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# Current forecast of HIV/AIDS using Bayesian inference

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In this study, we address the problem of fitting a mathematical model to the human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) data. We present a quantitative analysis of the formulated mathematical model by using Bayesian inference. The mathematical model consists of a suitable simple system of ordinary differential equations. We perform a local and global sensitivity analysis of parameters to determine which parameters of the model are the most relevant for the transmission and prevalence of the disease. We formulate the inverse problem associated to the parameter estimation of the model, and solve it using Bayesian statistics. Then, we estimate the basic reproductive number of the disease based on the estimation of the parameters of the model and its comparison with one is tested through hypothesis tests. The data set consist of HIV and AIDS data from Luxembourg, Czech Republic, Japan, Croatia, United Kingdom, and Mexico.
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Received: 31 December 2020
|
Accepted: 8 September 2021
DOI: 10.1111/nrm.12332
Current forecast of HIV/AIDS using Bayesian
inference
Kernel Prieto
1
|Jhoana P. RomeroLeiton
2
1
Insituto de Matemáticas, Universidad
Nacional Autónoma de México,
Cuernavaca, México
2
Cesmag, Pasto, Colombia
Correspondence
Jhoana P. RomeroLeiton, Facultad de
Ingeniería, Universidad Cesmag, Pasto,
Colombia.
Email: jpatirom3@gmail.com
Abstract
In this study, we address the problem of fitting a
mathematical model to the human immunodeficiency
virus (HIV)/acquired immunodeficiency syndrome
(AIDS) data. We present a quantitative analysis of the
formulated mathematical model by using Bayesian
inference. The mathematical model consists of a sui-
table simple system of ordinary differential equations.
We perform a local and global sensitivity analysis of
parameters to determine which parameters of the
model are the most relevant for the transmission and
prevalence of the disease. We formulate the inverse
problem associated to the parameter estimation of
the model, and solve it using Bayesian statistics. Then,
we estimate the basic reproductive number of the
disease based on the estimation of the parameters of
the model and its comparison with one is tested
through hypothesis tests. The data set consist of HIV
and AIDS data from Luxembourg, Czech Republic,
Japan, Croatia, United Kingdom, and Mexico.
KEYWORDS
Basic reproductive number, Bayesian sensitivity analysis, Data
driven, Hypothesis testing, Inverse problem, Sensitivity indices
Natural Resource Modeling. 2021;34:e12332. wileyonlinelibrary.com/journal/nrm
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https://doi.org/10.1111/nrm.12332
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and
reproduction in any medium, provided the original work is properly cited.
© 2021 The Authors. Natural Resource Modeling published by Wiley Periodicals LLC
1|INTRODUCTION
HIV (human immunodeficiency virus) is a virus that attacks the cells that help the body to
fight infections, making a person more vulnerable to other infections and diseases. It is
spread by contact with certain body fluids of a person with HIV, most commonly during
unprotected sex, or through sharing injection drug equipment. If left untreated, HIV can
lead to the disease AIDS (acquired immunodeficiency syndrome). The human body
cannot get rid of HIV and no effective HIV cure exists. So, once a person have HIV, it has
it for life. However, by taking HIV medicine (called antiretroviral therapy or ART), people
with HIV can live long and healthy lives and prevent transmitting HIV to their sexual
partners. In addition, there are effective methods to prevent getting HIV through sex or
drug use, including preexposure prophylaxis (PrEP) and postexposure prophylaxis (PEP)
(Barnett & Whiteside, 2002). The main characteristic of the HIV is its large incubation
period, several years on average.
First identified in 1981, HIV is the cause of one of humanity's deadliest and most
persistent epidemics. About 38 million people worldwide are estimated to be living with
HIV and around 8 hundred thousand people have died of AIDS in 2019 or 25 million have
died from AIDS since the first cases were identified in 1981. About 2.5 million children
under the age of 15 years are living with HIV and more than 12 million have been
orphaned by AIDS by 2004. An increase number of individuals infected with HIV are now
becoming ill and will die in the absence of intervention strategies in African countries.
The progression from HIV infection to AIDS occurs approximately over one or two dec-
ades. The regional analysis of the Joint United Nations Programme on HIV/AIDS (UN-
AIDS) shows that the main concentration of AIDS cases are in developing countries, 68%
of persons living with HIV are in SubSaharan Africa in 2010. In second place is Eastern
and Southeastern Asia, with an estimated number of 4 millions. Latin America, Eastern
Europe and Central Asia are in the third place. The use of antiretroviral drug have de-
creased the number of deaths related to AIDS.
Recently, several mathematical models for diseases transmission including compart-
mental epidemic models, have been proposed to predict and control the HIV/AIDS spread
(see, e.g., Cassels et al., 2008; Eaton et al., 2012;Johnson&White,2011;Sweilam,2020;
Van De Vijver et al., 2013;Wodarz&Nowak,2002). Other works have used Bayesian
inference to predict the future behavior of cases (see e.g., Huang et al., 2010,2011;Huang
&Wu,2006;Wuetal.,2008).
In this study, we propose a variation of a SItype compartmental model including anti-
retroviral treatment applied to Luxembourg, Czech Republic, Japan, Croatia, United King-
dom, and Mexico. We determine the basic reproductive number and it is compared with one
thorough hypothesis test. We also do a sensitivity analysis of parameters and their estimation
through Bayesian inference.
The remainder of this paper is organized as follows: Section 2describes the mathematical for-
mulation and the parameter description of the proposed model. In Section 3we determine the
sensitivity indices for some parameters of the model, whereas in Section 6we do a Bayesian sensi-
tivity analysis. Section 4describes the Bayesian inference framework to predict the dynamics of the
spread of HIV/AIDS. In Section 5we contrast the null hypothesis for
R
0
in each country. Each
section presents the mathematical framework and the numerical results. Discussion and conclusions
are presented in Section 7.
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2|MATHEMATICAL MODEL
To describe accurately the spread of the HIV/AIDS disease, a model that includes a combination of
an agechronological and an ageinfection structures must be considered, see Figure 1.Anage
chronological structure is due to there exists evidence that the HIV cases are concentrated in the age
of 1550 years because in general, the individual's sexual activity is higher in this life stage. This fact,
has been considered in some mathematical models (see e.g., Inaba, 2003;Luboobi,1994;Okongo
et al., 2013;Rongetal.,2007; Saxena & Hooda, 2015;Wangetal.,2015). An ageinfection structure is
because the HIV/AIDS disease has at least three main stages (see Figure 2; Hollingsworth et al., 2008;
Perelson & Nelson, 1999).Thesethreestagesareduetotheviralloadintheindividualchanges
during the period of time. At the beginning, the viral load starts to increase in the first 6 weeks until
achieves its maximum, then decrease until the 10 week where it stays steadily for many years, then it
increases again. Statistical data show that the average incubation period of the disease from HIV to
AIDS is 512 years (Cai et al., 2008; Granich et al., 2009). Therefore, mathematical models have been
proposed to incorporate these three stages of infection (Baryarama & Mugisha, 2007;Granich
et al., 2009), but also, some delayed models have been studied. On the other hand, the compartmental
structures SusceptibleInfectiousRemoved(SIR) and SusceptibleExposedInfectiousRemoved
FIGURE 1 Viral load of the HIV/AIDS disease. AIDS, acquired immunodeficiency syndrome; HIV, human
immunodeficiency virus. Source: Perelson and Nelson (1999)
FIGURE 2 Infection age of the HIV/AIDS disease. AIDS, acquired immunodeficiency syndrome; HIV,
human immunodeficiency virus. Source: Hollingsworth et al. (2008)
