Citation: Mukhamed Aleroev, Hedi Aleroeva, Temirkhan Aleroev. Proof of the completeness of the system of eigenfunctions for one boundary-value problem for the fractional differential equation[J]. AIMS Mathematics, 2019, 4(3): 714-720. doi: 10.3934/math.2019.3.714
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Human schistosomiasis, the third most devastating tropical disease in the world after malaria and intestinal helminthiasis, is a global public health problem [42]. According to World Health Organization (WHO), the number of people need preventive chemotherapy globally in 2013 was 262 million, of which 121.2 million were school-aged children [42,43,44].
The major forms of human schistosomiasis are caused by species of the water-borne flatworm or blood flukes called schistosomes [9,44]. Schistosomiasis in mainland China is caused by Schistosoma Japonicum (S. Japonicum). Though its transmission had been interrupted successively in five of the twelve formerly endemic provinces (see [8,9] and the references therein), schistosomiasis is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu and Jiangxi in the lake and marshland regions with vast areas of Oncomelania hupensis habita, Sichuan and Yunnan in the mountainous regions with diverse ecologies [8,9]. The monthly data of schistosomiasis cases in Hubei, Hunan and Anhui recorded by Chinese Center for Disease Control and Prevention (China CDC) [10] display a seasonal pattern. The cases in the late summer and early autumn are significantly higher than in the spring and winter (see Figure 1.).
Schistosomiasis is a parasitic disease caused by trematode flatworms of the genus schistosoma [6]. The reproductive cycle of schistosomiasis starts with parasitic eggs released into freshwater through faeces and urine, then some eggs hatch and became miracidia under appropriate conditions, those miracidia swim and penetrate snails as intermediate host. By escaping from the snail, the infective cercariae penetrate the skin of the human host. For more details on the life cycle of schistosome, we refer to [5,7,12,16,17,23,28,30,6]. To focus on the dynamics of S.Japonicum propagating between human and the intermediate host snails, we consider a simplified diagram for the life cycle given in Figure 2.
The earliest mathematical models for schistosomes were developed by Macdonald [32] and Hairston [25]. Since then, a good number of mathematical models involving the transmission dynamics of schistosomes have been proposed (see [8,12,16,17,30] and the references therein). Garira et al. [19] proposed a dynamic model of ordinary differential equations linking the within-host and between-host dynamics of infections with free-living pathogens in the water environment. Wang and Spear [48] explored the impact of infection-induced immunity on the transmission of S. Japonicum in hilly and mountainous environments in China, and underscored the need for improved diagnostic methods for disease control, especially in potentially re-emergent settings. Chen et al. [8] proposed an autonomous mathematical model for controlling schistosomiasis in Hubei Province, China, focusing on the disease spread among people, intermediate hosts snails and cattle. Feng et al. [16] estimated the parameters of a schistosome transmission system, which described the distribution of schistosome parasites in a village in Brazil.
Schistosomiasis often occurs in most tropical and some subtropical regions of the world. Environmental and climatic factors play an important role in the geographical distribution and transmission of schistosomiasis [44]. It was well known that seasonality can cause population fluctuations ranging from annual cycles to multi year oscillations, and even chaotic dynamics [2,22]. From an applied viewpoint, clarifying the mechanisms that link seasonal environmental changes to diseases dynamics may provide help in predicting the long-term health risks, in developing an effective public health program, and in setting objectives and utilizing limited resources more effectively (see [1,31,35] and the references therein). These considerations indicate that seasonal models are needed in order to describe the periodic incidence of schistosomiasis transmission. However, to the best of our knowledge, there are few studies modeling the seasonality influence on the transmission of schistosomiasis in mainland China [54].
More than 82% of infected persons lived in lake and marshland regions (such as Dongting Lake and Poyang Lake) along the Yangtze River, where interruption of transmission has been proven difficult [20,58,59]. The purpose of this paper is to propose a periodic schistosomiasis model to investigate the seasonal transmission dynamics and search for control strategies in these lake and marshland regions in China. We analyze the dynamical behavior, evaluate the basic reproduction number
The paper is organized as follows. In Section 2, we introduce the periodic schistosomiasis model. Some preliminary results are presented in Section 3, such as the positivity and boundedness of solutions and calculation of the basic reproduction number. The extinction and uniform persistence of the disease are discussed in Sections 4 and 5, respectively. Simulations of the schistosomiasis data from Hubei Province are presented in Section 6. Conclussion and discussion are given in Section 7.
