CN115437236A - Fractional order modeling method for new coronary pneumonia propagation process - Google Patents
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Abstract
The invention discloses a fractional order modeling method for a new coronary pneumonia propagation process, which comprises the following steps: the general population is divided according to the form of spreading the new coronavirus pneumonia epidemic: susceptible population, latent population, symptomatic infection population, asymptomatic infection population and rehabilitation population; the method comprises the steps of defining each group of people through Caputo-Fabrizio fractional order, establishing fractional order models of each group of people, spreading viruses to a certain area according to a symptomatic infection coefficient and an asymptomatic infection coefficient through joint seal of the rest symptomatic infection groups and the rest asymptomatic infection groups to form a new coronary pneumonia patient group in the area, and establishing the fractional order models of the new coronary pneumonia patient group in the area through Caputo-Fabrizio fractional order definition. The fractional order modeling method for the new coronary pneumonia propagation process improves the accuracy of modeling, accurately predicts the epidemic situation propagation trend and facilitates the follow-up epidemic situation control research.
Description
Technical Field
The invention relates to the technical field of mathematical modeling of a dynamic process of new coronary pneumonia propagation, in particular to a fractional order modeling method of the new coronary pneumonia propagation process.
Background
Coronaviruses, which have a unique corona or "corona" glycoprotein on their surface, were formally named coronaviruses in 1960, and have been responsible for fatal diseases such as middle east respiratory syndrome (MERS-CoV), severe acute respiratory syndrome (SARS-CoV), and the like. The novel coronavirus pneumonia (new coronavirus CoVID-19 for short) is a discovered novel coronavirus which has strong infectivity, high propagation speed and various propagation ways, symptoms such as fever, cough, tachypnea, dyspnea and the like can appear at the initial stage of infection, and the infection can cause pneumonia, severe acute respiratory syndrome, renal failure and even death along with the development of illness state. Because epidemic situation threatens the life and property safety of the nation and people seriously and restricts the development of human society, the research on the dynamic characteristics of the new coronary pneumonia and the prediction of the spreading trend of the new coronary pneumonia have extremely important practical significance for inhibiting the spread of the epidemic situation.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a fractional order modeling method for a new crown pneumonia propagation process, the advantages of the natural characteristics of a system can be better described according to a fractional order model, the defect of kernel singularity in the traditional fractional order definition is considered, the Caputo-Fabrizio fractional order definition is adopted to carry out mathematical modeling on the dynamic process of the new crown pneumonia propagation, the fractional order calculus operator under the definition can more accurately describe the memory effect of a real system, the modeling accuracy is improved, the epidemic situation propagation trend is accurately predicted, and the follow-up epidemic situation control research is facilitated.
In order to achieve the technical purpose, the invention adopts the following technical scheme: a fractional order modeling method for a new coronary pneumonia propagation process specifically comprises the following steps:
step S1, dividing the general population into: susceptible population, latent population, symptomatic infection population, asymptomatic infection population and rehabilitation population;
s2, establishing a fractional order mathematical model of the susceptible population by defining the susceptible population through a Caputo-Fabrizio fractional order according to the daily average growth number of the population of patients with the new coronary pneumonia;
s3, the susceptible population is converted into a latent population by an infection coefficient lambda (t) through contacting with the virus, and a fractional order mathematical model of the latent population is established through Caputo-Fabrizio fractional order definition;
s4, converting part of latent people into symptomatic infection groups according to the proportion of the symptomatic infection groups and the latent period of the symptomatic infection groups, and establishing a fractional order mathematical model of the symptomatic infection groups through Caputo-Fabrizio fractional order definition;
s5, converting the rest latent population into an asymptomatic infection population through the proportion of the asymptomatic infection population and the latent period of the asymptomatic infection population, defining the asymptomatic infection population through a Caputo-Fabrizio fractional order, and establishing a fractional order mathematical model of the asymptomatic infection population;
s6, through active treatment of the medical system, part of the symptomatic infected people can recover at the symptomatic recovery rate, part of the asymptomatic infected people can recover at the asymptomatic recovery rate, the recovery people are obtained, and a fractional order mathematical model of the recovery people is established through Caputo-Fabrizio fractional order definition;
and S7, respectively spreading the viruses to a certain area by the remaining symptomatic infection groups and the remaining asymptomatic infection groups through close contact according to the symptomatic infection coefficients and the asymptomatic infection coefficients to form a new coronary pneumonia patient group in the area, and establishing a fractional order mathematical model of the new coronary pneumonia patient group in the area through Caputo-Fabrizio fractional order definition.
