Nowadays, the problem of finding families of graphs for which one may ensure the existence of a vertex-labeling and/or an edge-labeling based on a certain class of integers, constitutes a challenge for researchers in both number and graph theory. In this paper, we focus on those vertex-labelings whose induced multiplicative edge-labeling assigns hyper totient numbers to the edges of the graph. In this way, we introduce and characterize the notions of hyper totient graph and restricted hyper totient graph. In particular, we prove that every finite graph is a hyper totient graph and we determine under which assumptions the following families of graphs constitute restricted hyper totient graphs: complete graphs, star graphs, complete bipartite graphs, wheel graphs, cycles, paths, fan graphs and friendship graphs.
Citation: Shahbaz Ali, Muhammad Khalid Mahmmod, Raúl M. Falcón. A paradigmatic approach to investigate restricted hyper totient graphs[J]. AIMS Mathematics, 2021, 6(4): 3761-3771. doi: 10.3934/math.2021223
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Nowadays, the problem of finding families of graphs for which one may ensure the existence of a vertex-labeling and/or an edge-labeling based on a certain class of integers, constitutes a challenge for researchers in both number and graph theory. In this paper, we focus on those vertex-labelings whose induced multiplicative edge-labeling assigns hyper totient numbers to the edges of the graph. In this way, we introduce and characterize the notions of hyper totient graph and restricted hyper totient graph. In particular, we prove that every finite graph is a hyper totient graph and we determine under which assumptions the following families of graphs constitute restricted hyper totient graphs: complete graphs, star graphs, complete bipartite graphs, wheel graphs, cycles, paths, fan graphs and friendship graphs.
In the United States of America, human papillomavirus (HPV) is the most common sexually transmitted infection (STI) in males and females [8]. Most sexually active males and females will get at least one type of HPV infection at some point in their lives [5]. In the United States, about 79 million are currently infected with HPV and about 14 million people become newly infected each year [8]. There are more than 150 different types of HPV [7]. Health problems related to HPV include genital warts and cancer. Most people infected with genital HPV do not know they are infected and never develop symptoms or health problems from it. Some people find out they have HPV when they get genital warts. Females may find out they have HPV when they get an abnormal Pap test result during cervical cancer screening. Others may only find out once they have developed more serious problems from HPV, such as cancer [5]. Most HPV infections cause no symptoms and are not clinically significant, but persistent infection can lead to disease or cancer.
Mathematical epidemic models have been used to study HPV infections in various populations. For example, Alsaleh and Gummel [1] in a recent paper, used a deterministic model to assess the impact of vaccination on both high-risk and low-risk HPV infection types. Ribassin-Majed and Clemencon [14] used a deterministic mathematical model to assess the impact of vaccination on non-cancer causing HPV (6/11) in French males and females. Lee and Tameru [13] used a deterministic model to assess the impact of HPV on cervical cancer in African American females (AAF). In all these studies, the HPV models have a constant recruitment function for the demographic equations.
