Schemes Comparison

Here we discuss how these vaccination schemes work, we use h1i, h2i and h3i to denote the schemes described in Sections (4.2.1), (4.2.2) and (4.2.3) for brevity. Scheme h1i has higher efficiency of reducing the severity of disease and subsequently the possibility of an epidemic compared with h2i. Such an example is shown at Figure 4.1(a) and 4.1(b).

Although h1i and h2i have used the same number of vaccines, h1i is more effective on reducing max(I_m) and R(∞). In Figure 4.1(a), the fraction of R(∞) = 0.15 in h1i that

is equal to the fraction of S_c(0). From this we find that no S_n is infected, that is, h1i effectively prevents the outbreak of an epidemic indeed.

(a)β = 2

Figure 4.1: Numerical simulation for the schemes with fixed S0 = 0.95, µ = 1.0,

² = 0.0001; r0= 0.7 for h1i and v = 10 for h3i. Scheme h2i uses vaccination rate with v = 0.1 in (a) and v = 0.3 in (b). We assume that the number of vaccines can provide 70 percent of the total population to be vaccinated.

Next we discuss the difference between h2i and h3i. Figure 4.2(a) shows the advantage of h2i on handling a less virulent disease case(smaller β). Scheme h2i has higher efficiency of reducing the severity of the disease, whereas h3i uses the same number of vaccines but has worse result of larger max(I_m) and R compared with h2i. Ideally, we would like to have ample amount of vaccines for preventing the disease. But sometimes, the availability of vaccines is a problem because of few vaccines are produced. From Figure 4.2(a) we notice that variation of the numbers of the infectives and the removed people R almost tends to a steady state at about time=7. In this case, it is possible to save vaccines using scheme

h2i by stopping the supplement of vaccines at time=5, the result is shown in Figure 4.2(b) with scheme h2i maintaining its efficiency while less vaccines are being used. However, there two schemes show similar effect on controlling the disease when β increases. Figure 4.3 represents a higher virulent case(large β) in which the speed of the epidemic spreading is too fast for h2i and h3i to take effect.

Usually we would prefer that there are less infected people, thus we will choose h1i to achieve this. But sometimes there are not enough vaccines available because of the lack of money or few vaccines have been produced, this will cause the vaccines to be administered differently under different circumstances.

Figure 4.2: Numerical simulation for the schemes with fixed S0 = 0.95, µ = 1.0,

² = 0.0001; r0= 0.7 for h1i, v = 0.1 for h2i and v = 10 for h3i. (a) has an additional restriction that vaccination of h2i is stopped if Rv = 0.58. Vaccination of h2i is stopped at time = 5 in (b).

β = 10

Figure 4.3: Numerical simulation for the schemes with fixed S0 = 0.95, µ = 1.0,

² = 0.0001; r0= 0.7 for h1i, v = 0.3 for h2i and v = 14.8 for h3i.

5 Conclusion

After the discussion of some compartmental models of epidemiology in Section 1, we ex-tend the standard SIR model to describe the dynamics of an infectious disease and use the avian influenza as an example. Different from the standard SIR model, our model indicates an epidemic time delay during which the disease may develop into a mutated form which is fast spreading among humans. Thus an epidemic subsequently occurs if the basic reproductive ratio R₀ > 1. In this model, a significant result is that R₀ ∼ βS₀ is different from that of the standard SIR model so that we can control R₀ through the manipulation of β and S₀, recalling that S₀ in the dimensional term is the number of the majority of the susceptibles, to lower the impact of the disease and subsequently prevent an epidemic.

In Section 4, we extend our model to accommodate the programmes of quarantine and some schemes of vaccination for eradication and control of the disease. The raised thresh-old of R₀ shows the effect of quarantine. The practicality of the three vaccination schemes is tested in different cases. The first scheme is most efficient in reducing the impact of the disease but can only be administered if sufficient amount of vaccines is available at t = 0.

Deciding which schemes to be applied depends on many factors such as the production cost of vaccination, the seriousness of the disease , and whether taking the vaccination is convenient, and so on.