Further, the susceptible population S (t) satisfies: s (t) -S (0) = f CF I α [Λ-λ(t)S(t)-μS(t)]
The latency group E (t) satisfies the following conditions: e (t) -E (0) = E (t) = CF I α [λ(t)S(t)-(θρ+(1-θ)ω+μ)E(t)]
The symptomatic infection group I (t) meets：I(t)-I(0)＝ CF I α [(1-θ)ωE(t)-(τ+μ)I(t)]
The asymptomatic infection group A (t) satisfies the following conditions: a (t) -A (0) = CF I α [θρE(t)-(τ a +μ)A(t)]
The rehabilitation population R (t) meets the following conditions: r (t) -R (0) = C CF I α [τ a A(t)+τI(t)-μR(t)]
The new coronary pneumonia patient population P (t) in the certain area meets the following requirements:
wherein the content of the first and second substances, CF I α is a Caputo-Fabrizio type fractional order integral operator, S (0) is susceptible population at the time 0, Λ is the daily increase number of new coronary pneumonia patients, E (0) is latent population at the time 0,
eta is contact rate, I (0) is symptom-infected population at time 0, psi is multiple inheritance rate, A (0) is asymptomatic-infected population at time 0, N (t) is general population at time t, eta w The disease spreading coefficient is mu is natural mortality, theta is the proportion of asymptomatic infection population, rho is the incubation period of asymptomatic infection population, omega is the incubation period of symptomatic infection population, tau is the recovery rate of symptomatic infection population, tau a The recovery rate of asymptomatic infection population, R (0) is the population recovering at time 0, P (0) is the population of patients with new coronary pneumonia in a certain area at time 0, zeta is the proportion of P (t) formed by introducing new coronary virus into a certain area through I (t),
v is the virus removal rate in a certain region P (t) in order to introduce new coronavirus into a certain region through a (t) to constitute the proportion of P (t).
Further, the establishing process of the fractional order mathematical model of the susceptible population is as follows:
CF D α S(t)＝Λ-λ(t)S(t)-μS(t) (1)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
Further, the establishment process of the fractional order mathematical model of the latent population is as follows:
CF D α E(t)＝λ(t)S(t)-(θρ+(1-θ)ω+μ)E(t) (2)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
Further, the establishment process of the fractional order mathematical model of the symptomatic infection population is as follows:
CF D α I(t)＝(1-θ)ωE(t)-(τ+μ)I(t) (3)
wherein the alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
Further, the establishment process of the fractional order mathematical model of the asymptomatic infection population is as follows:
CF D α A(t)＝θρE(t)-(τ a +μ)A(t) (4)
wherein the alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional differential operator.