In this paper, we use a two-sex HPV model with "fitted" logistic demographics to study the HPV disease dynamics in AAF and African American males (AAM) of 16 years and older. Using US Census Bureau data for AAF and AAM populations, we illustrate that the "fitted" logistic demographic equation captures the African American (AA) population better than the constant recruitment demographic equation. We compute the basic reproduction number,
The paper is organized as follows: In Section 2, we introduce a demographic equation for AAF (respectively, AAM) and we "fit" it to the US Census Bureau data of AAF (respectively, AAM) of 16 years and older. We introduce, in Section 3, a two-sex African American HPV model. In Section 4, we study disease-free equilibria and compute the basic reproduction number
In [14], Ribassin-Majed et al. used a HPV model with constant recruitment rate in the demographic equation to study HPV disease dynamics in male and female populations of France. In the absence of the HPV disease, the demographic equation of their model is the following ordinary differential equation:
dNdt=Λ−μN, | (1) |
where
In this paper, we use logistic models that are "fitted" to the
2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | |
AAF population 16 years and older [12,16] | 13,825,055 | 14,041,520 | 14,259,413 | 14,473,927 | 14,707,490 | 14,952,963 | 15,224,330 | 15,486,244 | 15,743,096 | 15,992,822 | 16,176,048 | 16,471,449 | 16,696,303 | 16,918,225 | 17,139,986 |
AAF total population [12,16] | 18,787,192 | 19,013,351 | 19,229,855 | 19,434,349 | 19,653,829 | 19,882,081 | 20,123,789 | 20,374,894 | 20,626,043 | 20,868,282 | 21,045,595 | 21,320,013 | 21,543,051 | 21,767,521 | 21,988,307 |
2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | |
AAM population 16 years and older [12,16] | 11,909,507 | 12,124,810 | 12,332,791 | 12,518,252 | 12,756,370 | 12,996,123 | 13,266,163 | 13,517,841 | 13,765,707 | 14,006,594 | 14,181,655 | 14,490,027 | 14,724,637 | 14,950,933 | 15,176,189 |
AAM total population [12,16] | 17,027,514 | 17,249,678 | 17,454,795 | 17,631,747 | 17,856,753 | 18,079,607 | 18,319,259 | 18,560,639 | 18,803,371 | 19,033,988 | 19,260,298 | 19,487,042 | 19,719,238 | 19,945,997 | 20,169,931 |
From Tables 1 and 2, we note that both male and female populations of African Americans of 16 years and older as well as the total populations exhibit increasing trends from 2000 to 2014. In the next section, we use a logistic differential equation model to "capture" the AA population data of Tables 1 and 2.
In [3], Brauer and Castillo-Chavez used the logistic equation "fitted" to United States Census Bureau data to model the total United States population. We use the same approach to "fit" the solution of the following logistic equation to the AAF population of 16 years and older (see Table 1) and the AAM population of 16 years and older (see Table 2).
dNidt=(ri−μi)Ni(1−NiKi(ri−μi)/ri),t≥0, | (2) |
where index
Ni(t)=0andNi(t)=Ki(ri−μi)/ri1+(Ki(ri−μi)/riN0i−1)e−(ri−μi)t. | (3) |
Let
Rdi=riμi. |
K∗i=Ki(ri−μi)ri,ast→∞, |
and the population persists. However, when
Equation (2) gives the per capita growth rate,
dNi/dtNi=(ri−μi)(1−NiKi(ri−μi)/ri). | (4) |
Using the
Fitting the line to the curve gives
Using our estimates, we express the nontrivial solution (3) of the logistic growth model for AAF of 16 years and older as
Nf(t)=52,967,1171+2.772176873e−0.021298978t, | (5) |
and for AAM of 16 years and older as
Nm(t)=99,484,6731+7.205050059e−0.019877926t, | (6) |
where year 2001 is taken as
The plot of the data of AAF of 16 years or older and solution (5) in Figure 1 show that our "fitted" model captures the AAF data of Table 1. Similarly, the plot of the data of AAM of 16 years or older and solution (6) in Figure 2 show that our "fitted" model captures the AAM data of Table 2.
When the population in equation (1) consists only of AAF (
Ni(t)=Λiμi+(N0i−Λiμi)e−μit. | (7) |
Using the initial condition
Nf(t)=2,315,038,811−2,300,997,291e−0.007266t, | (8) |
and
Nm(t)=1,887,409,019−1,875,284,209e−0.008227t. | (9) |
In Figure 1, we compare the "fitted" solution (8) of Model (1), and our "fitted" solution (5) of Model (2), to the US Census Bureau data in Table 1. Figure 1 shows that Model (2), the "fitted" logistic model, captures better the
Similarly, Figure 2 shows that, as in the female population, the "fitted" solution (9) of Model (1) over estimates the AAM US Census Bureau data while our "fitted" solution (5) of Model (2) captures it.