To study the seasonal transmission dynamics of schistosomiasis, we trace the life cycle of schistosome parasites in three different environments: human biological environment, physical water environment, and snail biological environment. The life cycle of schistosomiasis was given in Figure 2 and its transmission diagram among humans, snails, and miracidia and cercariae is illustrated in Figure 3.
We denote the total numbers of humans and snails by
(1) There is no vertical transmission of the disease.
(2) Susceptible humans are recruited at a positive constant rate
(3) There are no immigrations of infectious humans.
(4) People living near rivers and lakes are more likely going swimming and fishing in the summer and autumn, they are prone to infection for long contacting with contaminated water. The river, lake, pond water freezes or dry in winter, infected snail seldom or not produce larvae, then infection is not likely to happen. Due to these seasonal phenomena, we use two 12-month periodic functions
(5) It is clear that the snail population is seasonally changed in reality, the recruited rate
(6) There is no immune response in both snail and human populations.
(7) Several effective control strategies, such as drug treatment, improving sanitation and health education, the integrated strategies are considered here. We denote these strategies by the natural recovery and treatment rate
(8) We further assume that the human natural death rate
The model is described by the following system of ordinary differential equations:
{SH′(t)=ΛH−λH(t)SHP−μHSH+γHIH,IH′(t)=λH(t)SHP−(μH+γH)IH,M′(t)=λMIH−μMM,SV′(t)=ΛV(t)−λV(t)SVM−μV(t)SV,IV′(t)=λV(t)SVM−αV(t)IV,P′(t)=λPIV−μPP. | (1) |
We denote
S0H>0,I0H≥0,M0≥0,S0V>0,I0V≥0,P0≥0. | (2) |
When
Now, we deduce the basic reproduction number
Denote
F(t)=(000λH(t)^SH00000λV(t)^SV(t)000000),V(t)=(μH+γH000−λMμM0000αV(t)000−λPμp). |
Let
{dY(t,s)dt=−V(t)Y(t,s),∀t≥s,Y(s,s)=I, |
where
∫t−∞Y(t,s)F(s)ϕ(s)ds=∫+∞0Y(t,t−a)F(t−a)ϕ(t−a)da |
is the distribution of accumulative new infections at time
(Lϕ)(t)=∫+∞0Y(t,t−a)F(t−a)ϕ(t−a)da,∀t∈R,ϕ∈Cω. |
Applying the results obtained in [50], the basic reproduction number
Employing Theorem 2.1 and Theorem 2.2 in Wang and Zhao [50], we can deduce the following results with respect to
Lemma 3.1. On basic reproduction number
(ⅰ)
(ⅱ)
(ⅲ)
Then
On the positivity and boundedness of solutions of model (1) with nonnegative initial conditions (2), we have the following results.
Lemma 3.2. Let
Proof. In fact, by the continuous dependence of solutions with respect to initial values, we only need to prove that when
m(t)=min{SH(t),IH(t),M(t),SV(t),IV(t),P(t)},∀t>0. |
Clearly,
If
0=SH(t1)≥S0Hexp(−∫t10(λH(s)P+μH)ds)>0, |
which leads to a contradiction.
