Citation: Wisdom S. Avusuglo, Kenzu Abdella, Wenying Feng. Stability analysis on an economic epidemiological model with vaccination[J]. Mathematical Biosciences and Engineering, 2017, 14(4): 975-999. doi: 10.3934/mbe.2017051
[1] | Yi Jiang, Kristin M. Kurianski, Jane HyoJin Lee, Yanping Ma, Daniel Cicala, Glenn Ledder . Incorporating changeable attitudes toward vaccination into compartment models for infectious diseases. Mathematical Biosciences and Engineering, 2025, 22(2): 260-289. doi: 10.3934/mbe.2025011 |
[2] | Eunha Shim . Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1615-1634. doi: 10.3934/mbe.2013.10.1615 |
[3] | Alessia Andò, Dimitri Breda, Giulia Gava . How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology. Mathematical Biosciences and Engineering, 2020, 17(5): 5059-5084. doi: 10.3934/mbe.2020273 |
[4] | Jinliang Wang, Gang Huang, Yasuhiro Takeuchi, Shengqiang Liu . Sveir epidemiological model with varying infectivity and distributed delays. Mathematical Biosciences and Engineering, 2011, 8(3): 875-888. doi: 10.3934/mbe.2011.8.875 |
[5] | Thomas Torku, Abdul Khaliq, Fathalla Rihan . SEINN: A deep learning algorithm for the stochastic epidemic model. Mathematical Biosciences and Engineering, 2023, 20(9): 16330-16361. doi: 10.3934/mbe.2023729 |
[6] | Rama Seck, Diène Ngom, Benjamin Ivorra, Ángel M. Ramos . An optimal control model to design strategies for reducing the spread of the Ebola virus disease. Mathematical Biosciences and Engineering, 2022, 19(2): 1746-1774. doi: 10.3934/mbe.2022082 |
[7] | Suxia Zhang, Hongbin Guo, Robert Smith? . Dynamical analysis for a hepatitis B transmission model with immigration and infection age. Mathematical Biosciences and Engineering, 2018, 15(6): 1291-1313. doi: 10.3934/mbe.2018060 |
[8] | Antonella Lupica, Piero Manfredi, Vitaly Volpert, Annunziata Palumbo, Alberto d'Onofrio . Spatio-temporal games of voluntary vaccination in the absence of the infection: the interplay of local versus non-local information about vaccine adverse events. Mathematical Biosciences and Engineering, 2020, 17(2): 1090-1131. doi: 10.3934/mbe.2020058 |
[9] | Holly Gaff, Elsa Schaefer . Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences and Engineering, 2009, 6(3): 469-492. doi: 10.3934/mbe.2009.6.469 |
[10] | Ayako Suzuki, Hiroshi Nishiura . Transmission dynamics of varicella before, during and after the COVID-19 pandemic in Japan: a modelling study. Mathematical Biosciences and Engineering, 2022, 19(6): 5998-6012. doi: 10.3934/mbe.2022280 |
Due to their continued global prevalence, infectious diseases have been receiving great attention. While some developed nations have been affected by the adverse consequences of infectious diseases, their prevalence and impact are more profound in developing nations where prevention and treatment are not readily available. For instance, the spread of Sexually Transmitted Diseases (STDs) such as Human Immunodeficiency Syndrome (HIV) have had devastating effects on the socio-economic structure of many developing nations. In the past, it was believed that, with the introduction of effective antibiotics and vaccination programs as well as improved sanitation, infectious diseases will soon be eradicated. However, the world has witnessed the propagation and continued presence of infectious diseases at a global scale in spite of humanity's relentless effort to get rid of them. As a result of this alarming reality, the study of infectious diseases has become an area of significant scientific research. In particular, there has been immense scientific investigation focused in developing critical comprehension of the conditions or factors that contribute to the epidemic of diseases and the controlling measures that can be employed to curb this epidemic. These investigations attempt to address a number of vital scientific questions; how effective will an introduction of vaccination help decrease the impact of the epidemic? How will the behaviours of individuals affect the spread of the disease when vaccines are introduced?