To study the HPV dynamics in male and female African American populations of 16 years and older, we assume that the total AAF population (respectively, total AAM population) of 16 years and older is governed by Model (2) with
{dSfdt=rfNf(1−NfKf)−σfSfImNf+Nm+δIf−μfSf,dSmdt=rmNm(1−NmKm)−σmSmIfNf+Nm+δIm−μmSm,dIfdt=σfSfImNf+Nm−(δ+μf)If,dImdt=σmSmIfNf+Nm−(δ+μm)Im, | (10) |
where
Parameter (per day) | Description | Reference |
Death rate for AAF population | [6] | |
Death rate for AAM population | [6] | |
Clearance rate | [11] | |
Intrinsic growth rate for AAF population | Estimated | |
Intrinsic growth rate for AAM population | Estimated | |
Carrying capacity for AAF population | Estimated | |
Carrying capacity for AAF population | Estimated | |
Infection rate for AAF population | [1] | |
Infection rate for AAM population | [1] |
Notice that since
Nf(t)=Sf(t)+If(t)andNm(t)=Sm(t)+Im(t), |
adding the
Consequently, in Model (10), the total AAF and AAM populations, governed by our "fitted" logistic equations (5) and (6) are bounded. We will study Model (10) with the parameter values listed in Table 3 and with the initial conditions listed in Table 4.
= | 8,618,960 | |
= | 7,119,370 | |
= | 5,422,560 | |
= | 5,005,440 |
Notice that in Table 4,
In this section, we show that Model (10) is well-posed. In particular, we obtain that all orbits are nonnegative and there is no population explosion in Model (10).
Theorem 3.1. All solutions of Model (10) are nonnegative and bounded.
Proof. Consider the following nonnegative initial conditions
If
Similarly, if
If
Similarly, if
Recall that
Let
Then Model (10) is equivalent to
dXdt=F(X). |
In AAF population,
Unlike in [14], it is possible for Model (10) to exhibit up to four disease-free equilibrium points (DFEs), where
Since
Since
To determine the stability of
J|Pf0=[μf−rf02μf−rf+δ−σf0rm−μm02μm−rm+δ00−(δ+μf)σf000−(δ+μm)]. |
Similarly, to determine the stability of
J|Pm0=[rf−μf0rf+δ00μm−rm−σm2μm−rm+δ00−(δ+μf)000σm−(δ+μm)]. |
To determine the stability of
{dIfdt=σf(K∗f−If)ImK∗f+K∗m−(δ+μf)If,dImdt=σm(K∗m−Im)IfK∗f+K∗m−(δ+μm)Im,dSfdt=rfK∗f(1−K∗fKf)−σfSfImK∗f+K∗m+δIf−μfSf,dSmdt=rmK∗m(1−K∗mKm)−σmSmIfK∗f+K∗m+δIm−μmSm. | (11) |
Using the next generation matrix method [17] we obtain the following two matrices
F=[σf(K∗f−If)ImK∗f+K∗mσm(K∗m−Im)IfK∗f+K∗m00] |
andV=[(δ+μf)If(δ+μm)Im−rfK∗f(1−K∗fKf)+σfSfImK∗f+K∗m−δIf+μfSf−rmK∗m(1−K∗mKm)+σmSmIfK∗f+K∗m−δIm+μmSm]. |
Then, using the Jacobian matrices of
DF(Qfm)=[F000]andDV(Qfm)=[V0WU], |
where
Hence,
FV−1=1K∗f+K∗m[0σfK∗fδ+μmσmK∗mδ+μf0]. |
By the next generation matrix method [17], the reproduction number for Model (10),
R0=ρ(FV−1)=√R0fR0m,where |
R0f=σfK∗f(δ+μf)(K∗f+K∗m)andR0m=σmK∗m(δ+μm)(K∗f+K∗m). |
When
It is interesting to note that in [13], Lee and Tameru, obtained
To find an effective mitigation strategy that seeks to reduce HPV infection in AA population within the shortest time possible, in the next section, we use sensitivity analysis to study the impact of each model parameter on
Sensitivity indices are used to measure the relative change in a state variable when a parameter changes. Typically, 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 differential function of the parameter, the sensitive index may be alternatively defined using partial derivatives [4,10,18,19].
Definition 4.1([4,10,18,19]). The normalized forward sensitivity index of a variable,
Υuq:=∂u∂q×qu. |
We use Definition 4.1 to derive the sensitivity indices of the basic reproduction number
Increasing (respectively, decreasing) the clearance rate,
To illustrate the impact of HPV on AAF and AAM populations of 16 years and older, we simulate Model (10) with the parameter values listed in Table 3 and the initial conditions listed in Table 4.