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(SEIR) have been widely used to study the HIV/AIDS transmission dynamics (Mahato et al., 2014).
Besides, some mathematical models have incorporated the antiretroviral treatment of HIV/AIDS
patients (see e.g., Baggaley et al., 2005; Cai et al., 2009; Granich et al., 2009;Okongoetal.,2013;Rong
et al., 2007;Wazirietal.,2012). According to Brauer and others (Brauer et al., 2008;Ejigu,2010;Yan
&Lv,2016), more robust models of the spread of HIV/AIDS should be proposed using partial
differential equations (PDEs).
In this study, it is assumed that the disease is only transmitted horizontally (the horizontal
transmission can occur either by sexual contact or by indirect contact through drug syringe
exchange). Additionally, inspired in a full datadriven approach, we have tried to use all the
reliable data available to forecast the spread of the HIV/AIDS disease, keeping in mind that a
simple model may fit better than a complex one (Roda et al., 2020). Thus, we formulate a
singlestage SIAtype mathematical model similar to the proposed in Arazoza and Lounes
(2002), Biswas and Pal (2017), Cai et al. (2008), Eduafo et al. (2015). In our model, the total
population
N
, is divided into the following three epidemiological classes: susceptible in-
dividuals
S
, infective individuals who are the infected and infectious individuals that have not
yet developed AIDS symptoms
I
,andAIDSpatientswhoareinfectedandwithAIDS
symptoms
A
.Here,
β
denotes the disease transmission rate;
μ
and
ν
denotes the rates of birth
and death, respectively;
the death rate caused by the AIDS;
σ
denotes the transfer rate
between the compartment
A
to
I
;
ξ1
denotes the rate of infection due to drug syringe ex-
change; and
ξ2
denotes the rate at the AIDS patients (class
A
) obtain antiretroviral treatment.
With the previous hypotheses, our model is represented through the following ordinary
differential equations (ODEs) equations:
μβIνξS
βIξSσνξI
σξIκνA
=(++),
=( + ) (+ ),
=( )(+).
dS
dt
dI
dt
dA
dt
1
12
2
(1)
Since the vectorial field defined by the righthand side of the system (1) is continuously
differentiable (polynomial formulation), the existence and uniqueness of solutions is guaranteed.
It is easy to verify that the feasible region of (1)is
SIA S I A
Ω
={( , , ) :0 + + 1}
+
3
¦∈≤ ≤
,
where
+
3
¦
denotes the positive octant of
3
¦
. On the other hand, the system (1)hastwoequili-
brium points,
P
0
(the diseasefree equilibrium point) and P
*
(the endemic equilibrium point)
which are expressed as follows:
()
()
P
PRR
=,0,0,
*=,(1), ( 1) ,
μ
ν
μκ ν σ
k
μκ ν
k
μσ
k
0
(+ + ) (+)
00
11 1
(2)
where
kβκ ν κσ=(+)
.
1
To understand the spread of a disease, it is important to know certain epidemiological
thresholds, the basic reproductive number (denoted as
R
0
)isoneofthem.
R
0
,isthe
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expected number of secondary infectious cases generated by an infected person in an
entirely susceptible population. This threshold quantifies how many susceptible persons
are on average infected by one infected person. The estimation of
R
0
can be done in two
ways: either by inferring it from observed cases, or by following infection chains step by
step. For our mathematical model,
R
0
is the following:
Rβ
σν
=(+)
.
0(3)
The prevalence of HIV, that is, the accumulated cases of the HIV incidence between two times
of observation
t
t,
ii1
is
KβI
NξSdtΦ=+
,
i
t
t
11
i
i
1
(4)
and the prevalence of AIDS is given by
KσξISdt
Ξ
=()
,
i
t
t
22
i
i
1
(5)
where
K1
and
K2
account for the undetermined true number of HIVAIDS cases since some
reported HIVAIDS cases occur once the infected individuals start presenting symptoms
(Keeling & Rohani, 2008, p. 51; Ponciano & Capistrán, 2011). Table 1shows the
parameters of the SIA model (1) and the Bayesian inference framework. We estimate
the parameters
β
σK,,
1
,and
K2
for the countries: Luxembourg, Czech Republic, Japan, and
Croatia, and the parameters
β
σξ ξ K,, , ,
12
1
,and
K2
for the countries: United Kingdom
and Mexico. The natural birth
μ
and death rate
ν
, though they are assumed known
fixed values, they are different for the six countries that we are analyzing in this
manuscript.
TABLE 1 Parameters of the SIA model (1) and the Bayesian inference framework
Parameter Description Range
μ
Natural birth rate [0.0077, 0.0115]
β
Transmission rate of the disease [0, 1]
σ
Rate of progression from class
I
to
A
[0, 1]
ν
Natural death rate [0.0073, 0.0105]
κ
Death rate caused by AIDS [0.00001, 0.00003]
ξ
1
Rate of infection due to drug syringe exchange 0.00001
ξ
2
Rate at AIDS patients (class
A
) obtain treatment 0.000001
K
1
Factor of subrepresentation of HIV prevalence To be estimated
K
2
Factor of subrepresentation of AIDS prevalence To be estimated
Note: We estimated the parameters
β
σKK,, ,and
1
2
and for the countries: Luxembourg, Czech Republic, Japan, and Croatia,
and the parameters
β
σξ ξ KK,, , , ,and
12 1
2
and for the countries: United Kingdom and Mexico.
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3|SENSITIVITY INDICES FOR MODEL BUILDING
3.1 |Sensitivity indices of R
0
In determining how to reduce the number of susceptible population due to a disease outbreak,
it is necessary to know the relative importance of the different factors responsible for its
transmission and its prevalence. The initial transmission of a disease is directly related to
R
0
,
and its prevalence is directly related to the endemic equilibrium point, that for our case is given
by PSIA
*=( *,*,*)defined on the second equation of (2), more specifically to the magnitude of
I
*
, since it represents the people who may be clinically ill. In this section, we compute the
sensitivity indices of
R
0
and I
*
with respect to each parameter involved on the model (1). Those
indices allow us to measure the relative change in a variable when a parameter is changing.
Sensitivity analysis is commonly used to determine the robustness of a model predictions to the
parameter values, since there are usually errors in the data collection and the presumed
parameter values (Chitnis et al., 2008; Zi, 2011).
The normalized forward sensitivity index of a variable to a parameter, is defined as the ratio
of the relative change in the variable to the relative change in the parameter. When the variable
is a differentiable function of the parameter, the sensitivity index may be alternatively defined
using partial derivatives (Chitnis et al., 2008; Zi, 2011) as follows.
Definition 1. The normalized forward sensitivity index of a variable,
Q
, that depends
differentiably on a parameter,
p
, is defined as
Q
p
p
Q
ϒ:= ×
.
p
Q
(6)
Given that we have explicit formulas for
R
0
given on (3) and I
*
given on (2), we derive an
analytical expression for the sensitivity indices of both,
R
0
and I
*
, to each of the seven different
parameters described in Table 1. Those indices are given in Table 2.
3.2 |Global sensitivity indices of the total incidence of HIV
A global sensitivity analysis is done to identify relevant and noninfluential parameters when
the values of the parameters are not specified but they vary over the entire range of input
values. One objective of the global sensitivity analysis is to quantify how uncertainties in the
TABLE 2 Local sensitivity indices of
R
0
and
I
*
with respect each parameter involved on the model (1)
Parameter
R
0
I*
μ
0 1.0
β
1 1.17
ν
0.09 0.22
σ
0.9 1.94
κ
0 0.00046
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model outputs can be apportioned to uncertainties in the model inputs that are considered in
the parameter space. The global sensitivity analysis techniques can be broadly categorized as
regression, variance, or screeningbased methods. Variancebased indices are advantageous
over regression and correlationsbased indices since they do not require linearity or mono-
tonicity. For this reason, they are sometimes referred to as modelfree methods. In this section,
we used variancebased indices, more specifically Sobol indices. We also briefly describe for-
mulas for Sobol indices for uniform densities (Smith, 2013).
Let us consider the scalarvalued and nonlinear model YfQ=()
,where
Q
QQ=[ ,, ] Γ
p
p
1
¦∈⊂
. We initially assume that the random variables are independent and
uniformly distributed on
[
0, 1] so that
Q
~(0,1),Γ=[0,1]
.
ip
We consider the secondorder HighDimensional Model Representation (HDMR) or the Sobol
expansion
f
qf fq fqq()= + ( )+ ( , )
,
i
p
ii
ijp
ij ij
0
=1 1≤≤≤
 (7)
subject to the condition
f q dq f q q dq f q q dq()=(,)=(,)=0
,
iii ij ij i ij ij j
0
1
0
1
0
1