Further, the establishing process of the fractional order mathematical model of the rehabilitation group comprises the following steps:
CF D α R(t)＝τ a A(t)+τI(t)-μR(t) (5)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
Further, the determination process of the fractional order mathematical model of the new coronary pneumonia patient population in the certain region is as follows:
wherein, alpha is E(0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
Compared with the prior art, the invention has the following beneficial effects:
(1) The invention firstly proposes that the Caputo-Fabrizio fractional calculus theory is applied to mathematical modeling of the new crown pneumonia propagation process, the defect that a traditional fractional calculus operator has a singular kernel can be effectively overcome, the memory effect of a real system is better reflected, and the modeling accuracy is improved;
(2) The fractional order modeling method for the new coronary pneumonia in the spreading process fully considers the characteristics of several types of crowds appearing in the spreading process of the epidemic situation, improves the traditional infectious disease model, increases the fractional order models of asymptomatic infection crowds and crowds with patients with new coronary pneumonia in a certain area in the spreading process of the new coronary pneumonia, and can better describe the spreading characteristics of the epidemic situation;
(3) The fractional order modeling method for the new coronary pneumonia propagation process can verify the rationality of the modeling method through a fixed point theory, can calculate the numerical solution of the established model through a three-step Adams-Bashforth method, defines the relation between the fractional order and the number of various crowds, is convenient to predict the epidemic propagation trend, and provides reference for an epidemic prevention and control department to formulate an effective epidemic intervention scheme.
Drawings
FIG. 1 is a schematic diagram of a fractional order mathematical model of the novel process of coronary pneumonia propagation of the present invention;
FIG. 2 is a graph showing the time-dependent trend of a susceptible population at different fractional orders;
FIG. 3 is a graph showing the time-dependent trend of different fractional order sub-laterals of a population;
FIG. 4 is a graph of the trend of a population with symptomatic infections over time for different fractional orders;
FIG. 5 is a graph of the trend of asymptomatic infection in a population over time for different fractional orders;
FIG. 6 is a graph showing the time-dependent trend of a population of convalescent people in different fractional orders;
fig. 7 is a graph of the trend of new patients with coronary pneumonia in a certain area with different fractional orders over time.
Detailed Description
To more clearly illustrate the objects and advantages of the present invention, the following further description, taken in conjunction with the accompanying drawings, is to be understood to include, but not be limited to, the following detailed description.
As shown in fig. 1, the present invention provides a fractional order modeling method for a new coronary pneumonia propagation process, which specifically includes the following steps:
step S1, dividing the general population into: susceptible population, latent population, symptomatic infection population, asymptomatic infection population and rehabilitation population; generally, government mandated measures for susceptible populations include: wearing a mask, washing hands on duty, disinfecting on duty and the like, and the interaction between the latent population and the new coronary pneumonia diagnosis population is related to the susceptible population, and the latent population is converted from the susceptible population by a certain proportion after contacting with the virus; the latent population is converted into a symptomatic infected population and an asymptomatic infected population at a certain ratio after a certain latent time; part of symptomatic infection population and asymptomatic infection population become rehabilitation population after treatment.
The susceptible population S (t) meets the following requirements: s (t) -S (0) = f CF I α [Λ-λ(t)S(t)-μS(t)
The latent population E (t) satisfies: e (t) -E (0) = c CF I α [λ(t)S(t)-(θρ+(1-θ)ω+μ)E(t)]
The symptomatic infection group I (t) satisfies: i (t) -I (0) = CF I α [(1-θ)ωE(t)-(τ+μ)I(t)]
Asymptomatic infection group A (t) satisfies: a (t) -A (0) = CF I α [θρE(t)-(τ a +μ)A(t)]
The rehabilitation population R (t) meets the following requirements: r (t) -R (0) = CF I α [τ a A(t)+τI(t)-μR(t)]
The new coronary pneumonia patient population P (t) in a certain area meets the following requirements:
wherein, the first and the second end of the pipe are connected with each other, CF I α is a Caputo-Fabrizio type fractional order integral operator, S (0) is susceptible population at the time 0, Λ is the daily increase number of new coronary pneumonia patients, E (0) is latent population at the time 0, and λ (t) is infection coefficient,
eta is contact rate, I (0) is symptom-infected population at time 0, psi is multiple inheritance rate, A (0) is asymptomatic-infected population at time 0, N (t) is general population at time t, eta w Mu is natural mortality, theta is the proportion of asymptomatic infected population, rho is the incubation period of asymptomatic infected population, omega is the incubation period of symptomatic infected population, tau is the recovery rate of symptomatic infected population, tau is the spread coefficient of the disease a The recovery rate of asymptomatic infection population, R (0) is the population recovering at time 0, P (0) is the population of patients with new coronary pneumonia in a certain area at time 0, zeta is the proportion of P (t) formed by introducing new coronary virus into a certain area through I (t),
v is the virus removal rate in a certain region P (t) in order to transmit new coronavirus through a (t) into a certain region constituting the proportion of P (t).
