Malthu- sian growth of the population. This e�ect was in- corporated as a nonlinear correction to the temper- ature dependent larval mortality. Then, larval mortality can be written as: mlL(i,j) +αL(i,j)×(L(i,j)−1) where the value of α can be further decomposed as α = α0/BS(i,j) with α0 being associated with the carrying capacity of one (standardised) breeding site and BS(i,j) being the density of breeding sites in the (i,j) patch [8,9]. iii. Events related to the adult stage Aedes aegypti females (A1 and A2) require blood to complete their gonotrophic cycles. In this pro- cess, the female may ingest viruses with the blood meal from an infectious human during the human Viremic Period V P . The viruses develop within the mosquito during the Extrinsic Incubation Pe- riod EIP and then are reinjected into the blood stream of a new susceptible human with the saliva of the mosquito in later blood meals. The virus in the exposed human develops during the Intrin- sic incubation Period IIP and then begin to cir- culate in the blood stream (Viremic Period), the human becoming infectious. The �ow from sus- ceptible to exposed subpopulations (in the vector and the host) depends not only on the contact be- tween vector and host but also on the transmission probability of the virus. In our case, there are two transmission probabilities: the transmission proba- bility from host to vector ahv and the transmission probability from vector to host avh. The events related to the adult stage are shown in Table 5 to 8. Table 5 summarizes the events and rates related to adults during their �rst gonotrophic cycle and related to oviposition by �yers according to their disease status. Table 6 and Table 7 summarize the events and rates related to adult 2 gonotrophic cycles, exposed Adults 2 and exposed �yers becoming infectious and human contagion. Table 8 summarizes the events and rates related to non parous adult (Adult 2) and Flyer death. iv. Events related to �yer dispersal Some experimental results and observational stud- ies show that the Aedes aegypti dispersal is driven by the availability of oviposition sites [44�46]. Ac- cording to these observations, we considered that only the Flyers F(i,j) can �y from patch to patch in search of oviposition sites. The implementation of �yer dispersal has been described elsewhere [9]. 050002-17 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. Event E�ect Transition rate Egg death E(i,j) → E(i,j) − 1 me×E(i,j) Egg hatching E(i,j) → E(i,j)−1 L(i,j) → L(i,j) + 1 elr ×E(i,j) Larval death L(i,j) → L(i,j) − 1 ml×L(i,j) + α×L(i,j) × (L(i,j) − 1) Pupation L(i,j) → L(i,j)−1 P(i,j) → P(i,j) + 1 lpr ×L(i,j) Pupal death P(i,j) → P(i,j) − 1 (mp + par × (1 − (ef/2))) ×P(i,j) Adult emergence P(i,j) → P(i,j) − 1 A1(i,j) → A1(i,j) + 1 par × (ef/2) ×P(i,j) Table 4: Event type, e�ects on the populations and transition rates for the developmental model. The coe�cients are me: mortality of eggs; elr: hatching rate; ml: mortality of larvae; α: density-dependent mortality of larvae; lpr: pupation rate; mp: mortality of pupae; par: pupae into adults development coe�cient; ef: emergence factor. The values of the coe�cients are available in subsections vi. and vii.. Event E�ect Transition rate Adults 1 Death A1(i,j) → A1(i,j) − 1 ma×A1(i,j) I Gonotrophic cycle with virus exposure A1(i,j) → A1(i,j) − 1 Fe(i,j) → Fe(i,j) + 1 cycle1 × A1(i,j) × (Hi(i,j)/Nh(i,j)) × ahv I Gonotrophic cycle without virus exposure A1(i,j) → A1(i,j) − 1 Fs(i,j) → Fs(i,j) + 1 cycle1 × A1(i,j) × ((((Nh(i,j) − Hi(i,j))/Nh(i,j)) + (1 − ahv) × (Hi(i,j)/Nh(i,j))) Oviposition of suscep- tible �yers E(i,j) → E(i,j) + egn Fs(i,j) → Fs(i,j) − 