Because the avian influenza is complicated, our model has been simplified in order to carry

out asymptotics and hence some properties of the disease has been omitted. The model can be improved by including these properties. For example, the mutation process is com-monly a random process, and thus can be modified by using a stochastic method. Diseases may also have latent period, so the model will be more realistic if we introduce an exposed class. However, our model predicts a possible transmissible disease between humans, which is resulted from a disease that infects few humans. This model indicates an additional way of disease control which is given by the information of the basic reproductive ratio βS₀.

Reference

[1] DCD(Division of communicable disease control) Newsletter. (2005) (7).

[2] JD Murray, 2002. Mathematical biology, 3rd edition. Bainbridge Island, Washington.

[3] F. Brauer. Some simple epidemic models. Math. Biosci. Eng. (2006) 3(1), 1-15.

[4] H. W. Hethcote. The mathematics of infectious diseases. SIAM Review (2000) 42(4), 599-653.

[5] D. J. D. Earn. Mathematical modelling of recurrent epidemics. π in the sky (2004) 8, 14-17.

[6] R. M. Bush. Influenza as a model system for studying the cross-species transfer and evolution of the SARS coronavirus. Phil Trans. R. Soc. Lond. B (2004) 359, 1067-1073.

[7] M. J. Wonham, T. de-Camino-Beck and M. A. Lewis. An epidemiological model for West Nile virus: invasion analysis and control applications. Phil Trans. R. Soc. Lond. B (2004) 271, 501-507.

[8] F. Brauer. The Kermacl-McKendrick epidemic model revisited. Math. Biosi. (2005) 198, 119-131.

Appendix A: A more complete model

In this appendix, we present a more detailed model of the bird flu case. The model presented here is by no means a complete model, but will give indication on the complex-ity of the epidemic and the necesscomplex-ity to simplify the model in order to carry out analysis.

The whole population is divided into three main classes, namely susceptibles, infectives and removed. A small proportion of the susceptibles is considered to be the chicken-farmers who have direct contact with the birds and are at risk of catching bird flu from birds. The rest of the susceptibles are considered as the non-chicken-farmers who will not catch bird flu. Bird flu does not transmit among humans, so in this situation those infected with bird flu can not infect others. Once the bird flu virus mutates to a human-human transmissible form, the mutated disease can be spread between humans. There is a probable case that a chicken-farmer gets two diseases if he or she is infected by birds and humans before recovering from one of the diseases. If the infectives of the chicken-farmers recover from one type of the viruses, they are immune from the one but still are susceptible to the other disease. A schematic of the complete model is as follows,

2

where S, I, and R denote the classes of population: susceptibles, infectives, and removed, respectively. The subscripts {c,n} denote chicken-farmers and non-chicken-farmers, re-spectively; the superscripts {b, m, bm} denote the diseases that the person catch, namely bird flu, mutation flu and both diseases, respectively; the subscripts {b, m, bm} denote the diseases that the person is immune from. The parameter µ is the infection rate constant of bird flu from birds to humans, where the number of birds is assumed constant, β the infectious contact rate of the mutated disease, α the mutation rate from I_c^b to I_c^m while γ₁ and γ₂ are the recovery rates of bird flu and the mutated flu, respectively.

Population is assumed closed in this epidemic model, and therefore the governing equations are as follows:

d_bS_c

dt = γ₁ I_c^b− β _bS_c ( _bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) , dS_c

dt = −µ S_c− β S_c (_bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) , d_mS_c

dt = γ₂ I_c^m− µ _mS_c, dS_n

dt = −β S_n ( _bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) , d_bI_c^m

dt = β _bS_c( _bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) + γ₁ I_c^bm− γ_{2 b}I_c^m , dI_c^b

dt = µ S_c− γ₁ I_c^b− β I_c^b ( _bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) − α I_c^b , dI_c^bm

dt = β I_c^b (_bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) + µ I_c^m− γ₁ I_c^bm− γ₂ I_c^bm , dI_c^m

dt = β S_c( _bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) + α I_c^b− µ I_c^m− γ₂ I_c^m , d_mI_c^b

dt = µ_mS_c+ γ₂ I_c^bm− γ_{1 m}I_c^b , dI_n^m

dt = β S_n( _bI_c^m+ I_c^m+ I_c^bm+ I_n^m ) − γ₂ I_n^m , dR

dt = γ_{2 b}I_c^m+ γ_{1 m}I_c^b+ γ₂ I_n^m .