S2, establishing a fractional order mathematical model of the susceptible population by defining the susceptible population through a Caputo-Fabrizio fractional order according to the daily average growth number of the population of patients with the new coronary pneumonia; the establishing process of the fractional order mathematical model of the susceptible population comprises the following steps:
CF D α S(t)＝μ-λ(t)S(t)-μS(t) (1)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional differential operator.
S3, the susceptible population is converted into a latent population by an infection coefficient lambda (t) through contacting with the virus, and a fractional order mathematical model of the latent population is established through Caputo-Fabrizio fractional order definition; the establishment process of the fractional order mathematical model of the latent population comprises the following steps:
CF D α E(t)＝λ(t)S(t)-(θρ+(1-θ)ω+μ)E(t) (2)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
S4, converting part of latent population into symptomatic infection population according to the proportion of the symptomatic infection population and the latent period of the symptomatic infection population, and establishing a fractional order mathematical model of the symptomatic infection population through Caputo-Fabrizio fractional order definition; the establishment process of the fractional order mathematical model of symptomatic infection population comprises the following steps:
CF D α I(t)＝(1-θ)ωE(t)-(τ+μ)I(t) (3)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
S5, converting the rest latent population into an asymptomatic infection population through the proportion of the asymptomatic infection population and the latent period of the asymptomatic infection population, defining the asymptomatic infection population through a Caputo-Fabrizio fractional order, and establishing a fractional order mathematical model of the asymptomatic infection population; the establishment process of the fractional order mathematical model of asymptomatic infection population in the invention is as follows:
CF D α A(t)＝θρE(t)-(τ a +μ)A(t) (4)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional differential operator.
S6, through active treatment of the medical system, part of the symptomatic infected people can recover at the symptomatic recovery rate, part of the asymptomatic infected people can recover at the asymptomatic recovery rate, the recovery people are obtained, and a fractional order mathematical model of the recovery people is established through Caputo-Fabrizio fractional order definition; the establishing process of the fractional order mathematical model of the rehabilitation population comprises the following steps:
CF D α R(t)＝τ a A(t)+τI(t)-μR(t) (5)
wherein the alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
S7, spreading the viruses to a certain area by virtue of the symptomatic infection coefficient and the asymptomatic infection coefficient of the rest symptomatic infection groups and the rest asymptomatic infection groups respectively through joint sealing to form a new coronary pneumonia patient group in the area, and establishing a fractional order model of the new coronary pneumonia patient group in the area through Caputo-Fabrizio fractional order definition; the determination process of the fractional order mathematical model of the new coronary pneumonia patient population in a certain area in the invention is as follows:
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional differential operator.
The fractional order mathematical models of various crowds in the new coronary pneumonia spreading process are solved by the following method:
the Caputo-Fabrizio type fractional differentiation of various populations is defined as follows:
where f (t) is S (t), E (t), I (t), A (t), R (t) or P (t), and M (α) is a normalization function depending on α, the Caputo-Fabrizio type fractional order differential can be expressed as:
then the alpha-order Caputo-Fabrizio type fractional order integral of each population is defined as:
the rationality of various types of crowd fractional order mathematical models constructed by the invention is analyzed based on the theory of fixed points as follows:
for the convenience of subsequent analysis, the following formal kernels are defined:
the following formal function is defined:
according to practical conditions, S (t), E (t), I (t), A (t), R (t), P (t) and N (t) are all non-negative bounded functions, namely | | | S (t) | = epsilon | | 1 、||E(t)||＝ε 2 、||I(t)||＝ε 3 、||A(t)||＝ε 4 、||R(t)||＝ε 5 ，||P(t)||＝ε 6 And N (t) | = epsilon, as defined below:
calculating a Caputo-Fabrizio type fractional order integral for equations (1) - (6) can be found:
from equation (13) and using a recursive formula, the difference of consecutive terms in the recursive formula can be obtained:
from the formula (14):
based on the recursive inequality and the trigonometric inequality, the following can be obtained:
if there is a time t 0 So that omega (alpha) gamma i +σ(α)γ i t 0 < 1,i =1,2,3,4,5,6, then there are solutions for the fractional order models (1) - (6) of new coronary pneumonia, i.e. it is reasonable and feasible to establish the fractional order model of the spread of new coronary pneumonia.