Parameter | Sensitivity index of | Order of Importance |
-0.9915 | 1 | |
0.5000 | 2 | |
0.5000 | 3 | |
0.1526 | 4 | |
-0.1526 | 5 | |
-0.0631 | 6 | |
0.0586 | 7 | |
-0.0561 | 8 | |
0.0520 | 9 |
Simulations of our HPV Model (10) are performed using Matlab software, and are illustrated in Figure 4. Figure 4 (a) shows that susceptible population of AAF of 16 years and older,
To protect against HPV infections, HPV vaccines are available for males and females. Gardasil and Cervarix are two HPV vaccines that have market approval in many countries. Next, we introduce an extension of Model (10) with vaccinated male and female classes. We will use the extended model to study the impact of vaccination on Figure 4.
To introduce HPV vaccination in AAF and AAM populations of Model (10), we let
As in [14], we divide the AA population into eight compartments.
{dSfdt=rf(1−pf)Nf(1−NfKf)−σfSfIm+IvmNf+Nm+δIf−μfSf,dSvfdt=pfrfNf(1−NfKf)−(1−τ)σfSvfIm+IvmNf+Nm+δIvf−μfSvf,dSmdt=rm(1−pm)Nm(1−NmKm)−σmSmIf+IvfNf+Nm+δIm−μmSm,dSvmdt=pmrmNm(1−NmKm)−(1−τ)σmSvmIf+IvfNf+Nm+δIvm−μmSvm,dIfdt=σfSfIm+IvmNf+Nm−(δ+μf)If,dIvfdt=(1−τ)σfSvfIm+IvmNf+Nm−(δ+μf)Ivf,dImdt=σmSmIf+IvfNf+Nm−(δ+μm)Im,dIvmdt=(1−τ)σmSvmIf+IvfNf+Nm−(δ+μm)Ivm, | (12) |
where
= | 5,257,566 | |
= | 3,361,394 | |
= | 5,667,019 | |
= | 1,452,351 | |
= | 5,086,421 | |
= | 336,139 | |
= | 4,860,205 | |
= | 145,235 |
Note that in Table 6,
Proceeding exactly as in Theorem 3.1, we obtain the following result.
Theorem 5.1. All solutions of Model (12) are nonnegative and bounded.
Notice that when all vaccinated classes are missing
From Model (12), the demographic equations for the female and male total populations are respectively the following:
{dNfdt=rfNf(1−NfKf)−μfNf,dNmdt=rmNm1−NmKm)−μmNm. | (13) |
The equilibrium points of Model (13) are
As in Model (10), to state the "limiting" system of Model (12), we replace
{dSfdt=rf(1−pf)K∗f(1−K∗fKf)−σfSfIm+IvmK∗f+K∗m+δIf−μfSf,dSmdt=rm(1−pm)K∗m(1−K∗mKm)−σmSmIf+IvfK∗f+K∗m+δIm−μmSm,dIfdt=σfSfIm+IvmK∗f+K∗m−(δ+μf)If,dIvfdt=(1−τ)σf(K∗f−Sf−If−Ivf)Im+IvmK∗f+K∗m−(δ+μf)Ivf,dImdt=σmSmIf+IvfK∗f+K∗m−(δ+μm)Im,dIvmdt=(1−τ)σm(K∗m−Sm−Im−Ivm)If+IvfK∗f+K∗m−(δ+μm)Ivm, | (14) |
DFE of System (14) is
F=[σfSfIm+IvmK∗f+K∗m(1−τ)σf(K∗f−Sf−If−Ivf)Im+IvmK∗f+K∗mσmSmIf+IvfK∗f+K∗m(1−τ)σm(K∗m−Sm−Im−Ivm)If+IvfK∗f+K∗m00] |
andV=[(δ+μf)If(δ+μf)Ivf(δ+μm)Im(δ+μm)Ivm−rf(1−pf)K∗f(1−K∗fKf)+σfSfIm+IvmK∗f+K∗m−δIf+μfSf−rm(1−pm)K∗m(1−K∗mKm)+σmSmIf+IvfK∗f+K∗m−δIm+μmSm]. |
Let
DF(Q)=[F000]andDV(Q)=[V0WU], |
where
F=[00σf(1−pf)K∗fK∗f+K∗mσf(1−pf)K∗fK∗f+K∗m00(1−τ)σfpfK∗fK∗f+K∗m(1−τ)σfpfK∗fK∗f+K∗mσm(1−pm)K∗mK∗f+K∗mσm(1−pm)K∗mK∗f+K∗m00(1−τ)σmpmK∗mK∗f+K∗m(1−τ)σmpmK∗mK∗f+K∗m00], |
V=[δ+μf0000δ+μf0000δ+μm0000δ+μm],U=[μf00μm] |