which ensures that the functions
f
i
are orthogonal. The expansion terms: zeroth, first, and
secondorder terms have variance interpretations as follows:
f
f q dq dq Y q Y=()
~=[( )]=()
,
ii i
00
1
Γp1
 

(8a)
f
qYqf()= ( )
,
iii
0
(8b)
f
qq Yqdq fq fq f(, )= ( )()().
ij ij i j iiij0
(8c)
The total variance
D
of the response
Y
is given by
DYfqdqf=var( )= ()
Γ
2
0
2
since
f
Y=(
)
0
.
The total variance can be expressed as
DD D=+
,
i
p
i
ijp
ij
=1 1≤≤≤

where the partial variances are
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Dfqdq
D f q q dq dq
=(),
=(,).
iiii
ij ij ij i j
0
12
0
1
0
12

The Sobol indices are defined to be
S
D
DSD
Dij p=,= ,,=1,,
,
iiij
ij
so, by the definition, they satisfy
SS+=1
.
i
p
i
ijp
ij
=1 1≤≤≤

The terms
S
i
are often termed the importance measures or firstorder sensitivity indices,and
large values of
S
i
indicate parameters that strongly influence the response variance.
Similarly,
S
ij account for the influence of interaction terms. Because the number of
firstand secondorder Sobol indices is p+pp(1)
2, their analysis quickly becomes un-
tenable for large parameter dimensions. This motivates the consideration of total sensi-
tivity indices
S
SS=+
,
Ti
j
p
ij
=1
i
which quantify the total effect of the parameter
Q
ion the response
Y
.Usingtheequation(8),it
follows that
DYq=var[ ( )]
ii
and hence
S
Yq
Y
=var[ ( )]
var( )
.
ii
(9)
Similarly, one can show that
D Yq q Yq Yq=var[ ( , )]var [( )] var [( ) ]
,
ij ij i j

