USE OF A PERIODIC VACCINATION STRATEGY TO CONTROL THE SPREAD OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE. Islam A. Moneim.

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1 MATHEMATICAL BIOSCIENCES AND ENGINEERING Volume?, Number?,???????? pp.???? USE OF A PERIODIC VACCINATION STRATEGY TO CONTROL THE SPREAD OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE Islam A. Moneim Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt. David Greenhalgh Department of Statistics and Modelling Science, University of Strathclyde Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK. Communicated by Urszula Ledzewicz and Andrzej Swierniak) Abstract. In this paper a general periodic vaccination has been applied to control the spread and transmission of an ectious disease with latency. An SEIRS epidemic model with general periodic vaccination strategy is analysed. We suppose that the contact rate has period T and the vaccination function has period LT, where L is an integer. Also we apply this strategy in a model with seasonal variation in the contact rate. Both the vaccination strategy and the contact rate are general time dependent periodic functions. Also the same SEIRS models have been examined for a mixed vaccination strategy composed of both of the time dependent periodic vaccination strategy and the conventional one. A key parameter of the paper is a conjectured value R c for the basic reproduction number. R sup and R are respectively upper and lower bounds for R c. We prove that the disease free solution DFS) is globally asymptotically stable GAS) when R sup < 1. If R > 1 then the DFS is unstable and we prove that there exists a non trivial periodic solution of period the same as that of the vaccination strategy. Some persistence results are also discussed. Necessary and sufficient conditions for the eradication or control of the disease are derived. Threshold conditions for these vaccination strategies to ensure that R sup < 1 and R > 1 are also investigated. LIST OF SYMBOLS, ABBREVIATIONS AND MEDICAL TERMS St) - number of susceptibles at time t; Et) - number of exposed individuals at time t; It) - number of ected individuals at time t; Rt) - number of removed individuals at time t; N - total population size; 2 Mathematics Subject Classification. 92D3. Key words and phrases. mathematical modelling, disease control, periodic vaccination, basic reproduction number R, periodicity, childhood diseases, uniform strong repeller, uniform persistence. 1

2 2 I. A. MONEIM AND D. GREENHALGH βt) - total rate at which potentially ectious contacts occur between two individuals; rt) - general continuous periodic vaccination strategy; T - period of contact rate; L - LT is period of vaccination function; 1/) - average latent period conditional on survival to the end of it; 1/δ) - average immune period conditional on survival to the end of it; 1/γ) - average ectious period conditional on survival to the end of it; µ - common per capita birth and death rate; p - fraction of newborn children who are vaccinated; δt) - Dirac delta function; R - basic reproduction number; R c - conjectured value for basic reproduction number; R sup - upper bound for R c ; R - lower bound for R c ; τ = 1/µ + γ) - average length of ectious period; DFS - disease free solution; GAS - globally asymptotically stable; Ŝt) - susceptible population at DFS; ˆRt) - recovered population at DFS; E - lim sup t Et); I - lim sup t It). 1. Introduction. Mass immunisation is frequently used as a tool to control the spread of epidemics. The simplest vaccination strategy is to vaccinate all individuals at a constant rate. This may also be combined with vaccination of a fixed fraction of very young children at the smallest possible age where maternal antibodies no longer confound the effect of the vaccine, commonly 9-18 months for measles. In this paper we ignore the effect of maternal antibodies so these young children are in essence vaccinated at birth. In the absence of vaccination cases of many common childhood diseases show a regular periodic oscillation with period a whole number of years 9, 14. Much work has been done analysing seasonal periodic outbreaks of ectious diseases considering seasonal variation in the contact rate 4, 9, 14. Recently it has been postulated that in some circumstances a periodic vaccination strategy, for example pulse vaccination, can be a more efficient use of limited immunisation resources than continuous constant vaccination effort 2, 11, 15. In this paper we study a general continuous periodic vaccination strategy rt). This is combined with vaccination of a given proportion of newborn individuals. As in many real diseases there is a time delay between an individual becoming ected and becoming ectious we introduced an exposed or latent class into the model. We consider the model both with a periodic disease transmission rate and a constant one.