As a common tool in mathematical modelling, system of equations has always been effective in comprehending disease dynamics among population. See for instance [19,20,28,32,33]. However, most of these models do not explicitly take into account the impacts of behavioural responses of individuals on disease dynamics including whether there will be epidemic or not. Along with the rapid development of Internet and social network applications, social behaviour has become a new challenge in public health. As a result, infectious disease models incorporating social influence, risk perception and decision-making have attracted more and more interests of researchers from multiple areas. For example, in [16] (Science 2013), it was shown that when a social contagion is coupled to a biological contagion, the disease-behaviour system exhibits complex dynamics and social impact can be either positive or negative. In [34], injunctive social norms were added to an existing behaviour-incidence model to study the dynamics of vaccinating behaviour. To study the pandemic potential of influenza such as H1N1 or H5N1, an epidemiological game-theoretic model of an influenza pandemic was developed in [41]. The model compared the perspectives for antiviral coverage at the individual level (individual behaviour) and the population level to determine the optimal [41]. As examples, risk prediction in decision-making from psychological point of view was studied using the fuzzy trace theory [39] and later, risk perception for HIV transmission response through multiple pathways is discussed in [42]. Some recent progress on population dynamics involving decision-making under the consideration of both human-environment and biological conditions can be found in [1,2,17,34,42,43]. In particular, a review on coupled disease-behavior dynamics including social and disease perspectives from the approach of complex networks was given in [43].
At the meantime, as an emerging class of models, Economic Epidemiology (EE) is interdisciplinary and utilizes economic concepts to explicitly incorporate behavioural related responses. For some previous efforts in this direction see [8,10,18,35,37]. Also, a detailed account of some results in this field maybe found in [37] and recently in [5,36]. Different from the traditional mathematical epidemiology, EE models apply incentives for healthy behaviour and associated behavioural responses to offer unique insights to transmissibility of infectious diseases and thereby recommending optimal control strategies to contain their spread.
As classical models for epidemiology, SIR (susceptible-infected-removed), SI, SIS and SIRS models and their various extensions have been extensively studied [11]. For instance, [23] discusses continuous time mathematical epidemiological models integrated with utility functions and decision making and a discrete time EE models were discussed in [5], where equilibrium dynamics of EE under rational expectations were investigated.
In this paper, we introduce a discrete time EE model of vaccination incorporating social consideration and decision-making. Individuals are assumed to have control over the contacts they make. That is, it is assumed that contacts are made in order to maximize utility (satisfaction) subject to disease dynamics, health stock and probability of infection. It is also assumed that newborns and older susceptible individuals are vaccinated. The approach used in this study is similar to the one used in [4] but more detailed investigation on properties of system bifurcation and sensitivity analysis for the infection parameter and contact rates provides in-depth understanding to system dynamics. We also present results obtained from sensitivity analysis with specific disease parameters to show the behaviour of some particular diseases. Our results show that the system exhibits saddle-point and period-doubling bifurcation when older susceptible individuals are vaccinated. The converse also holds. That is the model does not show these bifurcation properties if there is no available vaccination for older susceptible individuals.
The organization of the paper is as follows: Section 2 describes the model, the various parameters and the underlining assumptions governing the formulation of the system. Section 3 describes the optimal behaviour of susceptible individuals. Parameter analysis is carried out in Section 4, where the endemic equilibria (when diseases is present in the population) of the prevalence rate of disease and sensitivity analysis on the equilibria of the number of contacts is carried out. Also, local stability analysis around the disease endemic equilibria as well as bifurcation analysis are discussed in this section. In Section 5, simulation of the system is presented and discussed. Conclusion on the paper is presented in Section 6.
The model is set in a discrete time interval such that individuals make decisions in discrete time given their disease status. The model considers a total population, say
● The population size is assumed constant with equal birth and death rates per year. This is denoted as
● Following the work in [26] and [44], the treatment function of infected individuals is modelled as
T(it)={σitif0≤it≤ic,σicifit>ic. |
Where
● There is no disease related death.
From the above, we have the following system of equations explaining the epidemiological model:
st+1−st=μ−mμ−(pt+μ+n)st+T(it)+νvt, | (1) |
it+1−it=ptst−μit−T(it), | (2) |
vt+1−vt=mμ+nst−(μ+ν)vt, | (3) |
where
pt=1−(1−λit)ct, |
where
{st+1−st=μ−mμ−(pt+μ+n)st+T(it)+ν(1−st−it),it+1−it=ptst−μit−T(it). | (4) |
To introduce behavioural influence of individuals into the dynamics of the model, suppose an individual k1 independently makes a decision by choosing a number of contacts
∞∑j=0βjU(ck,t+j,hk,t+j), |
1k = 1; 2; 3:: can be interpreted as individual 1, 2 and 3 etc..
where
U(ck,t,hi,t)=ck,t−δc2k,t+ϕhk,t, | (5) |
where
hk,t+1=h′+(1−ϵ)hk,t−it+1, | (6) |
where
Suppose that individuals are not altruistic (that is, they are not concerned about the welfare of the general public), then each individual will be willing to go for a contact level that will yield maximum satisfaction. Since individuals in the infected and vaccinated category are already infected and vaccinated respectively, we further assumed that unless they are altruistic, they will opt for the maximum number of contacts (that is
∂U∂ct=−β∂U∂ht+1∂ht+1∂it+1∂pt∂ct. | (7) |
2The maximum utility was calculated by substituting the value for
Eq. (7) measures the trade-off in the model in that an additional contact made by a susceptible individual may or may not result in an infection. Therefore, in order for the individual to be in equilibrium, the individual should make contacts that will satisfy the above equation.