Similar contradictions can be deduced in the cases of
Let
limsupt→∞NH(t)=ΛHμH:=BH, | (3) |
which implies that
M′(t)=λMIH−μMM≤λM(BH+ε)−μMM,∀t≥T0. |
Then, we have
limsupt→∞M(t)≤λMBHμM. | (4) |
Set
limsupt→∞(NV(t)−^SV(t))≤0, | (5) |
which implies that
Lastly, from the last equation of model (1), similar to the proof of (4) we obtain
Remark 3.3. Denote set
Ω={(SH,IH,M,SV,IV,P):0≤NH≤BH,0≤M≤λMBHμM,0≤NV≤BV,0≤P≤λPBVμP}. |
Lemma 3.2 implies that
Theorem 4.1. If
Proof. By considering the linearization system, we can prove that
N(t)=(000λH(t)00000λV(t)000000). | (6) |
Let
{I′H(t)≤λH(t)(ΛHμH+ε)P−(μH+γH)IH,M′(t)=λMIH−μM(t)M,I′V(t)≤λV(t)(^SV(t)+ε)M−αV(t)IV,P′(t)=λPIV−μP(t)P. | (7) |
Considering the following auxiliary system:
{˜I′H(t)=λH(t)(ΛHμH+ε)˜P−(μH+γH)˜IH,˜M′(t)=λM˜IH−μM(t)˜M,˜I′V(t)=λV(t)(^SV(t)+ε)˜M−αV(t)˜IV,˜P′(t)=λP˜IV−μP(t)˜P, |
that is
dh(t)dt=(F(t)−V(t)+εN)h(t),h(t)=(˜IH(t),˜M(t),˜IV(t),˜P(t))T. | (8) |
By Lemma 2.1 in Zhang and Zhao [56], it follows that there exists a positive
Let
When
˜R0=ρ(FV−1)=λHΛHλMλPμHμMμP(μH+γH). | (9) |
As a corollary of Theorem 4.1, we have the following result.
Corollary 4.2. If
Theorem 5.1. If
lim inft→∞(SH(t),IH(t),M(t),SV(t),IV(t),P(t))≥(ϵ,ϵ,ϵ,ϵ,ϵ,ϵ). |
Proof. By
For any small enough constant
U′ε(t)=ΛH−ελH(t)Uε(t)−μHUε(t) | (10) |
and
V′ε(t)=ΛV(t)−ελV(t)Vε(t)−μV(t)Vε(t). | (11) |
Applying Lemma 2 in [39] and Lemma 1 in [40], equations (10) and (11) admit globally uniformly attractive positive
U∗ε1(t)>ΛHμH−θ2,V∗ε1(t)>^SV(t)−θ2. | (12) |
Since model (1) is
X={(SH,IH,M,SV,IV,P):SH>0,IH≥0,M≥0,SV>0,IV≥0,P≥0} |
and
X0={(SH,IH,M,SV,IV,P)∈X:IH>0,M>0,IV>0,P>0}. |
Then
∂X0=X∖X0={(SH,IH,M,SV,IV,P)∈X:IHMIVP=0}. |
By Lemma 3.2,
Define
P(x0)=u(ω,x0),∀x0∈X, |
where
Let
M∂={(SH,0,0,SV,0,0):SH>0,SV>0}. | (13) |
If initial conditions
If
IH(t)≥I0He−(μH+αH+γH)t>0. |
Thus, by the third equation of model (1),
Similarly, when
Model (1) can be simplified as a subsystem
Secondly, let
‖u(t,x0)−u(t,M1)‖<ε1for allt∈[0,ω]. | (14) |
where
Now, we claim that
limsupn→∞∥Pn(x0)−M1∥≥δ1. | (15) |
Suppose (15) is not true, then we have
∥Pn(x0)−M1∥<δ1,∀n≥0. | (16) |
From (14) we obtain
S′H(t)≥ΛH−ε1λH(t)SH(t)−μHSH(t) |
and
S′V(t)≥ΛV(t)−ε1λV(t)SV(t)−μV(t)SV(t) |
for any
Since systems (10) and (11) with parameter
Uε1(t)≥U∗ε1(t)−θ2,Vε1(t)≥V∗ε1(t)−θ2 | (17) |
for all
{I′H(t)≥λH(t)(ΛHμH−θ)P−(μH+γH)IH,M′(t)=λMIH−μM(t)M,I′V(t)≥λV(t)(^SV(t)−θ)M−αV(t)IV,P′(t)=λPIV−μP(t)P. | (18) |
Considering the following auxiliary system:
{˜I′H(t)=λH(t)(ΛHμH−θ)˜P−(μH+γH)˜IH,˜M′(t)=λM˜IH−μM(t)˜M,˜I′V(t)=λV(t)(^SV(t)−θ)˜M−αV(t)˜IV,˜P′(t)=λP˜IV−μP(t)˜P, |
that is
dhdt=(F(t)−V(t)+θN(t))h(t),h(t)=(˜IH(t),˜M(t),˜IV(t),˜P(t))T. | (19) |
By Lemma 2.1 in Zhang and Zhao [56], there exists a positive
Denote
Lastly, since model (1) is periodic, we obtain that model (1) is uniformly persistent. From Remark 3.3, model (1) is also permanent. This completed the proof.