1 A2s(i,j) → A2s(i,j) + 1 ovr(i,j) ×Fs(i,j) Oviposition of exposed �yers E(i,j) → E(i,j) + egn Fe(i,j) → Fe(i,j) − 1 A2e(i,j) → A2e(i,j) + 1 ovr(i,j) ×Fe(i,j) Oviposition of infected �yers E(i,j) → E(i,j) + egn Fi(i,j) → Fi(i,j) − 1 A2i(i,j) → A2i(i,j) + 1 ovr(i,j) ×Fi(i,j) Table 5: Event type, e�ects on the subpopulations and transition rates for the developmental model. The coe�cients are ma: mortality of adults; cycle1: gonotrophic cycle coe�cient (number of daily cycles) for adult females in stages A1.; ahv: transmission probability from host to vector; ovr(i,j): oviposition rate by �yers in the (i,j) patch; egn: average number of eggs laid in an oviposition. The values of the coe�cients are available in Table 1, subsections vi., vii., viii. and ix.. The general rate of the dispersal event is given by: β × F(i,j), where β is the dispersal coe�cient and F(i,j) is the Flyer population which can be sus- ceptible Fs(i,j), exposed Fe(i,j) or infectious Fi(i,j) depending on the disease status. The dispersal coe�cient β can be written as β =   0 if the patches are disjoint diff/d2ij if the patches have at least a common point (3) where dij is the distance between the centres of the patches and diff is a di�usion-like coe�cient so that dispersal is compatible with a di�usion-like process [9]. v. Events related to human population Human contagion has been already described in Ta- ble 7. Table 9 summarizes the events and rates in which humans are involved. The human popula- 050002-18 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. Event E�ect Transition rate II Gonotrophic cycle of susceptible Adults 2 with virus exposure A2s(i,j) → A2s(i,j) − 1 Fe(i,j) → Fe(i,j) + 1 cycle2×A2s(i,j) ×(Hi(i,j)/Nh(i,j))× ahv II Gonotrophic cycle of susceptible Adults 2 without virus exposure A2s(i,j) → A1(i,j) − 1 Fs(i,j) → Fs(i,j) + 1 cycle2 × A2s(i,j) × ((((Nh(i,j) − Hi(i,j))/Nh(i,j)) + (1 − ahv) × (Hi(i,j)/Nh(i,j))) II Gonotrophic cycle of exposed Adults 2 A2e(i,j) → A2e(i,j) − 1 Fe(i,j) → Fe(i,j) + 1 cycle2 ×A2e(i,j) Table 6: Event type, e�ects on the subpopulations and transition rates for the developmental model. The coe�cients are cycle2: gonotrophic cycle coe�cient (number of daily cycles) for adult females in stages A2.; ahv: transmission probability from host to vector. The values of the coe�cients are available in Table 1, subsections vi., vii., viii. and ix.. Event E�ect Transition rate Exposed Adults 2 be- coming infectious A2e(i,j) → A2e(i,j) − 1 A2i(i,j) → A2i(i,j) + 1 (1/(EIP − (1/ovr(i,j))))A2e(i,j) Exposed �yers becom- ing infectious Fe(i,j) → Fe(i,j) − 1 Fi(i,j) → Fi(i,j) + 1 (1/(EIP − (1/ovr(i,j))))Fe(i,j) II Gonotrophic cycle of infectious Adults 2 without human conta- gion A2i(i,j) → A2i(i,j) − 1 Fi(i,j) → Fi(i,j) + 1 Hs(i,j) → Hs(i,j) − 1 He(i,j) → He(i,j) + 1 cycle2 × A2i(i,j) × (Hs(i,j)/Nh(i,j)) ×avh II Gonotrophic cycle of infectious Adults 2 without human conta- gion A2i(i,j) → A2i(i,j) − 1 Fi(i,j) → Fi(i,j) + 1 cycle2 × A2i(i,j) × ((((Nh(i,j) − Hs(i,j))/Nh(i,j)) + (1 − avh) × (Hs(i,j)/Nh(i,j))) Table 7: Event type, e�ects on the subpopulations and transition rates for the developmental model. The coe�cients are cycle2: gonotrophic cycle coe�cient (number of daily cycles) for adult females in stages A2; ovr(i,j): oviposition rate by �yers in the (i,j) patch; avh: transmission probability from vector to host; EIP : extrinsic incubation period. The values of the coe�cients are available in Table 1, subsections vi., vii., viii. and ix. tion was �uctuating but balanced, meaning that the birth coe�cient was considered equal to the mortality coe�cient mh. vi. Developmental Rate coe�cients The