which yields a variance interpretation for
S
ij. Finally, the total sensitivity index has the
interpretation
S
Yq
Y
Yq
Y
=1var[ ( ~)]
var( ) =[var( ~)]
var( )
.
Tii
i
 (10)
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Next, we describe the Sensitivity algorithm proposed in Smith (2013). The computation of
S
i
given by (9) requires the approximation of Yqvar[ ( )]
i
.Ifoneuses
M
Monte Carlo
evaluations to approximate the conditional mean Yq(
)
i
for fixed
q
i
and repeats the
procedure
M
times to approximate the variance, a total of
M2
evaluations will be required
to evaluate a single sensitivity index. For large parameter dimensions
p
, this approach is
prohibitive. The following algorithm of Saltelli, reduces the number of required function
evaluations to
M
p(+2
)
.Createtwo
Mp
×
sample matrices
A
qqq
qqq
B
qqq
qqq
=,=
ˆˆˆ
ˆˆˆ
,
ip
M
i
M
p
M
ip
M
i
M
p
M
1
111
1
1
111
1
⋯⋯
⋮⋮
⋯⋯
⋯⋯
⋮⋮
⋯⋯
where
q
i
j
and
q
ˆi
jare quasirandom numbers drawn from the respective densities.
Create
Mp
×
matrices
C
qqq
qqq
=
ˆˆˆ
ˆˆˆ
,
i
ip
M
i
M
p
M
1
111
1
⋯⋯
⋮⋮
⋯⋯
(11)
which are identical to
B
with the exception that the ith column is taken from
A
.
Compute
M
×
1
vectors of model outputs
yfAyfBy fC=(), =() =()
AB C
i
i(12)
by evaluating the model at the input values in
AB,
and Ci. The evaluation of y
Aand
y
B
requires
M2
model evaluations, whereas the evaluation of
yi
p
,=1,,
C
i, requires p
M
evaluations.
Hence the total number of model evaluations is
M
p(+2
)
.
The estimates for the firstorder sensitivity indices are
()
S
Yq
Y
yy f
yy f
yy f
yf
=var[ ( )]
var( ) =
=
,
iiMA
TC
A
TA
Mj
M
A
j
C
j
j
M
A
j
1
0
2
0
2
1
=1 0
2
=1
2
0
2
ii
where the mean is approximated by
()
fMyMy
SYq
Y
yy f
yy f
yy f
yf
=11
.
=1 var[ ( ~)]
var( ) =1
=
.
j
M
A
j
j
M
B
j
TiMB
TC
A
TA
Mj
M
B
j
C
j
j
M
A
j
0
2
=1 =1
1
0
2
0
2
1
=1 0
2
=1
2
0
2
i
i
i

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The intuition for the algorithm is the following. In the scalar product yy
A
TC
i
, the response
computed from values in
A
is multiplied by values for which all parameters except
q
i
have been
resampled. If
q
i
is influential, then large (or small) values of y
Awill be correspondingly mul-
tiplied by large (or small) values of
y
C
i
, yielding a large value of
S
i
.If
q
i
is not influential, large
and small values of y
Aand
y
C
i
will occur more randomly and
S
i
will be small. Table 3shows the
global sensitivity indices of the total incidence of HIV (Φ
T
), with respect to the model para-
meters of (1).
4|BAYESIAN INFERENCE
For the parameter estimation of the countries Croatia, Czech Republic and Luxembourg, we
used the yearly updated data UNAIDS. For the parameter estimation of Japan, Mexico, and
United Kingdom, we used the yearly updated data National Institute of Infectious Diseases,
Direccion General de Epidemiologia and HIV in the United Kingdom, respectively. From the
mathematical point of view, the parameter estimation of ODEs system is regarded as an inverse
problem. Fitting curve or estimation the parameters of a model is considered an inverse pro-
blem. Typically, an optimization method, for example, the Landweber in Prieto and Dorn
(2016), Prieto and IbarguenMondragon (2019), Smirnova et al. (2016), AlavezRamirez (2007),
and Capistrán et al. (2009), or faster methods such as the LevenbergMarquardt or Conjugate
Gradient methods, and regularization techniques, such as Tikhonov, Sparsity or Total Varia-
tion, are used to solve this inverse problem. In this manuscript, we use Bayesian inference to
solve the inverse problem since it is a tool which combines uncertainty propagation of mea-
sured data with available prior information of the parameters of the model, also, it is nu-
merically more stable approach than classical methods, since classical methods rely on the
starting parameter point must be relatively close to the true one, otherwise the solution ob-
tained corresponds to a local minimum. Moreover, classical methods give only a point estimate
solution instead of a band of the solutions using Bayesian inference, that is, in a Bayesian
framework, one works with credible intervals. Some references of works using Bayesian in-
ference are available in AcuñaZegarra et al. (2020), Argüedas et al. (2019), Bettencourt and
Ribeiro (2008), Bliznashki (2020), BoerschSupan et al. (2017), Brown et al. (2018), Capistrán
et al. (2020), Chatzilena et al. (2019), Chowell (2017), Grinsztajn et al. (2020), Luzyanina and
TABLE 3 Global sensitivity indices of the total incidence of HIV (
Φ
T
) with respect to the parameters
Parameter
S
i
S
T
i
μ
0.014370 0.143967
β
0.294391 0.637568
ν
0.069298 0.500846
σ
0.030566 0.091851
K
1
0.001263 0.717994
K
2
0.025874 0.753820
Note:S
i
and S
Ti
, represents the Sobol index and the total sensitivity index with respect to the ith parameter given by (9), and
given by (10), respectively.
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Bocharov (2018), Stojanovićet al. (2019). Using Bayesian inference, the solutions of the inverse
problem are obtained from the posterior distribution of the parameters of interest, and a
solution of interest is obtained using the Maximum a Posterior, called MAP. This MAP gives
the parameter value for which the posterior density is maximal. Also, one can compute the
median and quantiles from this posterior sample. As already mentioned, the Bayesian frame-
work provides a natural and formal way to quantify the uncertainty of the quantities of interest.
Denoting the state variable
x StItAt L T=( (), (), ()) ( ([0, ])
k
2
, that is,
denotes the number
of state variables, here
k=3
, and the parameters
θβσKK=( , , , )
m
12
¦
, that is,
m
denotes the
dimension number of parameters to estimate, here
m=
4
. Thus, we can write the model (1)as
the following Cauchy problem:
xφxθ
xx
˙=(,),
(0) = .
0
(13)
Problem (13), defines a mapping θ
x
Φ()= from parameters
θ
to state variables
x
,where
LTΦ:(([0,]))
m
k
+2
¦
, where +
¦
denotes the nonnegative real numbers. We assume that
Φ
has
a Fréchet derivative, that is, the mapping
FθLT(): ( ([0, ]))
m
k
+2
¦
, is injective, thus the for-
ward problem (13)hasauniquesolution
x
for a given
θ
. The Fréchet derivative of
Φ
, denoted by
Φ, results to be the usual derivative for the system (1) since the domain and range of Φare finite
dimensional spaces. Usually, not all states of the system can actually be directed measured, that
is, the data consists of measurements of some state variables at a discrete set of points
t
t,…,
n1
,for
example, in epidemiology, these data consist of number of cases of confirmed infected people.
This defines a linear observation mapping from state variables to data LTΨ:( ([0, ])) :
ns
k
¦,
where
s
k
is the number of observed variables and
n
is the number of sample points. Let
F:
ms
k
×
¦¦be defined by Fθθ()=Ψ(Φ())
, called the forward problem. The inverse problem
is formulated as a standard optimization problem
Fθymin ( )
,
θobs 2
m
¦

