Computational and Mathematical Methods in Medicine

Volume 2018, Article ID 6289681, 8 pages

https://doi.org/10.1155/2018/6289681

## A Note on the Risk of Infections Invading Unaffected Regions

^{1}School of Medicine, University of Sao Paulo, Sao Paulo, Brazil^{2}School of Applied Mathematics, Fundação Getúlio Vargas, Rio de Janeiro, Brazil^{3}College of Natural and Life Sciences, The University of Derby, Derby, UK^{4}London School of Hygiene and Tropical Medicine, London, UK

Correspondence should be addressed to Marcos Amaku; rb.psu@ukama

Received 14 March 2018; Revised 27 April 2018; Accepted 7 June 2018; Published 4 July 2018

Academic Editor: Chung-Min Liao

Copyright © 2018 Marcos Amaku et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present two probabilistic models to estimate the risk of introducing infectious diseases into previously unaffected countries/regions by infective travellers. We analyse two distinct situations, one dealing with a directly transmitted infection (measles in Italy in 2017) and one dealing with a vector-borne infection (Zika virus in Rio de Janeiro, which may happen in the future). To calculate the risk in the first scenario, we used a simple, nonhomogeneous birth process. The second model proposed in this paper provides a way to calculate the probability that local mosquitoes become infected by the arrival of a single infective traveller during his/her infectiousness period. The result of the risk of measles invasion of Italy was of 93% and the result of the risk of Zika virus invasion of Rio de Janeiro was of 22%.

#### 1. Introduction

Many countries where infectious diseases had been considered controlled in past decades are reporting the invasion of some exotic and frequently unknown infectious diseases that are spread by infected travellers/immigrants [1, 2]. Historical examples of disease invasion are numerous. A particular tragic invasion was the invasion of Europe by the Black Death in XIV century, which started probably in China and travelled by ship until reaching European shores where it decimated from a quarter to a half of the European population [3, 4]. Another tragic invasion occurred in the beginning of the last century when between 50 and 100 million individuals worldwide died victims of the Spanish Flu, which probably started in an American army barrack in the United States in 1918 and rapidly was spread by travellers to others areas of the world [5, 6]. Many years later SARS affected many countries, spread by infected travellers [7, 8]. In 2009 the swine flu pandemic (H1N1) frightened the world, exemplifying the dangers of international spread of communicable diseases [9, 10]. The recent outbreak of Ebola, which spilled over to the United States showed how individuals travelling from infected to uninfected areas of the world can be dangerous [11, 12]. Other recent examples of infectious diseases spreading by travellers include the Zika virus outbreak in Latin America, in particular in Brazil, which was imported from the French Polynesia by infected travellers [13]. Now, Europe is in the grip of a huge measles outbreak caused by the recent waves of nonvaccinated immigrants and the low vaccination coverage of some European countries [14–17].

Human mobility networks are playing an increasing role in the spread of communicable diseases [1, 19], at both the international and the national levels and even between different districts in the same city. Infected travellers can introduce infectious agents to new areas and populations [3]. The highest risk for global health now is the indisputable fact that a traveller with an infectious disease can reach virtually any part of the world within 24 h. The current staggering volume, speed, and reach of travel are unprecedented, showing a virtually uninterrupted growth—from 25 million in 1950 to 278 million in 1980, 528 million in 1995, and 1087 million in 2013 [20]. Asia and the Pacific recorded the fastest relative growth across all World Tourism Organization (UNWTO) regions, with a 6% increase in international arrivals per year in recent years and Asia is the epicentre of many infectious diseases [20]. Africa, another continent with many emerging infectious diseases, saw an increase of 5%. International tourist arrivals worldwide are expected to increase by 3.3% per year from 2010 to 2030 to reach 1.8 billion by 2030, according to UNWTO’s long-term forecast Tourism Towards 2030 [20].

Current methodological approaches to estimate the risk of diseases in travellers still have many shortcomings [9, 21–23]. For example, estimations based on notifications of imported cases underestimate the risk. This is so because many diseases are either not notifiable to authorities or even if legally notifiable are underreported, as not every traveller will report her/his condition to healthcare providers. Moreover, many imported diseases may go unnoticed because of a high frequency of asymptomatic cases that may also contribute to the transmission of diseases. In the absence of good data on importation and exportation of infectious diseases via international travellers, mathematical models can provide an additional tool for the estimating the risks involved.

Situations where a relatively small number of infected individuals travel to previously disease-free areas are particularly critical. In these cases, the deterministic formulation of risk is not sufficient and hence the necessity of new probabilistic models to estimate the risk of disease invasion. Here, a probabilistic model is developed, taking into account air travel volume, force of infection in the country of disembarkation, and herd immunity due to either background immunity or immunization coverage by vaccination. Two distinct situations related to the spread of infections by travellers are considered, namely, a directly transmitted infection and a vector-borne infection. We used the case of measles in Europe and the potential spread of Zika virus to disease-free but Aedes-infected areas (Rio de Janeiro, Brazil) as a proof of principle and to illustrate the methods. Uncertainty analysis was considered as confidence intervals and the simulation were carried out by solving the specific equations analytically.