The right hand side of Eq. (7) measures the marginal benefit associated with an additional contact made by a susceptible individual while the expression on the left hand side measures the damage the individual incurs as a result of an additional contact made. See [3] for detailed explanation on the above. By employing Eq. (4), (5) and (6) we have the Euler equation reducing to
1−2δct=βϕpc,t, | (8) |
where
pc,t=∂pt∂ct=−(1−λit)ctln(1−λit)=−(1−pt)ctln(1−pt). |
The value of
This section discusses the existence of the disease steady state. Let
{s∗=μ(1−m)+(σ−ν)i∗+νp+μ+n+ν,i∗=ps∗μ+σ,1−2δc∗=βϕpc,h∗=h′+(1−ϵ)h∗−i∗. | (9) |
And for
{s∗=μ(1−m)+ν+σic−νi∗p+μ+n+ν,i∗=ps∗μ+σic, |
where
{p=1−(1−λi∗)c∗,pc=−(1−λi∗)c∗ln(1−λi∗). | (10) |
It can be verified that there is no explicit solution for the endemic steady state. Numerical method is employed in examining the existence of positive endemic steady state(s). For example, Fig. 3 shows the relationship between the prevalence rate
The disease free equilibrium,
This section discusses the sensitivity analysis of the number of contacts to the parameter
S(c∗,λ)=∂c∗∂λλc∗=ζ1ζ2, |
where
{ζ1=βϕ(1−λi∗)c∗(c∗ln(1−λi∗)+1)λi∗,ζ2=(1−λi∗)(Δln(1−λi∗)−2δ)c∗, |
and
From the above expression, we can obtain results on the relationship between number of contacts and the infection parameter.
Theorem 4.1. (Positive relationship) If
{M1=2δeβϕ,M2=1−e−1c∗,M3=−1ln(1−λi∗). |
Proof.
Case 1. Assume that
c∗ln(1−λi∗)+1>0. |
Similarly,
ζ2=(1−λi∗)(βϕ(1−λi∗)c∗(ln(1−λi∗))2−2δ)c∗>(βϕe(ln(1−λi∗))2−2δ)(1−λi∗)c∗ | (11) |
since
However,
ζ2>(βϕe(λi)2−2δ)(1−λi∗)c∗>0 | (12) |
since
Therefore, if
Case 2. Suppose that
Since
ζ2=(1−λi∗)(βϕ(1−λi∗)c∗(ln(1−λi∗))2−2δ)c∗<(1−λi∗)(4βϕe2c∗2−2δ)c∗ |
due to
ζ2<(1−λi∗)(βϕeM1−2δ)=0. | (13) |
Hence,
Theorem 1 implies that an increase in the probability of infection with each infected contact
Theorem 4.2. (Negative relationship) If
{M1=2δeβϕ,M2=1−e−1c∗,M3=−1ln(1−λi∗). |
Proof.
Case 3. Assume
Since
c∗ln(1−λi∗)+1<0, |
which implies that
ζ2=(1−λi∗)(βϕ(1−λi∗)c∗(ln(1−λi∗))2−2δ)c∗>0 | (14) |
since
Hence, if
Case 4. Suppose
Similarly,
ζ2=(1−λi∗)(βϕ(1−λi∗)c∗(ln(1−λi∗))2−2δ)c∗<0 | (15) |
as shown in Eq. (13) above.