(14)
such that xθ=Φ(
)
holds, with
y
obs
is the data which has error measurements of size
η
.
Problem (13) may be solved using numerical tools to deal with a nonlinear leastsquares
problem or the Landweber method or the combination of both. As we mention before, we
implement Bayesian inference to solve the inverse problem (14) in this manuscript. From the
Bayesian perspective, all state variables
x
and parameters
θ
are considered as random variables
and the data
y
obs
is fixed. For random variables
x
θ
,
, the joint probability distribution density of
data
x
and parameters
θ
, denoted by πθx(,
)
, is given by πθxπxθπ θ(, )= ( ) (
)
, where πxθ()is
the conditional probability distribution, also called the likelihood function, and πθ()is the
prior distribution which involves the prior information of parameters
θ
. Given yIA=(
˜,˜
)
obs ,
which correspond to the diagnosed infected HIV cases and the notified sick AIDS cases,
respectively, the conditional probability distribution πθy(
)
obs
, called the posterior distribution
of
θ
is given by the Bayes' theorem:
πθyπyθπθ()()()
,
obs obs

(15)
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FIGURE 3 Credible intervals of the parameters of the model (1) within 95% highestposterior density. From
top to bottom: Luxembourg, Czech Republic, Japan, Croatia
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FIGURE 4 Credible intervals of the parameters of the model (1) within 95% highestposterior density of
United Kingdom
FIGURE 5 Credible intervals of the parameters of the model (1) within 95% highestposterior density of
Mexico
FIGURE 6 The fit for the diagnosed HIV cases and the notified AIDS cases of Mexico using the Stan
package (Carpenter et al., 2017). The blue and red semicontinuous lines represent the observed HIV and AIDS
data, respectively, the solid orange and violet lines represent the medians and the shaded area represents the
95% probability bands for the expected value for the state variables: Infected and Sick people, respectively. AIDS,
acquired immunodeficiency syndrome; HIV, human immunodeficiency virus
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If additive noise is assumed:
yFθη=()+
,
obs (16)
where
η
is the noise due to discretization, model error and measurement error. If the noise
probability distribution
πη()
H
is known,
θ
and
η
are independent, then
FIGURE 7 The fit for the diagnosed HIV cases and the notified AIDS cases of Luxembourg using the Stan
package (Carpenter et al., 2017). The blue and red semicontinuous lines represent the observed HIV and AIDS
data, respectively, the solid orange and violet lines represent the medians and the shaded area represents the
95% probability bands for the expected value for the state variables: Infected and Sick people, respectively. AIDS,
acquired immunodeficiency syndrome; HIV, human immunodeficiency virus
FIGURE 8 The fit for the diagnosed HIV cases and the notified AIDS cases of Czech Republic using the
Stan package (Carpenter et al., 2017). The blue and red semicontinuous lines represent the observed HIV and
AIDS data, respectively, the solid orange and violet lines represent the medians and the shaded area represents
the 95% probability bands for the expected value for the state variables: infected and sick people, respectively.
AIDS, acquired immunodeficiency syndrome; HIV, human immunodeficiency virus
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πyθπyFθ()=(())
.
H
obs obs
All the available information regarding the unknown parameter
θ
is codified into the a prior
distribution πθ()
, it specifies our belief in a parameter before observing the data. All the
available information regarding the way of how was obtained the measured data is codified into
the likelihood distribution πyθ(
)
obs. This likelihood can be seen as an objective or cost
FIGURE 9 The fit for the diagnosed HIV cases and the notified AIDS cases of Japan using the Stan
package (Carpenter et al., 2017). The blue and red semicontinuous lines represent the observed HIV and AIDS
data, respectively, the solid orange and violet lines represent the medians and the shaded area represents the
95% probability bands for the expected value for the state variables: infected and sick people, respectively. AIDS,
acquired immunodeficiency syndrome; HIV, human immunodeficiency virus
FIGURE 10 The fit for the diagnosed HIV cases and the notified AIDS cases of Croatia using the Stan
package (Carpenter et al., 2017). The blue and red semicontinuous lines represent the observed HIV and AIDS
data, respectively, the solid orange and violet lines represent the medians and the shaded area represents the
95% probability bands for the expected value for the state variables: Infected and Sick people, respectively. AIDS,
acquired immunodeficiency syndrome; HIV, human immunodeficiency virus
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function, as it punishes deviations of the model from the data. To solve the associated inverse
problem (15), one may use the maximum a posterior (MAP)
θπθxθπθx=max ( ), = [ ( )]
.
θ
MAP CM 
We used the data set yIA=(
˜,˜
)
obs , which correspond to the diagnosed infected HIV cases and
the notified sick AIDS cases, respectively. We mention that we have not used the data column
corresponding to the recovery people here because in a big range (from the beginning) of days
this data was not been collected. A Poisson distribution,
yμ(
)
, with respect to the time is
typically used to account for the discrete nature of these counts, where
μ
is the mean of the
random variable
y
, that is,
Yμ[]=
. In fact, the mean and variance of the Poisson distribution
coincide. We assume independent Poisson distributed noise
η
, that is, all dependency in the
data is codified into the model (1). In other words, the positive definite noise covariance matrix
η
is assumed to be diagonal. Therefore, using the Bayes' formula, the likelihood is
πθIA πIθπAθπθ(˜,˜)(
˜)(
˜)()
.