andW=[−δ0σf(1−pf)K∗fK∗f+K∗mσf(1−pf)K∗fK∗f+K∗mσm(1−pm)K∗mK∗f+K∗mσm(1−pm)K∗mK∗f+K∗m−δ0]. |
Hence,
FV−1=[00σf(1−pf)K∗f(δ+μm)(K∗f+K∗m)σf(1−pf)K∗f(δ+μm)(K∗f+K∗m)00(1−τ)σfpfK∗f(δ+μm)(K∗f+K∗m)(1−τ)σfpfK∗f(δ+μm)(K∗f+K∗m)σm(1−pm)K∗m(δ+μf)(K∗f+K∗m)σm(1−pm)K∗m(δ+μf)(K∗f+K∗m)00(1−τ)σmpmK∗m(δ+μf)(K∗f+K∗m)(1−τ)σmpmK∗m(δ+μf)(K∗f+K∗m)00]. |
By the next generation matrix method [17], the reproduction number for Model (12),
Rv0=ρ(FV−1)=√Rv0fRv0m,where |
Rv0f=(1−τpf)σfK∗f(δ+μf)(K∗f+K∗m)andRv0m=(1−τpm)σmK∗m(δ+μm)(K∗f+K∗m). |
Rv0f=(1−τpf)R0fandRv0m=(1−τpm)R0m. |
Since
Thus, adopting a HPV vaccination program decreases the basic reproduction number,
Using the parameter values of Table 3,
In the next section, we use sensitivity analysis to illustrate the impact of model parameters on
We use Definition 4.1 to derive the sensitivity indices of the basic reproduction number
From Table 7 and Figure 5, increasing (respectively, decreasing) the clearance rate,
Parameter | Sensitivity index of | Order of Importance |
-0.9915 | 1 | |
0.5000 | 2 | |
0.5000 | 3 | |
-0.3829 | 4 | |
-0.2704 | 5 | |
0.1526 | 6 | |
-0.1526 | 7 | |
-0.1124 | 8 | |
-0.0631 | 9 | |
0.0586 | 10 | |
-0.0561 | 11 | |
0.0520 | 12 |
To illustrate the impact of HPV on AAF and AAM populations of 16 years and older when a vaccination program is applied with
Simulations of our HPV Model (12) are performed using Matlab software, and are illustrated in Figure 6. Figures 6 (a-b) show that susceptible population of AAF of 16 years and older,
To study the impact of the presence of the vaccinated class on the results of Figure 4, we simulate Model (12) using initial conditions in Table 8. For these simulations of Model (12), we keep all the parameter values at their current values in Figure 4, where
![]() | |||
5,257,566 | 4,309,480 | 2,585,688 | |
3,361,394 | 4,309,480 | 6,033,272 | |
5,667,019 | 3,559,685 | 2,135,811 | |
1,452,351 | 3,559,685 | 4,983,559 | |
5,086,421 | 4,991,612 | 4,819,233 | |
336,139 | 430,948 | 603,327 | |
4,860,205 | 4,649,472 | 4,507,084 | |
145,235 | 355,969 | 498,356 |
Note that in Figure 6 and Table 6,
In AA population,
Furthermore, in both AAF and AAM populations, Figures 9 and 10 show that the number of infected populations is lower when the population is under a vaccination policy than when the population is not being vaccinated. Thus, HPV vaccines that provide partial immunity to both AAF and AAM populations of 16 years and older not only lower the number of HPV infectives but increase the number of susceptibles in both female and male populations.
Furthermore, we obtained in Figures 7-10 that the increase (respectively, decrease) in the susceptible (respectively, HPV infective) populations is larger when a bigger proportion of the population is vaccinated.