Therefore, if
Theorem 4.2 implies that a reduction in the value of
Also, we have
S(c∗,ϕ)=∂c∗∂ϕϕc∗=−ˉcβϕpcc∗. | (16) |
From Eq. (10), we have
Stability analysis of the system is carried out under two cases: When older susceptible individuals are not vaccinated (the proportion of vaccinated susceptible adults,
ˆst+1=(1−p−μ−ν−n)ˆst+T(^it)−(spi+ν)^it−spcˆct,ˆit+1=pˆst+(1−μ+spi)ˆit−T(ˆit)+spcˆct,ˆht+1=(1−ϵ)ˆht−ˆit+1,ˆct=(κc∗i∗)ˆit, |
where
T(ˆit)={σˆitif0≤it≤ic,0ifit>ic, |
and
κ=∂c∗∂i∗i∗c∗=βϕ(1−λi∗)c∗(c∗ln(1−λi∗)+1)λi∗(1−λi∗)(Δ(ln(1−λi∗))−2δ)c∗. |
[ˆst+1ˆit+1]=[1−p−μ−ν−nσ−ν−θp1−μ−σ+θ]⏟A[ˆstˆit], | (17) |
where
λ1,2=X1±√ψ2, | (18) |
where
{X1=2(1−μ)+θ−(p+ν+n+σ),ψ=(p+σ−ν−θ−n)2+4pn. | (19) |
From expression (18) the system is locally stable provided that
{λ1=1−σ−p−μ+θ,λ2=1−μ−ν. | (20) |
Proposition 1. The system does not exhibit stable or dampened cycle for both
The proof is straight forward by observing that the expression for
Proposition 2. If
σ+p+μ−2<θ<σ+p+μ. |
Proof. Suppose
−1<1−σ−p−μ+θ<1. |
This implies that
σ+p+μ−2<θ<σ+p+μ. |
Also for
−ν<μ<2−ν. | (21) |
Since
Proposition 2 implies that the system is stable if the sum effect of the change in disease prevalence on the probability of infection (number of contacts is held fixed) and the effect of a change in prevalence on the probability of infection due to a change in optimal number of contacts is less than the sum of the treatment rate (
It follows from case 2 that for
p+μ−2<θ<p+μ. | (22) |
Furthermore, if the endemic steady state for which disease prevalence is less than treatment capacity is stable it suffices to conclude that the endemic steady-state equilibrium for the system will be stable if the disease prevalence is greater than the treatment capacity.
Proposition 3. If
{θ1<θ<HL,L<2, |
where
{θ1=2(F+L)−H−42−L,H=LF−np,L=ν+μ+n,F=p+μ+σ. |
Proof. Stability requires that
|Tr(A)|<det(A)+1<2. | (23) |
Eq. (23) implies that
{−1−det(A)<Tr(A)<det(A)+1,det(A)<1, | (24) |
where
{Tr(A)=2+θ−F−L,det(A)=1−F−L+H−θL+θ. |
Case 5.
Tr(A)<det(A)+1, |
implies
θ<HL. | (25) |
And
−1−det(A)<Tr(A), |
implies
θ>θ1=2(F+L)−H−42−L. | (26) |
Also,
det(A)<1, |
implies
θ>θ2=H−F−LL−1. | (27) |
For
HL<F+(L−2)2L. | (28) |
Eq. (28) always holds, since
FL+L2−4L+4−H=pn+(n+μ+det(A)−2)2>0. |
Furthermore,
θ2−θ1=pn+(L−2)2(L−2)(L−1)<0 |
Hence, from Eq. (25), (26) and (27) we have
θ1<θ<HL. |
Case 6.
Tr(A)<det(A)+1, |
implies
θ<HL. |
And
{−1−det(A)<Tr(A),det(A)<1, |
implies that
θ2−HL=pn+L2L(1−L)>0. |
Hence,
θ1<θ<HL. |
Case 7.
θ2−θ1=pn+(L−2)2(L−2)(L−1)>0. |
Hence,
Proposition 3 implies that in case of full vaccination, we expect
It follows from Proposition 3 that System (17) is locally stable for
{θ1<θ<HL,L<2, | (29) |
where
{θ1=(μ+ν−2)(p+μ)+nμ−2(L−2)L−2,H=(ν+μ)(p+μ)+nμ,L=ν+μ+n. | (30) |
This section discusses conditions under which the model will exhibits saddle-point and period-doubling bifurcation. We have a saddle-point bifurcation if
θ=(ν+μ)(p+σ+μ)+n(σ+μ)ν+μ+n, |
and period-doubling bifurcation if
θ=(μ+ν−2)(σ+p+μ)+n(μ+σ)−2(ν+μ+n−2)ν+μ+n−2. |
Proposition 4. The model does not exhibit both saddle-point and period-doubling bifurcation if
Proof. It follows from the corresponding eigenvalues for case
{θ−(σ+p+μ)=0,μ+ν=0, | (31) |
and that of period-doubling bifurcation is
{θ−(σ+p+μ)=−2,μ+ν=2. | (32) |
Since
This section discusses some simulation results on the theorems and propositions of Section 4.