3 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 3 If the combined vaccination strategy is applied in the situation where no disease is present then the number of susceptibles eventually reaches a unique periodic solution. Our results lead us to conjecture that this combined periodic and fixed vaccination strategy is sufficient to eliminate disease from the population exactly when the weighted time-averaged disease-free susceptible population is less than a certain threshold value The pulse vaccination strategy. Pulse vaccination vaccinates susceptibles at discrete points in time, usually at regular intervals. One such example is the use of annual immunisation days which were successful in eradicating measles from The Gambia between 1967 and In recent times a pulse vaccination strategy has been applied in South and Central America and highly successfully applied to poliomyelitis in the same region 6, 13. This method is now used in Brazil and is both easier to arrange and has a greater uptake than the conventional continuous vaccination strategy. Pulse vaccination has been used in Africa recently albeit with only partial success. Agur et al. 2 discuss the possibility of implementation of the pulse vaccination method in Israel. Pulse vaccination has also been used in the United Kingdom. In November 1994 a single dose of combined measles and rubella MR) vaccine was given to children aged 5 to 16 years. In England and Wales an average of 92% of these children were vaccinated. This policy caused a significant fall in the number of cases of measles reported to the Office of Population Censuses and Surveys. It was concluded that application of pulse immunisation to all schoolchildren would probably prevent a large amount of morbidity and mortality and would have a marked effect on measles transmission for several years 11. Nokes and Swinton 11 use simple steady-state and age-structured dynamic models to extend the theory of the mechanism of action of pulse vaccination, and to explore the relationship between the maximum permitted interval between pulses and key epidemiological, demographic and vaccination variables. They develop the work of Agur et al. 2. An ordinary differential equation model is used to derive equilibrium expressions for the pulse interval and considers combined routine and pulse vaccination. Simulations using age-structured compartmental deterministic models illustrate complex epidemiological dynamics associated with pulse vaccination, particularly when there is age heterogeneity in contact rates in the population. Pulse vaccination in an SIR epidemic model with vaccination has been considered by Shulgin et al. 15. They consider a vaccination function rt) = n= pδt nt ). Here δt) is the Dirac delta function and p is a constant. This corresponds to a series of pulse vaccinations, each separated by time T. They found that a periodic disease free solution is possible where both the numbers of susceptibles and recovereds are periodic functions with period equal to that of the pulse vaccination. Shulgin et al. also discovered a threshold condition for this periodic ection-free solution to be locally stable, firstly in the case where β, the transmission rate of the ection, is a constant, and secondly in the more general case where the disease transmission rate βt) is a non-constant periodic function with the same period, T, as rt). If this threshold is not exceeded then the periodic DFS is locally stable and a serious epidemic will not occur. Now we shall investigate a more realistic and complicated SEIRS model with a general continuous periodic vaccination rate rt). 2. The SEIRS model with vaccination. Our SEIRS model of the spread of ectious diseases makes the following assumptions:

4 4 I. A. MONEIM AND D. GREENHALGH 1. The total population size is N and the per capita birth rate is a constant µ. As births balance deaths we must have that the per capita death rate is also µ. 2. The population is uniform and mixes homogeneously. 3. The population is divided into susceptible, exposed, ective and recovered individuals. The total number of individuals in each of these classes are respectively S St), E Et), I It) and R Rt). 4. The ection rate βt) is defined as the total rate at which potentially ectious contacts occur between two individuals. A potentially ectious contact is one which will transmit the disease if one individual is susceptible and the other is ectious, so the total rate at which susceptibles become exposed is βt)si. Biological considerations mean that βt) is continuous. We also assume that either i) βt) is not identically zero, positive, non-constant and periodic of period T or ii) βt) = β is a constant. 5. The susceptibles move from the exposed class to the ective class at a constant rate where 1/) is the average latent period conditional on survival to the end of it. 6. The ectives move from the ective class to the recovered class at a constant rate γ where 1/γ) is the average ectious period conditional on survival to the end of it. We assume that the disease does not give permanent immunity so individuals transfer back from the immune to the susceptible class at a constant per capita rate δ. 7. A fraction p p 1) of all new-born children are vaccinated. In addition all susceptibles in the population are vaccinated at a time dependent periodic rate rt). This is the periodic vaccination strategy. We shall suppose that rt) is periodic with period LT. The case where rt) has period T can be obtained by setting L = 1. Our SEIRS model with time dependent vaccination strategy can be written as a set of four coupled non-linear ordinary differential equations as follows: and ds de di dr = µn1 p) βt)si µ + rt))s + δr, 1) = βt)si µ + )E, 2) = E µ + γ)i, 3) = µnp + rt)s + γi µ + δ)r, 4) with S + E + I + R = N. 5) Here the disease transmission rate βt) and the vaccination rate rt) are nonzero, positive, continuous periodic functions. The system 1)-5) has no equilibrium points but a disease free solution DFS), with Et) = It) = is still possible. Consider the region D in R 4 defined by D = {S, E, I, R), N 4 S + E + I + R = N}.