Next, the established fractional order model of the spread of new coronary pneumonia was validated for rationality. According to formulas (15) and (17), it is possible to obtain:
only the proving function S is required n (t)，E n (t)，I n (t)，A n (t)，R n (t)，P n (t) converges to the solution of the model (16). Definition B n (t)，C n (t)，D n (t)，F n (t)，G n (t)，H n (t) are the remainder of each equation (18) after n iterations, respectively, then
All kernels K of the invention i (i =1,2,3,4,5,6) all satisfy the Lipschitz condition, and using the triangle inequality, we can obtain:
the same can be obtained:
||C n (t)||≤[(Ω(α)+ω(α)t)γ 2 ] n+1 ε 2 ,||D n (t)||≤[(Ω(α)+ω(α)t)γ 3 ] n+1 ε 3 ,
||F n (t)||≤[(Ω(α)+ω(α)t)γ 4 ] n+1 ε 4 ,||G n (t)||≤[(Ω(α)+ω(α)t)γ 5 ] n+1 ε 5 ,
||H n (t)||≤[(Ω(α)+ω(α)t)γ 6 ] n+1 ε 6 (20)
obviously, when n → ∞ is n | | | B n (t)||→0,||C n (t)||→0,||D n (t)||→0,||F n (t)||→0,||G n (t)||→0,||H n (t) | → 0, and therefore, solutions exist for the established fractional models (1) - (6) of the spread of new coronary pneumonia.
The uniqueness of the solution in the established fractional order model of the new coronary pneumonia propagation process is verified, and the correctness and the rationality of the fractional order model modeling method of the new coronary pneumonia propagation process are further verified.
To demonstrate the uniqueness of the solutions, it is assumed that there is another set of solutions to build a fractional order model (1) - (6) of the spread of new coronary pneumonia: s. the 1 ，E 1 ，I 1 ，A 1 ，R 1 ，P 1 And then:
according to the Lipschitz condition satisfied by S (t), the following can be obtained:
||S(t)-S 1 (t)||≤Ω(α)γ 1 ||S(t)-S 1 (t)||+ω(α)γ 1 t||S(t)-S 1 (t)|| (22)
further, the method can be obtained as follows:
||S(t)-S 1 (t)||(1-θ(α)γ 1 -ω(α)γ 1 t)≤0 (23)
if (1-omega (. Alpha.) gamma.) is satisfied i -ω(α)γ i t) > 0, i =1,2,3,4,5,6, the fractional order model (1) - (6) for establishing new coronary pneumonia transmission has unique solutions. The reason is that, under the given conditions, the assumption that equation (23) holds is | | | S (t) -S 1 (t)||(1-Ω(α)γ 1 -ω(α)γ 1 t) =0, i.e. | | S (t) -S 1 (t) | =0, then S (t) = S 1 (t), the same holds: e (t) = E 1 (t)，I(t)＝I 1 (t)，A(t)＝A 1 (t)，R(t)＝R 1 (t)，P(t)＝P 1 (t) of (d). The theoretical analysis results show that the new coronary pneumonia model established based on the Caputo-Fabrizio fractional order definition is reasonably feasible.