Fig. 6 confirms Theorems 1 and 2. Fig. 6(a) -6b satisfies theorem 1. Fig. 6(a) confirms the condition for which both
Fig. 6(c)-6(d) confirms theorem 2. Fig. 6(c) confirms the condition for which
In this section we present a general numerical sensitivity analysis in which model parameters are motivated by the outbreak of measles virus among 0-12 month old babies. Measles is a highly contagious and a serious respiratory disease caused by a virus. In spite of the availability of a safe and effective vaccine, the disease has remained one of the leading causes of death among young children globally [46,48,49]. Measles is prevalent in developing countries where per capita incomes are low and where their health care system is weak [49]. In this analysis, we consider parameter values related to children who received measles vaccination by the time they celebrate their first birthdays in the Kissii county in Kenya. Table 1 shows the relevant values for the model parameters employed:
Note that we assume
The resulting numerical solution for the above parameters yields eigenvalues
| | | |
0.533 | 0.266 | 0.200 | 0.214 |
Figures 5(a)-5(d) show the simulation output for the various disease categories and the number of contacts by babies.
For confirmation of some of the results on the stability of the system we chose the following parameters:
The parameters in Table 3 indicate that 80 % of the newborns in the population are vaccinated. The treatment rate is 60 % of the infected population, probability of a susceptible individual becoming infected by a single infected contact is 0.6, the value for
Parameters | m | | | | | |
Values | 0.8 | 0.6 | 0.6 | 0.05 | 0.96 | 3 |
Proposition 2. As a form of demonstration, we chose the values for death and birth rates as 5 %. The proportion of vaccinated individuals whose vaccination wear out per annum is chosen as 10 %. The proportion of older susceptible individuals who receive vaccination is set at 0 (that is
Proposition 3. Table 4 contains the respective parameters that satisfy the conditions under which the system is stable or unstable. As a form of demonstration, Fig. 8 shows the numerical simulation for System (4) and Eq. (8) for condition
Cases | Parameters | |
| | |
| | |
| ||
| | |
| | |
| ||
| | |
| | |
|
Furthermore, for
| | | | |
0.19 | 0.194 | 0.616 | 8.80 | 0.663 |
Based on the values for condition
To illustrate the condition where there is negative relationship between the contact levels and disease prevalence rate, we set
| | | | |
0.135 | 0.059 | 0.807 | 9.261 | 0.282 |
The values in Table 6 give the value for
Also, Fig. 10 shows the simulation result for
The parameter values in Table 7 are chosen to confirm the conditions for which the system exhibits bifurcation properties:
Parameter | | | n | m | | | | |
Value | 0.05 | 0.5 | 0.6 | 0.5 | 0.6 | 0.05 | 0.96 | 3 |
Fig. 11(a) and 11(b) show period-doubling bifurcation diagram for the system. The infection parameter
Fig. 12(a) and 12(b) show the period-doubling bifurcation diagram for the case where the number of contacts is fixed. In carrying out the numerical simulation, we fixed the number of contacts at 8. It is clear from the figures that the system makes a smooth transition of the system from an equilibrium path to a double equilibria paths is observed.
You will notice from Fig. 11 and 12 that there is a difference in the transitioning process. The implication of the above phenomenon is that if rational individuals are allowed to make choices on contact levels, the system can behave in a chaotic manner. This may be due to the unpredictable nature of the behaviours of these individuals. From policy perspective, policy makers should take into consideration the behaviour of individuals, in order to have a clear picture of the dynamics of the epidemiology of diseases, the infection parameter (
Fig. 13(a) and 13(b) show the period-doubling bifurcation diagram for
The epidemiological implication of the above dynamic is that the system has the tendency to switch from equilibria path to chaotic path. In other words, there are regions of multistability in which the disease can have a stabilizing effect as well as chaotic effect. This dynamic can become more complex as disease parameters are varied across some range of values. In [15] for instance, it is shown that as disease induced death rate with higher transmissibility, the system exhibits more complexities that give rise to period-doubling cascades couple with other dynamics. For other studies on possible secondary infection, period-doubling bifurcations and chaotic behaviour in epidemic models, we refer [12,14,30].
Decisions made by individuals in the presence of infectious disease(s) are most often done in a selfish manner. This is due to the risk and benefits associated with such decisions. Individuals are faced with either forgoing a number of contacts just to maintain their health stock or risk their health stock by making some number of contacts. Associated with this choices are benefits(utilities). Also, associated with this is the cost of becoming infected. These phenomena affect disease dynamics. That is individual decisions determine whether a population will be faced with an epidemic or not.