#### 2. Case 1. Directly Transmitted Infection

The model assumes that a density of infected individual, (see [24]), arrived at and remains infective for a period of weeks, that is,where , , and are the natural mortality rate, the recovery rate from infection, and the disease-induced mortality rate, respectively, and is the Heaviside step function. If the region where these infected travellers arrive had an area the number of them is .

The total number of measles cases, , infected by these travellers, one year after its introduction ( weeks), is given bywhere is the potentially infective contact per unit time between infected and susceptible individuals and and are the susceptible and total population, respectively.

The risk of measles invasion of a previously unaffected country, , can be defined as the probability that at least one autochthonous case be produced by the arrival of one single infected individual at the area during his/her infectiousness period. To calculate this risk, we assumed a nonhomogeneous simple birth process [25], which describes the introduction of the disease. Note that we are interested only in the estimation of the risk of infection and, therefore, recovery from the infection is not considered in the risk estimation.

Let be probability of cases. The probability generating function of such process is . After some calculation [25] we obtainwhere is a dummy variable, , and is the “birth rate of the process”; in this case the incidence of the infection and is defined as , where is obtained by solving (A.1) from the Appendix using the parameters obtained in the next section and described in Table 2. In (3), , is a dummy variable and is the Heaviside step function, introduced to simulate the moment the infected traveller arrives. Note that has no physical meaning and was introduced to simplify the notation.

We assumed that the region to be studied has an area and that the number of infected travellers arriving at is . We set from now on.

Expanding (3) in powers of we find that the risk, that is, the probability of having infected individual at time , denoted as

The risk (probability) of having no infected individuals is

Note that neither nor have any physical meaning and were introduced to simplify the notation. Note also that both and increase with time reaching a constant value denoted and , respectively.

The probability of at least 1 autochthonous case in a previously unaffected region can be calculated as follows:

The upper and lower values of 95% confidence intervals for the number of secondary cases are given by

##### 2.1. The Case of Measles in Europe

In 2017, Europe reported more than 21,000 cases of measles, including 35 deaths (a mortality rate of 6.0×10^{−5} month^{−1})[14]. This represents a fourfold increase from 2016 when just 5,273 cases were reported. This outbreak was due to two factors: first, there was an overall decline in routine immunization coverage, with particularly low coverage among marginalised groups, in addition to interruption in vaccine supply or underperforming disease surveillance; second, large numbers of nonvaccinated immigrants flooded Europe in previous years (5,136,383 since 2011, according to the European Asylum Support Office, [17]).

According to the Communicable Disease Threat Report [14], the highest numbers of cases of measles since 1 January 2017 were reported in Romania (8,274), Italy (4,885), and Germany (919), the three countries which received the highest numbers of immigrants from undervaccinated areas [17].

Let us analyse the case of Italy where, according to the ECDC [14], the measles vaccination coverage is less than 84% (well below the herd immunity threshold of 95%). If we assume for 2017 the same proportion of immigrants from measles undervaccinated countries to Europe as the one observed in 2015 (3% see [17]), then of the around 700 thousand immigrants that arrived in Europe in 2017, around 21 thousand settled in Italy. In that year, Italy reported 4,885 measles cases, as mentioned above. Assuming a vaccination coverage of 80%, it is possible to estimate the risk of measles introduction by considering the effective reproduction number , where is the basic reproduction number [26]. So, the effective reproduction number reaches a value of 5, a value compatible with the 20% susceptible individuals, the proportion of nonvaccinated individuals in Italy. This is because the basic reproduction number of measles is around 25 and therefore . Assuming a recovery rate month^{−1}, a natural mortality rate of 1.10×10^{−3} month^{−1}, a disease-induced mortality of = 6.0×10^{−5} month^{−1}, and using the expression for , we estimated that the potentially infective contact rate is month^{−1}. By using these parameters it is possible to calculate , and of (6). From this equation it follows that the probability that one single infective immigrant, arriving in Italy on January , could generate at least one secondary, autochthonous measles case with a probability of around 93%, which is reasonably high risk. The 95% CI for the number secondary cases is (0.89 - 1.40). In other words, one secondary case will be produced by a single traveller with confidence interval between 0.89 and 1.40 cases per unit area.

As we estimated the basic reproduction number of measles in Italy at the time of the outbreak as equal to 5, it would be interesting to estimate the number of infected individuals necessary to arrive to guarantee that at least one secondary infection would be generated. In other words, what would be the value of in (4a) that would make . The result is that seven infected immigrants would guarantee the invasion of measles in Italy.

We have no way to know how many infective individuals arrived with measles in Italy in 2017 but with the above assumptions and the method proposed here, it is possible, although very laborious, to backcalculate how many infected immigrants would be necessary to generate the observed 4,885 cases in 2007.

Table 1 shows the variables and parameters used in this section as well as their biological meanings.