5 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 5 The system of differential equations 1)-4) with initial conditions in D obviously starts off in the region D. The right-hand sides of these equations are differentiable with respect to S, E, I and R with continuous derivatives. It is straightforward to show using standard techniques 7 and considering separately the cases E) = I) = and E) > or I) > ) that the equations 1)-4) with initial conditions in D have a unique solution that remains in D for all time and moreover S + E + I + R = N. 3. The disease free solution. In the case that rt) is a non-constant bounded continuous periodic function, there is no equilibrium point for the system 1)-5). So there is no disease free equilibrium point. But still there is a periodic DFS corresponding to the case that Et) = It) =. In this case 1) becomes ds = µn1 p) µ + rt))s + δr, = Nµ1 p) + δ) µ + rt) + δ)s. 6) If Et) = It) =, 6) has a solution for St). We shall examine the behaviour of this solution. Integrating 6) we find that Hence t St) = St ) exp µ + δ)t t ) rτ)dτ t t + Nµ1 p) + δ exp µ + δ)t t ) rτ)dτ t t t exp µ + δ)ζ t ) + St + n + 1)LT ) = St + nlt ) exp + Nµ1 p) + δ exp t+lt t exp ζ t rτ)dτ µ + δ)lt µ + δ)lt µ + δ)ζ t ) + dζ. 7) ζ t+lt t t t+lt t rτ)dτ rτ)dτ rτ)dτ dζ. 8) Equation 8) gives a recursive relationship between the number of susceptibles at time t + nlt, n = 1, 2, 3,.... If we let S n = St + nlt ) then 8) defines a mapping F such that F S n ) = S n+1. If S 1 and S 2 are different values of S then we have that F S 1 ) F S 2 ) S 1 S 2 exp µlt ).

6 6 I. A. MONEIM AND D. GREENHALGH So F is a contraction mapping 3 and has a unique fixed point S t ) such that S t ) = Nµ1 p) + δ exp t+lt t exp µ + δ)lt µ + δ)ζ t ) + ζ t rτ)dτ rτ)dτ 1 1 exp µ + δ)lt. 9) LT rτ)dτ Hence S t + LT ) = S t ). So S is a periodic function of t. Differentiating 9) S t ) is continuously differentiable with respect to t and Ŝt) = S t), Ê = Î = and ˆRt) = R t) = N S t) is a disease free periodic solution of the system 1)-5) which repeats itself every LT years. We have the following result: Theorem 3.1. Equations 1)-5) have a disease free periodic solution of period LT which is continuously differentiable and this is the only disease free periodic solution to 1)-5), and any disease free solution to 1)-5) approaches this one as time becomes large. Proof A straightforward adaption of the SIRS model considered in 1 for the case L = 1. Recall that R, the basic reproduction number of the disease is defined as the expected number of secondary cases caused by a single ected case entering the disease-free population at equilibrium 1. Anderson and May 1 call this the basic reproduction rate, but it is a number not a rate. Consider a single newly ected person entering the population at the DFS. During the latent period this person suffers a death rate µ and leaves for the ectious class at rate. Assuming that the time taken for these two events to happen follow independent exponential distributions the probability that the individual survives his or her incubation period is /µ + ). Similarly the average length of the ectious period is τ = 1/µ + γ). The average value taken over a cycle of the expected number of secondary cases produced by a single ected person entering the population at the DFS is our conjectured value for R, namely R c = 1 LT Define R sup = and µ + γ)µ + ) R = µ + γ)µ + ) βτ)ŝτ)dτ µ + )µ + γ). 1) sup t,lt t,lt µ + )βt ζ)ŝt ζ) exp µ + )ζdζ, 11) 1 exp µ + )LT µ + )βt ζ)ŝt ζ) exp µ + )ζdζ. 12) 1 exp µ + )LT Later we shall show that R sup R c R with both of the inequalities being strict if βt)ŝt) is non-constant on, LT. We expect that if Rc > 1 the disease will take off whereas if R c 1 the disease will die out. However we have been able to show only the result that if R sup < 1 the disease will die out and if R > 1 dζ )