developmental rates that correspond to egg hatching, pupation, adult emergence and the gonotrophic cycles were evaluated using the results of the thermodynamic model developed by Sharp and DeMichele [47] and simpli�ed by Schoo�eld et al. [48]. According to this model, the maturation process is controlled by one enzyme which is ac- tive in a given temperature range and is deacti- vated only at high temperatures. The development is stochastic in nature and is controlled by a Poisson process with rate RD(T). In general terms, RD(T) takes the form RD(T) = RD(298 K) (4) × (T/298 K) exp((∆HA/R)(1/298 K− 1/T)) 1 + exp(∆HH/R)(1/T1/2 − 1/T)) where T is the absolute temperature, ∆HA and ∆HH are thermodynamics enthalpies characteris- tic of the organism, R is the universal gas constant, and T1/2 is the temperature when half of the en- zyme is deactivated because of high temperature. 050002-19 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. Event E�ect Transition rate Susceptible �yer Death Fs(i,j) → Fs(i,j) − 1 ma×Fs(i,j) Exposed �yer Death Fe(i,j) → Fe(i,j) − 1 ma×Fe(i,j) Infectious �yer Death Fi(i,j) → Fi(i,j) − 1 ma×Fi(i,j) Susceptible Adult 2 Death A2s(i,j) → A2s(i,j) − 1 ma×A2s(i,j) Exposed Adult 2 Death A2e(i,j) → A2e(i,j) − 1 ma×A2e(i,j) Infectious Adult 2 Death A2i(i,j) → A2i(i,j) − 1 ma×A2i(i,j) Table 8: Event type, e�ects on the subpopulations and transition rates for the developmental model. The coe�cients are ma: adult mortality. The values of the coe�cients are available in subsection vii. Event E�ect Transition rate Born of susceptible humans Hs(i,j) → Hs(i,j) + 1 mh×Nh(i,j) Death of susceptible humans Hs(i,j) → Hs(i,j) − 1 mh×Hs(i,j) Death of exposed humans He(i,j) → He(i,j) − 1 mh×He(i,j) Transition from exposed to vi- raemic He(i,j) → He(i,j) − 1 Hi(i,j) → Hi(i,j) + 1 (1/IIP) ×He(i,j) Death of Infectious humans Hi(i,j) → Hi(i,j) − 1 mh×Hi(i,j) Transition from infectious humans to humans in remission state Hi(i,j) → Hi(i,j) − 1 Hr(i,j) → Hr(i,j) + 1 (1/V P) ×Hi(i,j) Death of humans in remission state Hr(i,j) → Hr(i,j) − 1 mh×Hr(i,j) Transition from humans in remis- sion to toxic humans Hr(i,j) → Hr(i,j) − 1 Ht(i,j) → Ht(i,j) + 1 ((1 −rar)/rP) ×Hr(i,j) Recovery of humans in remission Hr(i,j) → Hr(i,j) − 1 HR(i,j) → HR(i,j) + 1 (rar/rP) ×Hr(i,j) Death of removed humans HR(i,j) → HR(i,j) − 1 mh×HR(i,j) Death of toxic humans Ht(i,j) → Ht(i,j) − 1 Hd(i,j) → Hd(i,j) + 1 (mt/tP) ×Ht(i,j) Recovery of toxic humans Ht(i,j) → Ht(i,j) − 1 HR(i,j) → HR(i,j) + 1 ((1 −mt)/tP) ×Ht(i,j) Table 9: Event type, e�ects on the subpopulations and transition rates for the developmental model. The coe�cients are mh: human mortality coe�cient; V P : human viremic period; mh: human mortality coe�cient; IIP: intrinsic incubation period; rP: remission period; tP: toxic period; rar: recovery after remission probability; mt: mortality probability for toxic patients. The values of the coe�cients are available in Table 1. Table 10 presents the values of the di�erent coef- �cients involved in the events: egg hatching, pupa- tion, adult emergence and gonotrophic cycles. The values are taken from Ref. [30] and are discussed in Ref. [8]. vii. Mortality coe�cients Egg mortality. The mortality coe�cient of eggs is me = 0.01 1/day, independent of temperature in the range 278 K ≤ T ≤ 303 K [49]. Larval mortality. The value of α0 (associated to the carrying capacity of a single breeding site) is α0 = 1.5, and was assigned by �tting the model to observed values of immatures in the cemeteries of Buenos Aires [8]. The temperature dependent larval death coe�cient is approximated by ml = 0.01 + 0.9725 exp(−(T−278)/2.7035) and it is valid