It is therefore imperative that decision makers take into consideration the effects of private choices on epidemiological processes in order to better understand the disease dynamics for better policy on "disease-epidemic-control". In this paper, behavioural responses by individuals is incorporated into a "disease vaccination model". This gave an explicit fashion in which we can analyze how individuals respond to diseases with available vaccination. In our analyses, we were able to establish that, with the introduction of vaccination the system did not exhibit dampened cycles. And this also holds when adults are not vaccinated. We have established that the system exhibits period-doubling and saddle-point bifurcation where there is transitioning from unstable to stable endemic steady-state paths when adult individuals are vaccinated. With regards to the case where adult susceptible individuals are not vaccinated, the system exhibits either stability or instability; there is no bifurcation.
In carrying out the numerical simulations, the parameter values are chosen to reflect the case where susceptible individuals increase their number of contacts for an initial decrease in disease prevalence. Also, we chose the values of the parameters to reflect the opposite: susceptible individuals respond positively in terms of choices on contact levels to disease prevalence, a situation Aadland et. al. refers to as dynamic resonance [7]. In our case, by choosing high values for the proportion of the vaccinated individuals who lose vaccination (all other parameters are constant), we established a positive relationship between contacts levels and disease prevalence; A situation that causes disease prevalence to increase in the long run. This behaviour can be attributed to social factors such as those that contributed to the Ebola outbreak; where people for instance, consider health condition of their infected relatives is more important than their own health condition and thus make contacts with these relatives. In [4] for instance where "syphilis cycles" is studied, Aadland et. al. pointed out that this dynamics produces cycles in syphilis infection such that an increase in disease prevalence can cause susceptible individuals to increase their partners and vice-versa, and a decrease in the disease prevalence will opt to reduce partners. This phenomenon amplifies the disease cycles in the population.
In a nutshell, this paper is an attempt to further emphasize the importance of considering private choices in formulating policies on tackling infectious diseases. And that instead of public policy makers imposing policies on individuals in the advent of disease spread, they should rather understand how their behaviours can affect the epidemiological process of the disease.
The authors thank the referee for valuable comments. The project was partly supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
[1] | [ M. Andrews,C. T. Bauch, The impacts of simultaneous disease intervention decisions on epidemic outcomes, Journal of Theoretical Biology, 395 (2016): 1-10. |
[2] | [ M. Andrews and C. T. Bauch, Disease interventions can interfere with one another through disease-behaviour interactions PLOS Computational Biology 11(2015), e1004291. |
[3] | [ D. Aadland, D. Finnof and X. D. K. Huang, Syphilis Cycles University Library of Munich, Germany in its series MPRA Paper with number 8722. http://ideas.repec.org/p/pra/mprapa/8722.html, 2007. |
[4] | [ D. Aadland,D. Finnoff,X. D. K. Huang, Syphilis cycles, The B.E, Journal of Economic Analysis and Policy, De Gruyter, 14 (2013): 297-348. |
[5] | [ D. Aadland, D. Finnoff and K. X. D. Huang, The Equilibrium Dynamics of Economic Epidemiology (2011) https://www.researchgate.net/publication/50310816. |
[6] | [ D. Aadland, D. Finnof and X. D. K. Huang, The Dynamic of Economics Epidemiology Equilibria Association of Environmental and Resource Economists 2nd Annual Summer Conference, Asheville, NC, June 2012. |
[7] | [ D. Aadland, D. Finnof and X. D. K. Huang, The Equilibrium Dynamics of Economic Epidemiology Vanderbilt University Department of Economics Working Paper Series 13-00003, http://ideas.repec.org/p/van/wpaper/vuecon-sub-13-00003.html, March 2013. |