As mentioned above, we approximate the likelihood probability distribution corresponding to
diagnosed HIV individuals and AIDS patients with a Poisson distribution
IθAθ
˜~(Φ()),˜~(Ξ())
,
iiii

where
Φ,Ξ
ii
are given by (4) and (5), respectively, and the index
i
denotes the number time, in
our case the number of years. For independent observations, the likelihood distribution πyθ(),
is given by the product of the individual probability densities of the observations
FIGURE 11 ThefitforthediagnosedHIVcasesandthenotifiedAIDScasesofUnitedKingdomusing
the Stan package (Carpenter et al., 2017). The blue and red semicontinuous lines represent the observed
HIV and AIDS data, respectively, the solid orange and violet lines represent the medians and the shaded
area represents the 95% probability bands for the expectedvalueforthestatevariables:Infectedand
Sick people, respectively. AIDS, acquired immunodeficiency syndrome; HIV, human immunodeficiency
virus
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πyθπIθπAθ()= (
˜)(
˜)
,
i
n
ii
obs
=1

where the mean
μ
of the Poisson distribution Kθ(Φ())
i1
, is given by the product of the parameter
K1
and
Φ
iat time
t
t=
i
. Analogously, the mean for the Poisson distribution Kθ(Ξ()
)
i2
is given by the
product of the parameter
K2
and
Ξ
i
at time
t
t=
i
. For the prior distribution, we select Lognormal
distribution for the
β
parameter and Uniform distributions for the rest of parameters to estimate:
σ
KK,,
1
2
using the Stan package (Carpenter et al., 2017) (Gamma distributions using the Twalk
package Christen & Fox, 2010). The description of the parameters in (1) and their corresponding
hyperparameters' (range column) are given on Table 1.
FIGURE 12 Joint probability density distributions of the estimated parameters within 95%
(highestposterior density) of Luxembourg. The blue lines represent the medians
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πθ ab ab ab ab()= N( , )( , )( , )( , )
.
i
n
ββ σσ KK KK
=1
12 12

(17)
The posterior distribution πθy(
)
obs
given by (15) does not have an analytical closed form since
the likelihood function, which depends on the solution of the nonlinear SIA model, does not
have an explicit solution. Then, we explore the posterior distribution using the Stan Statistics
package (Carpenter et al., 2017), general purpose Markov Chain Monte Carlo Metropolis
Hasting (MCMCMH) algorithm to sample it, the package Twalk (Christen & Fox, 2010). Both
algorithms generate samples form the posterior distribution πθy(
)
obs
that can be used to
estimate marginal posterior densities, mean, credible intervals, percentiles, variances, and
others. We refer to House et al. (2016) for a more complex MCMCMH algorithms. We point
out that the parameter estimation of the countries: Luxembourg, Czech Republic, Japan,
Croatia were done using the package Stan, and for the countries United Kingdom and Mexico
were done using the package Twalk.
We have used the interface in Python (PyStan) (Carpenter et al., 2017), specifically, we
have used the NoUTurnSampler (NUTS) method. Figure 3shows the credible intervals of the
parameters of the model (1) within 95% highestposterior density (HPD). From top to bottom:
FIGURE 13 Joint probability density distributions of the estimated parameters within 95% (highest
posterior density) of Czech Republic. The blue lines represent the medians
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Luxembourg, Czech Republic, Japan, Croatia. Figures 4and 5show the credible intervals of
parameters estimated within 95% HPD. Figures 611 show the fit for the diagnosed HIV cases
and the notified AIDS cases of Luxembourg, Czech Republic, Japan, Croatia, United Kingdom,
and Mexico. The blue and red semicontinuous lines represent the observed HIV and AIDS
data, respectively, the solid orange and violet lines represent the medians and the shaded area
represents the 95% probability bands for the expected value for the state variables: Infected class
I
and Sick class
A
, respectively, for the countries Luxembourg, Czech Republic, Japan, Croatia,
United Kingdom and Mexico, respectively. Figures 1217 show the joint probability density
distributions of the estimated parameters within 95% (HPD) of the countries Luxembourg,
Czech Republic, Japan, Croatia, United Kingdom and Mexico, respectively. The blue lines
represent the medians. Table 4shows
R
0
given by (3) for the Luxembourg, Czech Republic,
Japan, Croatia, United Kingdom, and Mexico. We performed 600,000 iterations with 300,000 of
them as burnin. Using both packages, we have done predictions 5 years in advance. Some
future work will correspond to analyze the identifiability of the parameters of model (1), as
suggested in Chowell (2017), Magal and Webb (2018), Roosa and Chowell (2019), specifically
the
β
and
K1
parameters since these parameters are multiplied in the HIV prevalence Equation
(4), thus, estimating both parameters simultaneously may lead to nonidentifiability difficulty.
FIGURE 14 Joint probability density distributions of the estimated parameters within 95%
(highestposterior density) of Japan. The blue lines represent the medians
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Similarly, it happens with
σ
and
K2
parameters since these parameters are multiplied in the
AIDS prevalence Equation (5). We have uploaded all the codes used in this paper to the
following Github link: https://github.com/kernelprieto/HIV-AIDS, for a detailed review.
5|HYPOTHESIS TESTING
In this section we address the hypothesis testing on the basic reproductive number
R
0
.To
simplify notation, we use
θR=
0
. Suppose we observe
n
i.i.d. pairs of
yIA
θ
={(
˜,˜)} ,
i
n
obs =1 some
unknown value on the parameter space
Θ
. A onesided test is defined by
Hθvs H θ:1.:>1
.
01
(18)
The hypotheses H
0
and H
1
can be replaced by models
M
0
and
M1
. A Bayesian version of th
standard pvalue can be produced once the posterior distribution is obtained. For the onesided
case in (18), the specified prior distribution of
θ
provides an a priori probability over the two
FIGURE 15 Joint probability density distributions of the estimated parameters within 95%
(highestposterior density) of Croatia. The blue lines represent the medians
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regions of the sample space of
θHp θπHp θππ::(0 1)=,:(1<<)==1
001 10
≤≤ ∞
.
The distribution
π
0
can take on an large number of forms, but the uninformative uniform
distribution is particularly useful, and many authors have suggested that lacking specific in-
formation
πpH=( is true)=
00
1
2is a useful value (Gelman et al., 2014; Shikano, 2019). One
way to decide between H
0
and H
1
is to compare pHy(
)
0obs
and pHy(
)
1obs
, and accept the
hypothesis with the higher posterior probability. To be more specific, according to the MAP
test, we choose H
0
if and only if
pHy pHy()()
,
0obs 1obs

(19)
FIGURE 16 Joint probability density distributions of the estimated parameters within 95%
(highestposterior density) of United Kingdom. The blue lines represent the medians
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FIGURE 17 Joint probability density distributions of the estimated parameters within 95%
(highestposterior density) of United Kingdom. The blue lines represent the medians
TABLE 4 Basic reproductive number
R
0
given by (3) for the countries analyzed in this manuscript
Country
R
0
Credible interval (95%)
Luxembourg 0.8383 [0.8374, 0.8455]
Czech Republic 1.8405 [1.3997, 3.5471]
Japan 1.1635 [1.1540, 1.1667]
Croatia 1.2124 [1.1200, 1.3300]
United Kingdom 0.7310 [0.6104, 0.8782]
Mexico 0.9080 [0.9067, 0.9100]
or equivalently, we choose H
0
if and only if
py H pH py HpH()()()()
.
obs 00 obs 11