in the range 278 K ≤ T ≤ 303 K [50�52]. Pupal mortality. The intrinsic mortality of a pupa has been considered as mp = 0.01 + 050002-20 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. Develop. Cycle (4) RD(T) RD(298 K) ∆HA ∆HH T1/2 Egg hatching elr 0.24 10798 100000 14184 Larval develop. lpr 0.2088 26018 55990 304.6 Pupal Develop. par 0.384 14931 -472379 148 Gonotrophic c. (A1) cycle1 0.216 15725 1756481 447.2 Gonotrophic c. (A2) cycle2 0.372 15725 1756481 447.2 Table 10: Coe�cients for the enzymatic model of maturation [Eq. (4)]. RD is measured in day−1, enthalpies are measured in (cal / mol) and the temperature T is measured in absolute (Kelvin) degrees. 0.9725exp(−(T −278)/2.7035) [50�52]. Besides the daily mortality in the pupal stage, there is an ad- ditional mortality contribution associated to the emergence of the adults. We considered a mortality of 17% of the pupae at this event, which is added to the mortality rate of pupae. Hence, the emergence factor is ef = 0.83 [53]. Adult mortality. Adult mortality coe�cient is ma = 0.091/day and it is considered independent of temperature in the range 278 K ≤ T ≤ 303 K [2,50,54]. viii. Fecundity and oviposition coe�cient Females lay a number of eggs that is roughly proportional to their body weight (46.5 eggs/mg) [55, 56]. Considering that the mean weight of a three-day-old female is 1.35 mg [2], we estimated the average number of eggs laid in one oviposition as egn = 63. The oviposition coe�cient ovr(i,j) depends on breeding site density BS(i,j) and it is de�ned as: ovr(i,j) = { θ/tdep if BS(i,j) ≤ 150 1/tdep if BS(i,j) > 150 (5) where θ was chosen as θ = BS(i,j)/150, a linear function of the density of breeding sites [9]. ix. Dispersal coe�cient We chose a di�usion-like coe�cient of diff = 830 m2/day which corresponds to a short dispersal, ap- proximately a mean dispersal of 30 m in one day, in agreement with short dispersal experiments and �eld studies analyzed in detail in our previous ar- ticle [9]. x. Mathematical description of the stochastic model The evolution of the subpopulations is modeled by a state dependent Poisson process [41,57] where the probability of the state: (E,L,P,A1,A2s,A2e,A2i,Fs,Fe,Fi, Hs,He,Hi,Hr,Ht,HR,Hd)(i,j) evolves in time following a Kolmogorov forward equation that can be constructed directly from the information collected in Tables 4 to 9 and in Eq. 3. xi. Deterministic rates approximation for the density-dependent Markov process Let X be an integer vector having as entries the populations under consideration, and eα,α = 1 . . .κ the events at which the populations change by a �xed amount ∆α in a Poisson process with density-dependent rates. Then, a theorem by Kurtz [57] allows us to rewrite the stochastic pro- cess as: X(t) = X(0) + κ∑ α=1 ∆αY ( ∫ t 0 ωα(X(s))ds) (6) where ωα(X(s) is the transition rate associated with the event α and Y (x) is a random Poisson process of rate x. The deterministic rates approximation to the stochastic process represented by Eq. (6) consists of the introduction of a deterministic approxima- tion for the arguments of the Poisson variables Y (x) in Eq. (6) [34,58]. The reasons for such a proposal is that the transition rates change at a slower rate than the populations. The number of each kind of 050002-21 Papers in Physics, vol. 5, art. 050002 (2013) / M L Fernández et al. event is then approximated as independent Poisson processes with deterministic arguments satisfying a di�erential equation. The probability of nα events of type α having oc- curred after a time dt is approximated by a Poisson distribution with parameter λα. 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