Examples
In the present embodiment, the population N (t) =8266000, the system parameter Λ =294.92, η =0.05, ψ =0.02, η w ＝0.000001231，
θ＝0.1243，ρ＝0.005，ω＝0.00047876，τ＝0.09871，τ a ＝0.854302，ζ＝0.000398，
ν =0.01, initial conditions S (0) =8065518, e (0) =200000, i (0) =282, a (0) =200, r (0) =0,P (0) =50000.
As shown in fig. 2, when α =0.8,0.85,0.9,0.95 in the fractional modeling method of the new coronary pneumonia transmission process proposed by the present invention is α = 5363, the variation trend of the susceptible population S (t) can be seen from fig. 2, and as the time increases, the number of the susceptible population S (t) will gradually decrease with the spread of the virus infection under the premise that the general population is fixed.
Fig. 3-6 show the time-dependent changes of the latent group E (t), the symptomatic infection group I (t), the asymptomatic infection group a (t), and the convalescent group R (t) when the fractional order α =0.8,0.85,0.9,0.95 and the remaining system parameters are unchanged, and the time-dependent changes of the above groups of people at the same time with the increase of the fractional order.
Fig. 7 shows that when the fractional order α =0.8,0.85,0.9,0.95 and the rest system parameters are unchanged, the population P (t) of new coronary pneumonia patients in a certain area changes with time, and obviously, the population P (t) of new coronary pneumonia patients in a certain area gradually decreases with the increase of the population of recovery patients.
The above are only preferred embodiments of the present invention, and the scope of the present invention is not limited to the above examples, and all technical solutions that fall under the spirit of the present invention belong to the scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.
Claims (8)
1. A fractional order modeling method for a new coronary pneumonia propagation process is characterized by comprising the following steps:
step S1, dividing the general population into: susceptible population, latent population, symptomatic infection population, asymptomatic infection population and rehabilitation population;
s2, establishing a fractional order mathematical model of the susceptible population by defining the susceptible population through a Caputo-Fabrizio fractional order according to the daily average growth number of the population of patients with the new coronary pneumonia;
s3, the susceptible population is converted into a latent population by an infection coefficient lambda (t) through contacting with the virus, and a fractional order mathematical model of the latent population is established through Caputo-Fabrizio fractional order definition;
s4, converting part of latent population into symptomatic infection population according to the proportion of the symptomatic infection population and the latent period of the symptomatic infection population, and establishing a fractional order mathematical model of the symptomatic infection population through Caputo-Fabrizio fractional order definition;
s5, converting the rest latent population into an asymptomatic infection population through the proportion of the asymptomatic infection population and the latent period of the asymptomatic infection population, defining the asymptomatic infection population through a Caputo-Fabrizio fractional order, and establishing a fractional order mathematical model of the asymptomatic infection population;
s6, through active treatment of the medical system, part of symptomatic infected people recover at a symptomatic recovery rate, part of asymptomatic infected people recover at an asymptomatic recovery rate to obtain recovered people, and a fractional order mathematical model of the recovered people is established through Caputo-Fabrizo fractional order definition;
and S7, respectively spreading the viruses to a certain area by the remaining symptomatic infection groups and the remaining asymptomatic infection groups through close contact according to the symptomatic infection coefficients and the asymptomatic infection coefficients to form a new coronary pneumonia patient group in the area, and establishing a fractional order mathematical model of the new coronary pneumonia patient group in the area through Caputo-Fabrizio fractional order definition.