(20)
Once prior probabilities are assigned, the Bayesian posterior probability is derived from the
nonnormalized region defined by the null hypothesis divided by the total nonnormalized
region, which can be derived as follows:
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pHy pH θydθ
py H θpH θ
py dθ
py H θpH θdθ
py H θpH θpy H θpH θdθ
py θπdθ
py θπdθpy θπdθ
py θdθ
py θπdθ
()=(,)
=(,)(,)
()
=
(,)(,)
[( ,)( ,)+ ( ,)( ,)]
=
()
() + ( )
=
()
()
,
0obs 0
0obs
0
obs 00
obs
0obs 00
0obs 00 obs 11
0
1
obs 0
0
1
obs 0
1obs 1
0
1
obs
0obs 0




(21)
where the part of the integral in the numerator from 1 to
contributeszerotothiscalcu-
lation since H
0
is on the righthandside of the conditionals, similar argument holds for the H
1
part. The terms inside the integrals are modified using the definition of conditional prob-
ability:
py H θpH θpy H θpθHpH py HθpθHπ( ,)( ,)= ( ,)( )( )= ( ,)( )
obs 00 obs 000
obs 000

.Inthe
last step, the simplification because ππ=
0
1
.ThisposteriorvaluepHy(
)
0obs
isapvalue
for: the probability that the null hypothesis is true, given the data and the model. Conversely,
the standard pvalue is the less revealing probability of seeing these or more extreme
data, given model and assumed true null hypothesis. The value
pHy(
)
0obs
corresponds the
area numerator of (21) and the entire probability density function is the denominator as
illustrated in fig. 7.1 of Gill (2014). Table 5shows the hypothesis testing of the Basic re-
productive number
R
0
for the countries analyzed in this manuscript. From Table 5,weobtain
that all the null hypothesis are accepted with pvalue equal to 1. The results are consistent
with Table 4sincethe95%ofthecredibleintervalwereinsidetheintervalofthenull
hypothesis.
6|BAYESIAN SENSITIVITY ANALYSIS
In this section we present an analysis of how the prior distribution selection impacts on the
posterior distribution through the Bayes Theorem for Japan's case. If the posterior does not
depend a lot on the prior distribution selection, we say that our model is robust, otherwise, we
say that our model is very sensitive to prior selection. We analyze the posterior distribution as
compromise between data and prior information (Gelman et al., 2014). We make reasonable
modifications to the assumptions in question, recompute the posterior distribution of para-
meters of interest, and observe how much the results on these posterior distributions have
changed. If the posterior distribution change highly, that is, it is very sensitive to prior dis-
tribution changes, we should collect more data in general (Carlin & Louis, 2009). We first have
done a comparison between diffuse, weakinformative and informative prior distributions.
We used uniform distributions for all the parameters to be estimated as diffuse, for the
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weakinformative, we used the LogNormal distribution for the
β
parameter (Capistran
et al., 2021), and uniform distributions for the rest of parameters, and finally, for the in-
formative distributions we used Gamma distributions for all the parameters since the Gamma
distribution is conjugate of the Poisson distribution. Table 6shows the results of the posterior
distributions of the parameters when different type (diffuse, weakinformative, informative) of
prior distributions are used. Table 7shows the result of the posterior distributions of the
parameters using different hyperparameters values for a Gamma (informative) Prior Dis-
tribution. Table 8shows the result of the posterior distributions of the parameters using dif-
ferent hyperparameters values for a Uniform (diffuse) Prior Distribution.
7|DISCUSSION AND CONCLUSIONS
In this study we formulated a mathematical model for HIV/AIDS spread keeping in mind a
simple formulation, but enough to accurately to fit the HIV/AIDS data of six countries at hand:
Luxembourg, Czech Republic, Japan, Croatia, United Kingdom, and Mexico. We presented a
shortterm (3 years) forecast of transmission of the HIV/AIDS disease using Bayesian inference
based on two software: the Stan package (Carpenter et al., 2017) and the Twalk package
(Christen & Fox, 2010). We showed credible intervals, bands projections with medians and
the joint probability distributions given as a corner. From Figures 611, we could observe that the
SIA model proposed in Section 2fit adequately the transmission of the HIV/ADIS disease.
The assumption of independence for data collected of communicable diseases over time
have been questioned in Chowell et al. (2009), Koopman and Longini (1994). This phenomenon
TABLE 5 Hypothesis testing of the Basic reproductive number
R
0
for the countries analyzed in this
manuscript
Country
H
0
(null hypothesis) pvalue Confidence level (%)
Luxembourg
R
<
1
0
1.0 100
Czech Republic
R
>
1
0
1.0 100
Japan
R
>
1
0
1.0 100
Croatia
R
>
1
0
1.0 100
United Kingdom
R
<
1
0
1.0 100
Mexico
R
<
1
0
1.0 100
TABLE 6 Posterior distribution of all the parameters to be estimated using different type of prior
distributions within 95% highestposterior density
Prior
\
posterior
β
σ
K
1
K
2
J (misfit)
Diffuse [1.05251, 1.09537] [0.89715, 0.93471] [0.03417, 0.03864] [0.01549, 0.01763] 797.92749
Weakinformative [1.05151, 1.09589] [0.89634, 0.93484] [0.03421, 0.03868] [0.01614, 0.01765] 798.14841
Informative [1.05166, 1.09528] [0.89696, 0.93458] [0.03431, 0.03874] [0.01555, 0.01768] 797.90774
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is called dependent happening, that is, observation of a single infected individual is not in-
dependent of observing other individuals in a population of interest. This phenomenon is not
significant in noncommunicable diseases. Thus, we point out that the independent happening
assumption is a limitation of our current approach to identify the risk of this infectious disease.
A solution for dependent happening to assess vaccine efficacy is addressed in Chowell et al.
(2009), called the Household Secondary Attack Rate Method. Another limitation of our current
model is that we have not included the incubation period of the disease, defined as the time
from infection with the microorganism to symptom development, which in the HIV/AIDS case
is rather long, may be one or a few decades. A study in Brookmeyer and Gail (1994) obtained
shortterm predictions of the progress disease of HIV/AIDS using the incubation period and the
backcalculation method. We will try to incorporate the incubation period in a future work.