As a consequence of Theorem 5.1 and Remark 3.3, from the main results obtained in [38] on the existence of positive periodic solutions for general population dynamical systems, we have the following result.
Corollary 5.2. If
By Corollary 5.2, model (1) has at least one positive
Example 5.3. Set
The initial values are
The monthly reported human schistosomiasis data in Hubei from January 2008 to December 2014 from the China CDC [10] show a seasonal fluctuation, with a peak in late summer to early autumn and a nadir in late winter. We use model (1) to simulate these cases and estimate the values of parameters in
We explain the parameter values as follows:
(a) The average human lifespan is about 74 years in Hubei in 2008, which is obtained from the National Bureau of Statistics of China [34]. Thus, the monthly average death rate
Similarly, the natural death rate of snails is
(b) The total number of population was
(c) Due to the lack of information about periodic functions
(d) Parameters
Parameter | Interpretation | Value | Unit | Source |
| Recruiting of susceptible humans | | month | [34] |
| Natural death rate of humans | | month | [34] |
| The baseline transmission rate | | month | Estimated |
| The magnitude of forcing | 0.6 | none | [54] |
| The initial phase | | none | Estimated |
| Cure rate | 0.131 | month | [44] |
| Migration rate | 209 | month | [5], [33] |
| Natural death rate of miracidia | 27 | month | [18], [36] |
| Recruiting of susceptible snails | | month | [8], [27], [53] |
| Natural death rate of snails | | month | [33] |
| Disease induced death rate of snails | 0.012 | month | [18], [33] |
| The baseline transmission rate | | month | Estimated |
| The magnitude of forcing | 0.6 | none | [54] |
| The initial phase | | none | Estimated |
| Migration rate | 78 | month | [18], [33] |
| Natural death rate of cercariae | 0.12 | month | [18], [36] |
Based on these known parameter values, the fitted values are
R2=∑841(ˆyi−ˉy)2∑841(yi−ˉy)2+∑841(ˆyi−ˉy)2=0.6463, |
where
The cure rate in 2008 was about
We carry out some sensitivity analysis to investigate the influence of parameters
Traditional strategies in controlling schistosomiasis include chemotherapy, health education, livestock chemotherapy, and snail control in risk areas [8], relying more on treating humans and animals. The sensitivity analysis demonstrate that these are all important measures to control schistosomiasis infection in Hubei, see Figure 8. and Figure 9.
From (a) in Figure 8, we can see that
The simulation results (b) and (c) in Figure 8. and (c-e) in Figure 9. also show that virus migration rate
Increasing cure rate
It was observed that the number of schistosomiasis cases arrives at peak in late summer to early autumn, and reaches nadir in winter and spring in Hubei, Hunan, Anhui and other regions with similar geographic characteristics and environmental factors in China (see Figure 1), which display a seasonal pattern in these epidemic provinces. For the sake of simplicity and convenience, we only list data of three provinces in Figure 1. To investigate the human schistosomiasis transmission dynamics and explore effective control and prevention measures in these lake and marshland regions along the Yangtze River, we developed a nonautonomous model to describe seasonal schistosomiasis incidence rate by incorporating periodic transmission rates
Based on the data from China CDC [10], we used our model (1) to simulate the monthly infected human data from January 2008 to December 2014 in Hubei, the parameters in transmission functions
Hubei, Hunan, Jiangxi, Jiangsu and Anhui provinces are located along the Yangtze River in central China, where climate changes clearly all the year round. Rivers and lakes water level rise in rainy spring and summer, then the area of snails increase, farmers and students have more chances to contact with contaminated water for agriculture work or routine life, so epidemics occur naturally in this period. With temperature declining in winter, people have less opportunities to contact with water. Sun et.al [37] estimated that the lowest critical temperature for the infection of snails with miracidia is
To prevent and control the disease, the most basic work is to increase residents' knowledge of schistosomiasis, including harm of the disease, the transmission through feces of infected people and livestock, how people contract the disease (infection route), the snails as the intermediate host, etc. The best way to prevent infection is to avoid contacting infested water, and once infected, drug treatment of praziquantel is recommended [26].