We use a two-sex HPV model with "fitted" logistic demographics to study HPV disease dynamics in AAF and AAM populations of 16 years and older. In agreement with Lee and Tameru [13], we obtained that in AA population,
Using sensitivity analysis on
● Increasing (respectively, decreasing) the clearance rate,
● Increasing (respectively, decreasing) the infection rate of the AAF population,
● Increasing (respectively, decreasing) the infection rate of the AAM population,
In the second part of the paper, we extended our model to include vaccination classes in both male and female AA populations of 16 years and older. We obtained that in AA population when the vaccination program is implemented,
● Increasing (respectively, decreasing) the clearance rate,
● Increasing (respectively, decreasing) the infection rate of the AAF population,
● Increasing (respectively, decreasing) the infection rate of the AAM population,
● Increasing (respectively, decreasing) the success rate of vaccination,
● Increasing (respectively, decreasing) the proportion of HPV vaccinated females,
Furthermore, using the extended model with vaccination we obtained the following results:
● Adopting a vaccination policy lowers HPV infections in both AAF and AAM populations.
● Vaccinating a larger proportion of AAF and AAM populations leads to fewer cases of HPV infections in the vaccinated population.
This research was partially supported by National Science Foundation under grant DUE-1439758.
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3. | Xinwei Wang, Haijun Peng, Boyang Shi, Dianheng Jiang, Sheng Zhang, Biaosong Chen, Optimal vaccination strategy of a constrained time-varying SEIR epidemic model, 2019, 67, 10075704, 37, 10.1016/j.cnsns.2018.07.003 | |
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2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | |
AAF population 16 years and older [12,16] | 13,825,055 | 14,041,520 | 14,259,413 | 14,473,927 | 14,707,490 | 14,952,963 | 15,224,330 | 15,486,244 | 15,743,096 | 15,992,822 | 16,176,048 | 16,471,449 | 16,696,303 | 16,918,225 | 17,139,986 |
AAF total population [12,16] | 18,787,192 | 19,013,351 | 19,229,855 | 19,434,349 | 19,653,829 | 19,882,081 | 20,123,789 | 20,374,894 | 20,626,043 | 20,868,282 | 21,045,595 | 21,320,013 | 21,543,051 | 21,767,521 | 21,988,307 |
2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | |
AAM population 16 years and older [12,16] | 11,909,507 | 12,124,810 | 12,332,791 | 12,518,252 | 12,756,370 | 12,996,123 | 13,266,163 | 13,517,841 | 13,765,707 | 14,006,594 | 14,181,655 | 14,490,027 | 14,724,637 | 14,950,933 | 15,176,189 |
AAM total population [12,16] | 17,027,514 | 17,249,678 | 17,454,795 | 17,631,747 | 17,856,753 | 18,079,607 | 18,319,259 | 18,560,639 | 18,803,371 | 19,033,988 | 19,260,298 | 19,487,042 | 19,719,238 | 19,945,997 | 20,169,931 |
Parameter (per day) | Description | Reference |
Death rate for AAF population | [6] | |
Death rate for AAM population | [6] | |
Clearance rate | [11] | |
Intrinsic growth rate for AAF population | Estimated | |
Intrinsic growth rate for AAM population | Estimated | |
Carrying capacity for AAF population | Estimated | |
Carrying capacity for AAF population | Estimated | |
Infection rate for AAF population | [1] | |
Infection rate for AAM population | [1] |
= | 8,618,960 | |
= | 7,119,370 | |
