Impact of HCV antiviral therapy on the Liver Transplantation Waiting-List assessed by mathematical models

In this paper we analyze, through a mathematical model, the potential impact of HCV antiviral therapy on the liver transplantation waiting list (LTWL) in the State of Sao Paulo, Brazil. In previous papers, we projected the size of the waiting list by taking into account the incidence of new patients per year, the number of transplantations carried out in that year, and the number of patients that died in the waiting list. In the present work, we projected the LTWL size for the next 30 years and we introduced the anti-HCV treatment, which was assumed to half the incidence of patients in the LTWL and that the recovery of patients in the list would triple. The liver transplantation rate was assumed to not be affected by the anti-HCV treatment. Our mathematical model demonstrates that anti-HCV therapy would have a remarkable impact on the size of the LTWL, in the State of Sao Paulo, dropping from twenty-four thousand to approximately twelve hundred patients in the next 30 years. Research Article Impact of HCV antiviral therapy on the Liver Transplantation WaitingList assessed by mathematical models Eleazar Chaib1*, João Luis Erbs Pessoa2, Marizete Medeiros2 and Eduardo Massad3 1Department of Gastroenterology, Division of Liver Transplantation, University of São Paulo, 01246-903, Sao Paulo, Brazil 2State Secretary of Health, 05403-000, Sao Paulo, SP, Brazil 3School of Applied Mathematics, Fundação Getulio Vargas, 22250-900, Rio de Janeiro, RJ, Brazil Received: 05 December, 2018 Accepted: 02 January, 2019 Published: 03 January, 2019 *Corresponding author: Eleazar Chaib, M.D, PhD, FRCS, Associate Professor of Surgery, School of Medicine, University of São Paulo, Av. Dr. Arnaldo 455, 3rd fl oor, suite 3208, 01246-903, São Paulo, Brazil, Tel: +55 11 30618319; Fax: +55 11 30617270; E-mail:


Introduction
Prevalence of hepatitis C virus (HCV) infection is found worldwide however country prevalence ranging from less than 1% to greater than 10%. The highest prevalence has been reported in Africa and Middle-East, with a lower prevalence in the Americas, Australia, and Northern and Western Europe [1].
Sao Paulo is the fi rst Brazilian state to perform liver transplantation in 1968 [2]. Since then the recipient waiting list has increased; now approximately 150 new cases per month are referred to the single list at the central organ procurement organization [3] .
HCV infection is considered a major public health problem [4] with a global prevalence rate of 2.8%, equating to over 185 million infections, and more than 350,000 deaths annually.
An estimated 3 million to 4 million new cases of HCV infection emerge every year, worldwide [5]. Furthermore, the HCV-related mortality is increasing and HCV infection is projected to be the most important leading cause of viral hepatitis-related mortality in the near future [4,6].
End-stage liver disease due to HCV is currently the leading indication for liver transplantation (LT) in both the United State of America (USA) and Brazil, mainly in the State of Sao Paulo accounting for over 30% and 40% of all transplants annually, respectively (8,9). However, treatment for chronic HCV infection, with elimination of HCV infection, has revolutionized in the past 5 years with the approval of secondgeneration direct-acting antiviral agents.
The number of patients on the liver transplantation waiting list (LTWL) in the State of Sao Paulo jumped 2.71-fold in the past ten years, almost 50% of them due to HCV consequently the number of deaths on LTWL moved to a higher level increasing 2.09-fold [7][8][9][10][11].
Our aim is to analyze, through a mathematical model, the potential impact of HCV antiviral therapy on the LTWL in the

Materials and Methods
This is a theoretical work and we used mathematical models designed to mimic the LTWL's dynamics and which represent improvements on works previously published. In previous papers Chaib et al. [3][4][5], projected the size of the waiting list,  In this paper we improved the list dynamics by considering a continuous-time model as follows: where  is the incidence rate of patients with the model for end-stage liver disease (MELD) criteria to get into the LTWL,  is the death rate and r T is the transplantation rate of patients in the LTWL, respectively. We used the Latin Hypercube sampling method [7], to fi nd the values of the parameters that would explain the observed data.
Equation (2) has the following solution: Data used in the work has been collected in the Service of

Ethical issues
This work has been approved by the Institutional Review  Table 1 shows the value of the variables that entered the model. As can be observed in the fi gure, the set of parameters used retrieves the actual data with the same accuracy as the exponential fi tting. This should be expected since the solution of equation (1) is also an exponential function (equation (2)).

Results
However, the remarkable tally of the models output with the exponential fi tting was obtained by optimizing the value of the parameters through the latin hypercube sampling technique used.
We the projected the result of equation (3) for the next 30 years, under the assumption that all the conditions would remain the same. Next, we introduced the anti-HCV treatment, which was assumed to halve the incidence of patients in the LTWL and that the recovery of patients in the list would triple.
The liver transplantation rate was assumed to not be affected by the anti-HCV treatment. Figure 2 shows the results of this simulation.
It can be seen from fi gure 2 that the anti-HCV treatment would have a remarkable impact on the size of the LTWL, dropping from around 24 thousand patients to around 1.2  thousand patients in 30 years. This reduction would obviously more accentuated if the transplantation capacity of the system would increase, such that by doubling the transplantation rate from year 12, the LTWL would be reduced to zero after less than ten years.

Discussion
In this work we propose an improvement of mathematical models aimed to mimicking the LTWL's dynamics previously published in order to assess the theoretical impact of anti-HCV treatment on the size of the list. It represents an important contribution to the understanding of the possible impact of anti-HCV treatment on the number and rate of new liver failure cases in an affected community because HCV-related end-stage cirrhosis is currently the fi rst cause of liver transplantation [5]. The risk for developing cirrhosis 20 years after initial HCV infection among those chronically infected varies between studies, but is estimated at around 10-15% for men and 1-5% for women. Once cirrhosis is established, the rate of developing hepatocellular carcinoma (HCC) is at 1-4% per year [12].
The health care burden caused by hepatitis C is projected to increase signifi cantly in the next 20 years, on the basis of modeling estimates of cirrhosis, hepatic decompensation, and HCC likely to be seen in this population in the future [13].

Currently, chronic hepatitis C virus-infection-related
cirrhosis is the most common indication for liver transplantation in the USA and most parts of the world. While the incidence of new hepatitis C virus cases has decreased, the prevalence of infection will not peak until the year 2040. In addition, as the duration of infection increases, the proportion of new patients with cirrhosis will double by 2020 in an untreated patient population. In previous papers our group [14][15][16][17][18][19], proposed a series of mathematical models dealing with distinct aspects of liver transplantation. Some of the models were simple like the present one and some more complex.