In schistosomiasis epidemic seasons (April-October), schistosomiasis prevention and control work is very hard. In addition to routine control approaches such as chemotherapy, molluscicide treatment of snail habitats and health education, other major interventions including agriculture mechanization (phasing out the cattle for ploughing and other field work), prohibiting pasture in the grasslands along lake and rivers, building safe grassland for grazing, raising livestock in herds, improving sanitation through supplying safe water, constructing marsh gas pools, building lavatories and latrines, and providing fecal matter containers for fishermen's boats, etc., could decrease the prevalence of schistosomiasis to a very low level (see Figure 9. and [46,49]).
Duo to the increasing migration population and the changes in environments and diet habits, schistosomiasis rebounded in some areas where it had formerly been controlled or eliminated (see [60] and the references therein). Moreover, another threat is that traveling causes new infections of other species of schistosomiasis, for example, an increasing in the cases infected with S. haematobium or S. mansoni is reported in those returning to China after the China-aided projects in Africa and labor services export to Africa [47]. So highly sensitive surveillance and response system for those from overseas is necessary.
The model we set up is used to study the transmission dynamics and control of schistosomiasis in the lake and marshland areas. For mountainous regions, such as Sichuan and Yunnan provinces, the corresponding model needs further research. It is widely acknowledged that the transmission processes of S. japonica is considerably more complex in comparison to other schistosome species because its definitive hosts include more than 40 animal reservoirs, such as cattle, dogs, pigs and rodents [25]. The model should include the role of these hosts. We leave these for future consideration.
We are grateful to Dr. Daozhou Gao and two anonymous reviewers for their valuable comments and suggestions that greatly improved the presentation of this paper. This research was partially supported by the National Natural Science Foundation of China [11271312,31670656,11501498] and National Science Foundation (DMS-1412454).
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1. | Viktor A. Rukavishnikov, Alexey V. Rukavishnikov, On the Properties of Operators of the Stokes Problem with Corner Singularity in Nonsymmetric Variational Formulation, 2022, 10, 2227-7390, 889, 10.3390/math10060889 |
Parameter | Interpretation | Value | Unit | Source |
| Recruiting of susceptible humans | | month | [34] |
| Natural death rate of humans | | month | [34] |
| The baseline transmission rate | | month | Estimated |
| The magnitude of forcing | 0.6 | none | [54] |
| The initial phase | | none | Estimated |
| Cure rate | 0.131 | month | [44] |
| Migration rate | 209 | month | [5], [33] |
| Natural death rate of miracidia | 27 | month | [18], [36] |
| Recruiting of susceptible snails | | month | [8], [27], [53] |
| Natural death rate of snails | | month | [33] |
| Disease induced death rate of snails | 0.012 | month | [18], [33] |
| The baseline transmission rate | | month | Estimated |
| The magnitude of forcing | 0.6 | none | [54] |
| The initial phase | | none | Estimated |
| Migration rate | 78 | month | [18], [33] |
| Natural death rate of cercariae | 0.12 | month | [18], [36] |
Parameter | Interpretation | Value | Unit | Source |
| Recruiting of susceptible humans | | month | [34] |
| Natural death rate of humans | | month | [34] |
| The baseline transmission rate | | month | Estimated |
| The magnitude of forcing | 0.6 | none | [54] |
| The initial phase | | none | Estimated |
| Cure rate | 0.131 | month | [44] |
| Migration rate | 209 | month | [5], [33] |
| Natural death rate of miracidia | 27 | month | [18], [36] |
| Recruiting of susceptible snails | | month | [8], [27], [53] |
| Natural death rate of snails | | month | [33] |
| Disease induced death rate of snails | 0.012 | month | [18], [33] |
| The baseline transmission rate | | month | Estimated |
| The magnitude of forcing | 0.6 | none | [54] |
| The initial phase | | none | Estimated |
| Migration rate | 78 | month | [18], [33] |
| Natural death rate of cercariae | 0.12 | month | [18], [36] |