= | 5,422,560 | |
= | 5,005,440 |
Parameter | Sensitivity index of | Order of Importance |
-0.9915 | 1 | |
0.5000 | 2 | |
0.5000 | 3 | |
0.1526 | 4 | |
-0.1526 | 5 | |
-0.0631 | 6 | |
0.0586 | 7 | |
-0.0561 | 8 | |
0.0520 | 9 |
= | 5,257,566 | |
= | 3,361,394 | |
= | 5,667,019 | |
= | 1,452,351 | |
= | 5,086,421 | |
= | 336,139 | |
= | 4,860,205 | |
= | 145,235 |
Parameter | Sensitivity index of | Order of Importance |
-0.9915 | 1 | |
0.5000 | 2 | |
0.5000 | 3 | |
-0.3829 | 4 | |
-0.2704 | 5 | |
0.1526 | 6 | |
-0.1526 | 7 | |
-0.1124 | 8 | |
-0.0631 | 9 | |
0.0586 | 10 | |
-0.0561 | 11 | |
0.0520 | 12 |
![]() | |||
5,257,566 | 4,309,480 | 2,585,688 | |
3,361,394 | 4,309,480 | 6,033,272 | |
5,667,019 | 3,559,685 | 2,135,811 | |
1,452,351 | 3,559,685 | 4,983,559 | |
5,086,421 | 4,991,612 | 4,819,233 | |
336,139 | 430,948 | 603,327 | |
4,860,205 | 4,649,472 | 4,507,084 | |
145,235 | 355,969 | 498,356 |
2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | |
AAF population 16 years and older [12,16] | 13,825,055 | 14,041,520 | 14,259,413 | 14,473,927 | 14,707,490 | 14,952,963 | 15,224,330 | 15,486,244 | 15,743,096 | 15,992,822 | 16,176,048 | 16,471,449 | 16,696,303 | 16,918,225 | 17,139,986 |
AAF total population [12,16] | 18,787,192 | 19,013,351 | 19,229,855 | 19,434,349 | 19,653,829 | 19,882,081 | 20,123,789 | 20,374,894 | 20,626,043 | 20,868,282 | 21,045,595 | 21,320,013 | 21,543,051 | 21,767,521 | 21,988,307 |
2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | |
AAM population 16 years and older [12,16] | 11,909,507 | 12,124,810 | 12,332,791 | 12,518,252 | 12,756,370 | 12,996,123 | 13,266,163 | 13,517,841 | 13,765,707 | 14,006,594 | 14,181,655 | 14,490,027 | 14,724,637 | 14,950,933 | 15,176,189 |
AAM total population [12,16] | 17,027,514 | 17,249,678 | 17,454,795 | 17,631,747 | 17,856,753 | 18,079,607 | 18,319,259 | 18,560,639 | 18,803,371 | 19,033,988 | 19,260,298 | 19,487,042 | 19,719,238 | 19,945,997 | 20,169,931 |
Parameter (per day) | Description | Reference |
Death rate for AAF population | [6] | |
Death rate for AAM population | [6] | |
Clearance rate | [11] | |
Intrinsic growth rate for AAF population | Estimated | |
Intrinsic growth rate for AAM population | Estimated | |
Carrying capacity for AAF population | Estimated | |
Carrying capacity for AAF population | Estimated | |
Infection rate for AAF population | [1] | |
Infection rate for AAM population | [1] |
= | 8,618,960 | |
= | 7,119,370 | |
= | 5,422,560 | |
= | 5,005,440 |
Parameter | Sensitivity index of | Order of Importance |
-0.9915 | 1 | |
0.5000 | 2 | |
0.5000 | 3 | |
0.1526 | 4 | |
-0.1526 | 5 | |
-0.0631 | 6 | |
0.0586 | 7 | |
-0.0561 | 8 | |
0.0520 | 9 |
= | 5,257,566 | |
= | 3,361,394 | |
= | 5,667,019 | |
= | 1,452,351 | |
= | 5,086,421 | |
= | 336,139 | |
= | 4,860,205 | |
= | 145,235 |
Parameter | Sensitivity index of | Order of Importance |
-0.9915 | 1 | |
0.5000 | 2 | |
0.5000 | 3 | |
-0.3829 | 4 | |
-0.2704 | 5 | |
0.1526 | 6 | |
-0.1526 | 7 | |
-0.1124 | 8 | |
-0.0631 | 9 | |
0.0586 | 10 | |
-0.0561 | 11 | |
0.0520 | 12 |
![]() | |||
5,257,566 | 4,309,480 | 2,585,688 | |
3,361,394 | 4,309,480 | 6,033,272 | |
5,667,019 | 3,559,685 | 2,135,811 | |
1,452,351 | 3,559,685 | 4,983,559 | |
5,086,421 | 4,991,612 | 4,819,233 | |
336,139 | 430,948 | 603,327 | |
4,860,205 | 4,649,472 | 4,507,084 | |
145,235 | 355,969 | 498,356 |