[8] | [ A. Ahituv,V. Hotz,T. Philipson, Is aids self-limiting? evidence on the prevalence elasticity of the demand for condoms, Journal of Human Resources, 31 (1996): 869-898. |
[9] | [ J. Arino,K. L. Cooke,P. Van Den Driessche,J. Velasco-Hern{á}ndez, An epidemiology model that includes a leaky vaccine with a general waning function, Discrete and Continuous Dynamical Systems Series B, 4 (2004): 479-495. |
[10] | [ M. C. Auld, Choices, beliefs, and infectious disease dynamics, Journal of Health Economics, 22 (2003): 361-377. |
[11] | [ L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci, 124 (2003): 83-105. |
[12] | [ J. L. Aron,I. B. Schwartz, Seasonality and period-doubling bifurcations in an epidemic model, Journal of Theoretical Biology, 110 (1984): 665-679. |
[13] | [ W. S. Avusuglo,K. Abdella,W. Feng, Stability analysis on an economic epidemiology model on syphilis, Communications in Applied Analysis, 18 (2014): 59-78. |
[14] | [ M. Aguiar,B. Kooi,N. Stollenwerk, Epidemiology of dengue fever: A model with temporary cross-immunity and possible secondary infection shows bifurcations and chaotic behaviour in wide parameter regions, Math. Model. Nat. Phenom., 3 (2008): 48-70. |
[15] | [ A. M. Bate,F. M. Hilker, Complex dynamics in an eco-epidemiological model, Bulletin of Mathematical Biology, 75 (2013): 2059-2078. |
[16] | [ C. T. Bauch,A. P. Galvani, Social factors in epidemiology, Science, 342 (2013): 47-49. |
[17] | [ C. T. Bauch and R. McElreath, Disease dynamics and costly punishment can foster socially imposed monogamy Nature Communications 7 (2016), 11219. |
[18] | [ S. M. Blower,A. R. McLean, Prophylactic vaccines, risk behaviour change, and the probability of eradicating HIV in San Francisco, Science, 265 (1994): 1451-1454. |
[19] | [ F. Brauer, Models for the spread of universally fatal diseases, Journal of Mathematical Biology, 28 (1990): 451-462. |
[20] | [ F. Brauer, Epidemic models in populations of varying size, Mathematical Approaches to Problems in Resource Management and Epidemiology, 81 (1989): 109-123. |
[21] | [ R. O. Baratta,M. C. Ginter,M. A. Price,J. W. Walker,R. G. Skinner,E. C. Prather,J. K. David, Measles (rubeola) in previously immunized children, Pediatrics, 46 (1970): 397-402. |
[22] | [ M. P. Do Carmo and M. P. Do Carmo, Differential Forms and Applications Translated from the 1971 Portuguese original, Universitext, Springer-Verlag, Berlin, 1994. |
[23] | [ E. P. Fenichel,C. Castillo-Chavez,M. G. Ceddia,G. Chowell,P. A. G. Parra,G. J. Hickling,G. Holloway,R. Horan,B. Morin,C. Perrings,M. Springborn,L. Velazquez,C. Villalobos, Adaptive human behavior in epidemiological models, PNAS, 108 (2011): 6306-6311. |
[24] | [ M. O. Fred,J. K. Sigey,J. A. Okello,J. M. Okwoyo,G. J. Kang'ethe, Mathematical Modeling on the Control of Measles by Vaccination: Case Study of KISII County, Kenya, The SIJ Transactions on Computer Science Engineering and its Applications (CSEA), The Standard International Journals (The SIJ), 2 (2014): 61-69. |
[25] | [ M. Grossman, On the concept of health capital and the demand for health, Journal of Political Economy, 80 (1972): 223-255. |
[26] | [ Z. Hu,W. Ma,S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Mathematical Biosciences, 238 (2012): 12-20. |
[27] | [ Kenya National Bureau of Statistics. 2013. Kisii County Multiple Indicator Cluster Survey 2011 Final Report. Nairobi, Kenya: Kenya National Bureau of Statistics, pp. 33. |
[28] | [ L. Marcos,R. Jesus, Multiparametric bifurcations for a model in epidemiology, J. Mathematical Biology, 35 (1996): 21-36. |
[29] | [ M. Mark, Mathematical Modelling (4th Edition), ISBN 978-0-12-386912-8, ScienceDirect, 2012. |
[30] | [ R. M. May, Nonlinear phenomena in ecology and epidemiology, Annals of the New York Academy of Sciences, 357 (1980): 267-281. |
[31] | [ R. E. Mickens, Analysis of a discrete-time model for periodic diseases with pulse vaccination, Journal of Difference Equations and Applications, 9 (2003): 541-551. |
[32] | [ Z. Mukandavire,A. B. Gumel,W. Garira,J. M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection, Mathematical Biosciences and Engineering, 6 (2009): 333-362. |