2. The fractional order modeling method of the new coronary pneumonia propagation process according to claim 1, wherein the susceptible population S (t) satisfies the following condition: s (t) -S (0) = CF I α [Λ-λ(t)S(t)-μS(t)]
The latent population E (t) satisfies: e (t) -E (0) = c CF I α [λ(t)S(t)-(θρ+(1-θ)ω+μ)E(t)]
The symptomatic infection group I (t) satisfies: i (t) -I (0) = CF I α [(1-θ)ωE(t)-(τ+μ)I(t)]
The asymptomatic infection group A (t) satisfies the following conditions: a (t) -A (0) = CF I α [θρE(t)-(τ a +μ)A(t)]
The rehabilitation population R (t) meets the following requirements: r (t) -R (0) = CF I α [τ a A(t)+τI(t)-μR(t)]
The new coronary pneumonia patient population P (t) in the certain area meets the following requirements:
wherein the content of the first and second substances, CF I α is a Caputo-Fabrizio type fractional order integral operator, S (0) is susceptible population at the time 0, Λ is the daily average increment of new coronary pneumonia patient population, and E (0) is latent population at the time 0The population of people is provided with the health-care tea,
eta is contact rate, I (0) is symptom-infected population at time 0, psi is multiple inheritance rate, A (0) is asymptomatic-infected population at time 0, N (t) is general population at time t, eta w The disease spreading coefficient is mu is natural mortality, theta is the proportion of asymptomatic infection population, rho is the incubation period of asymptomatic infection population, omega is the incubation period of symptomatic infection population, tau is the recovery rate of symptomatic infection population, tau a The recovery rate of asymptomatic infected people, R (0) is the recovery people at time 0, P (0) is the new coronary pneumonia patients in a certain area at time 0, zeta is the proportion of P (t) formed by importing new coronary viruses into a certain area through I (t),
v is the virus removal rate in a certain region P (t) in order to introduce new coronavirus into a certain region through a (t) to constitute the proportion of P (t).
3. The method according to claim 2, wherein the establishing process of the fractional order mathematical model of the susceptible population is as follows:
CF D α S(t)＝Λ-λ(t)S(t)-μS(t) (1)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
4. The method according to claim 2, wherein the establishing of the fractional mathematical model of the latent population comprises:
CF D α E(t)＝λ(t)S(t)-(θρ+(1-θ)ω+μ)E(t) (2)
wherein alpha epsilon (0,1) is the system fractional order, CF D α as a Caputo-FabriziAnd (4) an o-type fractional order differential operator.
5. The fractional order modeling method for the new coronary pneumonia propagation process according to claim 2, wherein the fractional order mathematical model of the symptomatic infection population is established by:
CF D α I(t)＝(1-θ)ωE(t)-(τ+μ)I(t) (3)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
6. The fractional order modeling method for the new coronary pneumonia propagation process according to claim 2, wherein the fractional order mathematical model of the asymptomatic infection group is established by the following steps:
CF D α A(t)＝θρE(t)-(τ a +μ)A(t) (4)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
7. The method for fractional order modeling of a new coronary pneumonia propagation process according to claim 2, wherein the fractional order mathematical model of the convalescent population is established by:
CF D α R(t)＝τ a A(t)+τI(t)-μR(t) (5)
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
8. The method of claim 2, wherein the fractional order model of the new coronary pneumonia propagation process is determined by a fractional order mathematical model of a population of patients with new coronary pneumonia in a certain region:
wherein alpha epsilon (0,1) is the system fractional order, CF D α is a Caputo-Fabrizio type fractional order differential operator.
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| CN117174324A (en) * | 2023-10-27 | 2023-12-05 | 中国人民解放军总医院第一医学中心 | Respiratory system modeling method based on hybrid model and electronic equipment |
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| CN116911212A (en) * | 2023-07-31 | 2023-10-20 | 中国人民解放军总医院第一医学中心 | Respiratory system modeling method based on fractional calculus |
| CN116911212B (en) * | 2023-07-31 | 2024-03-19 | 中国人民解放军总医院第一医学中心 | Respiratory system modeling method based on fractional calculus |
| CN117174324A (en) * | 2023-10-27 | 2023-12-05 | 中国人民解放军总医院第一医学中心 | Respiratory system modeling method based on hybrid model and electronic equipment |
| CN117174324B (en) * | 2023-10-27 | 2024-02-02 | 中国人民解放军总医院第一医学中心 | Respiratory system modeling method based on hybrid model and electronic equipment |
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