7 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 7 then the disease will take off if it is initially present. In the following sections we shall formally investigate these results. 4. The stability analysis of the disease free solution. In this section we concentrate on the stability analysis of the DFS, and we try to extend the results obtained in 5 for an SIRS model with constant vaccination of individuals of all ages Stability of the DFS when R sup < 1. Our first result is to show that the DFS Ŝt),,, ˆRt)) is globally asymptotically stable GAS) when R sup < 1. We need the following lemma: Lemma 4.1. lim sups Ŝ)t). t Proof From equation 1) ds = µn1 p) βt)si µ + rt))s + δr, Nµ1 p) + δ) µ + rt) + δ)s. As Ŝt),,, ˆRt)) is a solution of 1) then we have that Therefore dŝ = Nµ1 p) + δ) µ + rt) + δ)ŝ. ds Ŝ) µ + rt) + δ)s Ŝ). Integrating this inequality we find that t ) S Ŝ)t) S Ŝ)t ) exp µ + δ)t t ) rτ)dτ t. Lemma 4.1 now follows. Now we can prove the global stability of the DFS when R sup < 1. Theorem 4.1. If R sup < 1, the DFS, Ŝ,,, ˆR) is GAS for the system 1)-5). Proof As we can easily show that, di = E µ + γ)i, I E µ + γ. The idea behind the proof of Theorem 4.1 is as follows: By Lemma 4.1 we know that given ɛ > there exists t 1 such that St) Ŝt) + ɛ and I I + ɛ for all t t 1. From 2) we bound Et) above and from this upper bound we deduce that E =. Suppose that E >. Integrating 2) we find that for t t > t 1, Et) Et ) exp µ + )t t ) + I + ɛ) exp µ + )t t t βτ)ŝτ) + ɛ) expµ + )τdτ.

8 8 I. A. MONEIM AND D. GREENHALGH Define yτ) = βτ)ŝτ). Note that yτ) is a non zero positive periodic function with period LT, so exp µ + )t t t yτ) expµ + )τdτ = t t Suppose that k + 1)LT t t klt. We have that t t yt u) exp µ + )udu yt u) exp µ + )udu. = yt u) exp µ + )udu 1+ exp µ+)lt ) + exp µ+)2lt exp µ + )k 1)LT t t + yt u) exp µ + )udu, klt yt u) exp µ+)udu 1+ exp µ+)lt ) + exp µ + )2LT exp µ + )klt, < Therefore for t t we find that yt ζ) exp µ + )ζdζ. 1 exp µ + )LT Et) Et ) exp µ + )t t ) + I + ɛ µ + )βt ζ)ŝt ζ) + ɛ) exp µ + )ζdζ, µ + 1 exp µ + )LT N exp µ + )t t ) + I + ɛ LT µ + )yt ζ) exp µ + )ζdζ sup µ + t,lt 1 exp µ + )LT ) LT µ + )βt ζ) exp µ + )ζdζ + ɛ. 1 exp µ + )LT Now choose t 2 > t large enough so that for t t 2, N exp µ + )t t ) < ɛ. Then for t t 2, Et) R sup E + ɛ 1 + sup t,lt yt ζ) exp µ + )ζdζ 1 exp µ + )LT + I ) ) + ɛ β max, µ + where β max = sup u,lt βu). Now choose ɛ small enough so that yt ζ) exp µ + )ζdζ I ) ) + ɛ ɛ 1 + sup + β max < ψe, t,lt 1 exp µ + )LT µ + where R sup +ψ < 1 and ψ >. Hence for t t 2 we have that Et) R sup +ψ)e. Thus E R sup +ψ)e. So E =. Hence also I = and Et) and