[33] | [ A. M. Niger,A. B. Gumel, Mathematical analysis of the role of repeated exposure on malaria transmission dynamics, Differential Equations and Dynamical Systems, 16 (2008): 251-287. |
[34] | [ T. Oraby and C. T. Bauch, The influence of social norms on dynamics of paediatric vaccinating behaviour, Proc. R. Soc. B. 281 (2014), 20133172. |
[35] | [ T. Philipson and R. A. Posner, Private Choices and Public Health: An Economic Interpretation of the AIDS Epidemic Harvard University, Cambridge, MA, 1993. |
[36] | [ C. Perrings,C. Castillo-Chavez,G. Chowell,P. Daszak,E. P. Fenichel,D. Finnoff,R. D. Horan,A. M. Kilpatrick,A. P. Kinzig,N. V. Kuminoff,S. Levin,B. Morin,K. F. Smith,M. Springborn, Merging economics and epidemiology to improve the prediction and management of infectious disease, EcoHealth, 11 (2014): 464-475. |
[37] | [ T. Philipson, Economic epidemiology and infectious diseases, Handbook of Health Economics, 1 (2000): 1761-1799. |
[38] | [ S. A. Plotkin, W. A. Orenstein and P. A. Offit, Vaccines 5th ed. (2008), Pennsylvania: Elsevier Inc. |
[39] | [ V. F. Reyna, How people make decisions that involve risk, American Psychological Society, 13 (2004): 60-66. |
[40] | [ L. W. Rauh,R. Schmidt, Measles immunization with killed virus vaccine. Serum antibody titers and experience with exposure to measles epidemic, Bulletin of the World Health Organization, 78 (2000): 226-231. |
[41] | [ E. Shim,G. B. Chapman,A. P. Galvani, Medical decision making, Decision Making with Regard to Antiviral Intervention during an Influenza Pandemic, 30 (2010): E64-E81. |
[42] | [ S. Tully,M. Cojocaru,C. T. Bauch, Sexual behaviour, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways, Scientific Reports, 5 (2015): 15411. |
[43] | [ Z. Wang,M. A. Andrews,Z.-X. Wu,L. Wang,C. T. Bauch, Coupled disease-behavior dynamics on complex networks: A review, Physics of Life Reviews, 15 (2015): 1-29. |
[44] | [ W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006): 58-71. |
[45] | [ Z. Yicang,L. Hanwu, Stability of periodic solutions for an SIS model with pulse vaccination, Mathematical and Computer Modelling, 38 (2003): 299-308. |
[46] | [ M. T. Caserta, ed. (September 2013), http://www.merckmanuals.com/professional/pediatrics/miscellaneous-viral-infections-in-infants-and-children/measles, Merck Manual Professional. Merck Sharp and Dohme Corp. Retrieved 15 January 2017. |
[47] | [ Government of Canada, http://healthycanadians.gc.ca/publications/healthy-living-vie-saine/4-canadian-immunization-guide-canadien-immunisation/index-eng.php?page=12 (accessed January 12,2017). |
[48] | [ http://www.saskatchewan.ca/residents/health/diseases-and-conditions/measles, Government of Saskatchewan, Retrieved 15 January 2017. |
[49] | [ http://www.who.int/mediacentre/factsheets/fs286/en/, November 2016, Retrieved 15 January 2017. |
[50] | [ http://www.immune.org.nz/duration-protection-efficacy-and-effectiveness |
1. | Wisdom S. Avusuglo, Nicola Bragazzi, Ali Asgary, James Orbinski, Jianhong Wu, Jude Dzevela Kong, Leveraging an epidemic–economic mathematical model to assess human responses to COVID-19 policies and disease progression, 2023, 13, 2045-2322, 10.1038/s41598-023-39723-0 |
| | | |
0.533 | 0.266 | 0.200 | 0.214 |
Parameters | m | | | | | |
Values | 0.8 | 0.6 | 0.6 | 0.05 | 0.96 | 3 |
Cases | Parameters | |
| | |
| | |
| ||
| | |
| | |
| ||
| | |
| | |
|
| | | | |
0.19 | 0.194 | 0.616 | 8.80 | 0.663 |
| | | | |
0.135 | 0.059 | 0.807 | 9.261 | 0.282 |
Parameter | | | n | m | | | | |
Value | 0.05 | 0.5 | 0.6 | 0.5 | 0.6 | 0.05 | 0.96 | 3 |
Parameters | Values | Sources |
92.9 % | [27] | |
| 0.0 | Assumed |
40 % | Assumed | |
| 0.09091 per day | [24] |
| 0.05 | Assumed |
| 0.96 | [4] |
1 | Assumed | |
| 0.02755 per year | [24] |
| 10 % | Assumed |
| | | |
0.533 | 0.266 | 0.200 | 0.214 |
Parameters | m | | | | | |
Values | 0.8 | 0.6 | 0.6 | 0.05 | 0.96 | 3 |
Cases | Parameters | |
| | |
| | |
| ||
| | |
| | |
| ||
| | |
| | |
|
| | | | |
0.19 | 0.194 | 0.616 | 8.80 | 0.663 |
| | | | |
0.135 | 0.059 | 0.807 | 9.261 | 0.282 |
Parameter | | | n | m | | | | |
Value | 0.05 | 0.5 | 0.6 | 0.5 | 0.6 | 0.05 | 0.96 | 3 |