We point out that for Luxembourg, Mexico and UK fits, we estimated more parameters,
ξ
ξ,
1
2
, than the cases Czech Republic, Japan, Croatia, to obtain a better fit for them. Inclusive,
we performed a forecast of 2 years for the UK case, again, to obtain a more accurate forecast.
The UK database contains only 20 data. Thus, we should try to collect more data points to
increase the projection period of time.
We also did a sensitivity analysis of the main quantities of a epidemic, being the Basic
Reproductive Number and the component of HIV cases of the endemic point, that is,
RI,
*
0
and
the prevalence of HIV (Φ
T
). We could observe in Tables 2and 3that the most relevant
parameters are
β
and
σ
for I
*
and the
β
and
K1
for Φ
T
. We also performed a Bayesian Sensitivity
Analysis for Japan case in Section 6, observing the Tables 68, we concluded that the Bayesian
forecast is robust. An interesting result was that our estimation of
R
0
for each country given on
Table 4was supported in Section 5, where we found that an HIV/AIDS certain outbreak will be
expected for Czech Republic and Japan, as is shown in Table 5.
Finally, in modeling problems of real phenomena with ODEs, most attention has been paid
to the problem of simulating the state variables with given parameters, but it is also often
required to study the inverse problem, that is, to estimate or predict the parameters using some
TABLE 7 Posterior distribution of all the parameters to be estimated using different hyperparameters
values for a Gamma (informative) Prior Distribution within 95% highestposterior density
Prior
\
posterior
β
σ
K
1
K
2
J (misfit)
Gamma(2,2) [1.05166, 1.09528] [0.89696, 0.93458] [0.03431, 0.03874] [0.01555, 0.01768] 797.90774
Gamma(3, 1) [1.05135, 1.09555] [0.89658, 0.93510] [0.03431, 0.03884] [0.01555, 0.01769] 797.71300
Gamma(1, 3) [1.05099, 1.09461] [0.89632, 0.93377] [0.03423, 0.03871] [0.01551, 0.01766] 797.73887
TABLE 8 Posterior distribution of all the parameters to be estimated using different hyperparameters
values for a Uniform (diffuse) Prior Distribution within 95% highestposterior density
Prior\posterior
β
σ
K
1
K
2
J (misfit)
(0, 2) (0, 1.5
)
βσ
 [1.05201, 1.09526] [0.89694, 0.934467] [0.03415, 0.03865] [0.01547, 0.01763] 797.77474
(0, 1) (0, 1
)
KK
12
 [1.05151, 1.09589] [0.89634, 0.93484] [0.03420, 0.03869] [0.01549, 0.01764] 797.96628
(0, 2) (0, 1
)
βK2
 [1.05206, 1.09539] [0.89695, 0.934574] [0.03419, 0.03869] [0.01549, 0.01764] 798.65047
PRIETO AND ROMEROLEITON Natural Resource Modeling
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measures of state variables. In this second case, before applying rigorous parameter estimation
methods such as the ones we have used in this study, a first challenge is to verify whether the
model parameters are identifiable based on the measurements of the output variables. This first
challenge can be overcome by an identifiability analysis before deal with the inverse problem.
Having said that, some important and recent references on ODEs identifiability analysis can be
found in Calvetti and Somersalo (2018), Girolami (2008), Miao et al. (2008,2011), Wu et al.
(2008). In particular, in this study, the identifiability of the parameters
β
σξ,,
1
, and
ξ2
could be
proved using the Multiple Time Points Method (MTP), which is based on the assumption that
the measurements of one of the state variables and its higherorder derivatives are known in a
small number of points (usually
t
tt,,
12 5
), since the structure of our HIV/AIDS model is
similar to the model proposed in Wu et al. (2008). The challenge presented here, is that we also
estimate the parameters
K1
and
K2
, from equations (4) and (5) (which are used to determine the
prevalence of both HIV and AIDS) and they cannot be explicitly entered within model (1).
Nevertheless, we plan to address this identifiability problem as our current formulation in a
future work. The biological importance of estimating
K1
and
K2
is that they involve the infected
individuals' detection probability, that is, they account for the undetermined true number of
HIV/AIDS cases since some reported HIV/AIDS cases occur once the infected individuals start
presenting symptoms. In addition, if the infection period of a disease happens to be different to
the resolution of the data as it occurs for HIV/AIDS; the period of infection are some weeks
(Hollingsworth et al., 2008; Perelson & Nelson, 1999) and the resolution of the data are years,
then the expressions (4) and (5) of the current manuscript are required (Keeling &
Rohani, 2008; Ponciano & Capistrán, 2011). Same assumptions and estimations have been
considered in Capistran et al. (2021) and Ponciano and Capistrán (2011).
We have uploaded all the codes used in this paper to the following Github link: https://
github.com/kernelprieto/HIV-AIDS, for a detailed review.
CONFLICT OF INTERESTS
The authors declare that there are no conflict of interests.
AUTHOR CONTRIBUTIONS
Kernel Prieto: conceptualization (equal); data curation (equal); formal analysis (equal);
funding acquisition (equal); investigation (equal); methodology (equal); project administration
(equal); resources (equal); software (equal); supervision (equal), validation (equal); visualiza-
tion (equal); writingoriginal draft (equal); and writingreview and editing (equal). Jhoana P.
RomeroLeiton: conceptualization (equal); formal analysis (equal); investigation (equal);
methodology (equal); supervision (equal); validation (equal); writingoriginal draft (equal); and
writingreview and editing (equal).
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available in the Joint United Nations Pro-
gramme on HIV/AIDS, the National Institute of Infectious Diseases, Centro Nacional para la
Prevención y Control del VIH y el SIDA form the government of Mexico and the government of
United Kingdom. These data were derived from the following resources available in the public
domain: https://www.unaids.org/en;https://www.niid.go.jp/niid/en/865-iasr/10489-488te.html;
https://www.gob.mx/censida/documentos/epidemiologia-registro-nacional-de-casos-de-sida;
https://www.gov.uk/government/publications/hiv-in-the-united-kingdom
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How to cite this article: Prieto, K., & RomeroLeiton, J. P. (2021). Current forecast of
HIV/AIDS using Bayesian inference. Natural Resource Modeling, 34, e12332.
https://doi.org/10.1111/nrm.12332
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