9 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 9 It) as t. It remains to show that S Ŝ)t) and R ˆR)t) as t. As ˆRt) is a solution of equation 4) when Et) = It) = we have that, dr ˆR) = rt)s Ŝ) + γi µ + δ)r ˆR). 13) Given ɛ 1 > using Lemma 4.1 there exists t 3 such that Et) + It) ɛ 1 and St) Ŝt) + ɛ 1 for all t t 3. So for t t 3, dr ˆR) r max + γ)ɛ 1 µ + δ)r ˆR), where r max = sup u,lt ru). Integrating this inequality we find that, R ˆR)t) R ˆR)t 3 ) exp µ + δ)t t 3 ) ) rmax + γ + ɛ 1 1 exp µ + δ)t t 3 )), µ + δ ) rmax + γ N exp µ + δ)t t 3 ) + ɛ 1. µ + δ It is now straightforward to show that given ɛ 2 > there exists t 4 such that Rt) ˆRt) ɛ 2 for t t 4. Hence St) = N Rt) It) Et) N ˆRt) ɛ 1 ɛ 2 for t t 4. Using Lemma 4.1 we deduce that St) Ŝt) as t. As Rt) = N St) It) Et) then we must have that Rt) N Ŝt) = ˆRt) at t. This completes the proof of Theorem 4.1. Thus if R sup < 1 the disease free solution is globally asymptotically stable. 5. The existence of periodic solutions. The existence of a periodic solution of 1)-5) can be proved in two stages. The first is to prove that if R > 1 then there exists a minimum threshold value for It) such that if I) > or E) > then It) will rise above this threshold value and moreover from then on the time spent continuously beneath it can be bounded above by a bound that depends only on the model parameters not on the initial conditions. Moreover the time taken to rise initially above the threshold can be bounded above by a bound that depends only on the initial value I) and the model parameters. The second is using fixed point theory we shall prove that the system 1)-5) has a LT -periodic solution. These results are also true for the same model when the contact rate is constant if the vaccination strategy is a general continuous periodic function. To prove the existence of periodic solutions of 1)-5) we need the following notations: Definition 5.1. R λ) = Define µ + γ)µ + ) ft, λ) = t,lt µ + )yt ζ) exp µ + + λ)ζdζ. 1 exp µ + + λ)lt yt u) exp µ + + λ)u du. 1 exp µ + + λ)lt Note that ft, λ) is monotone decreasing in λ for a fixed t and hence monotone decreasing in λ. R λ) is

10 1 I. A. MONEIM AND D. GREENHALGH Now consider the equation λ 2 + d 1 λ + d 2 λ) = where d 1 = µ + ) + µ + γ) and d 2 λ) = 1 1 R λ))µ + )µ + γ). 2 If R ) = R > 1 then d 2 ) < and the equation λ 2 + d 1 λ + d 2 λ) = has a unique positive root, say λ 1 >. Then R λ 1 ) + 1)µ + )µ + γ) = 1. 14) 2µ + + λ 1 )µ + γ + λ 1 ) Definition 5.2. β sup = sup t,lt µ + )βt ζ) exp µ + + λ 1 )ζdζ. 1 exp µ + + λ 1 )LT Definition 5.3. K 1 η) = 1 4 R λ 1 ) 1)µ + )µ + γ)1 η) 1 + βmaxn µ+) + 2γ µ+δ) β, sup where η < 1 is an arbitrarily small positive number and ψη) is a positive number such that { K1 η) < ψη) < min, N 1 β maxɛ η), N 2γɛ } η), 2 µ + ) µ + δ) where ɛ η) is a positive number such that < ɛ η) < min { µ + ) β max, } Nµ + δ), K 1 η). 2γ Here K 1 η) is a well defined positive number and ɛ η) and ψη) are in well defined ranges. Given η > we suppose that I) > and find an upper bound for the time for which It) can spend continuously beneath ɛ η). We do this by supposing that It) remains indefinitely continuously beneath ɛ η) and deduce a contradiction. For the moment we shall suppose that η > is fixed and write K 1, ɛ and ψ for K 1 η), ɛ η) and ψη) respectively. We need four preliminary lemmas. We use these to show that It) must eventually rise again above ɛ and the time taken to do this can be bounded above by a time depending only on ɛ, ψ and the model parameters. If the solution has the initial value I) > ɛ suppose that the solution It) drops beneath ɛ for the first time at time ζ. Then without loss of generality we can assume that ζ =. Our first result is that Et) also becomes small. Lemma 5.1. Suppose that I) = ɛ and It) ɛ for all t. Then there exists a time T > such that Et) < β maxnɛ µ + ) + ψ for all t > T, where T depends only on ψ, ɛ and the model parameters. Proof This is straightforward.

11 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 11 Lemma 5.2. Suppose that It) ɛ for all t. Then there exists a time T 1 > such that Rt) ˆRt) + 2γɛ µ + δ + ψ for all t > T 1 and T 1 depends only on ψ, ɛ and the model parameters. Proof Given ɛ > there exists t, depending only on ɛ and the model parameters, such that S Ŝ)t) ɛ for all t t. So for t > t, from 13) dr ˆR) r max ɛ + γɛ µ + δ)r ˆR). Lemma 5.2 now follows straightforwardly. Now suppose that I) = ɛ, our next aim is to find a time T 2 and a strictly positive lower bound E 1 for ET 2 ) where E 1 and T 2 depend only on the model parameters and ɛ. We shall deal with the two cases p < 1 and p = 1 separately in the following two lemmas: Lemma 5.3. If p < 1 define a time T 2 = x + 1)LT where x is the smallest integer such that xlt T 3 = ln 2)/β max N +µ+r max ) and E 1 = E 2 exp µ+)t 2 >, E 2 = ɛ S 1 βτ) exp γτdτ exp γxlt > and S 1 = µn1 p)/2β max N +µ+ r max ). Then ET 2 ) E 1 > and E 1 and T 2 depend only on the model parameters and ɛ. Proof From 1) we have that ds µn1 p) β max N + µ + r max )S. Integrating this inequality we have that St) As I) = ɛ, It) exp µ + γ)t. Hence for t T 3, µn1 p) β max N + µ + r max 1 exp β max N + µ + r max )t). de + µ + )E βt)s 1ɛ exp µ + γ)t. Multiplying by expµ + )t and integrating between xlt and x + 1)LT, the required result follows. Lemma 5.4. If p = 1 define a time T 2 = x 1 +1)LT where x 1 is the smallest integer such that x 1 LT T 4 + T 5. Here T 4 = ln 2)/µ + δ) and T 5 = ln 2)/β max N + µ + r max ). Define E 1 = E 3 exp µ + )T 4 >, E 3 = ɛ S 2 βτ) exp γτdτ exp γx 1 LT >, S 2 = δr 1 )/2β max N + µ + r max )) and R 1 = µn)/2µ + δ). Then ET 2 ) E 1 > and T 2 and E 1 depend only on the model parameters and ɛ. Proof From equation 4) we have that dr µnp µ + δ)r.

12 12 I. A. MONEIM AND D. GREENHALGH Integrating this inequality and arguing similarly to Lemmas 5.2 and 5.3 we deduce that for t T 4 = ln 2)/µ + δ), Rt) µn)/2µ + δ)) = R 1. Again from 1) we have that for t T 4 ds δr 1 β max N + µ + r max )S. Integrating this inequality for t T 4 we have that St) δr 1 β max N + µ + r max 1 exp β max N + µ + r max )t T 4 )). Hence for t T 4 + T 5, St) S 2 and de + µ + )E βt)s 2I. Lemma 5.4 is now straightforward arguing as in Lemma 5.3. These results allow us to proceed to the first theorem in this section which gives a lower bound η for I and an upper bound on the initial time for which the value of It) remains beneath η. Theorem 5.1. If R > 1 then there exists η > such that for all η 1 > if I) η 1 then It) η for all t T η 1 ) where T η 1 ) depends only on η 1 and the model parameters. Proof First of all suppose that I) = ɛ and It) ɛ for t. We shall show that R > 1 forces It) to rise to at least the level ɛ by a time T which depends only on ψ, ɛ and the model parameters. From Lemmas 5.1 and 5.2 we have shown that for t > T 6 = max T, T 1 ), St) = N Et) It) Rt), Ŝt) 2ψ ɛ 1 + β maxn µ + ) + 2γ ). µ + δ) By Definition 5.3 we have that 2ψ + ɛ 1 + β maxn µ + ) + 2γ µ + δ) < K β maxn µ + ) + 2γ µ + δ) < K 1 + <, as ψ < K 1 /2) and ɛ < K 1, 1 4 R λ 1 ) 1)µ + )µ + γ)1 η) β, using Definition 5.3, sup R λ 1 ) 1)µ + )µ + γ)1 η) 2 β sup. 15) From 2) choose t > T 7 = maxt 4, T 6 ), then Et ) E 1 exp µ+)t T 4 ) > using Lemmas 5.3 and 5.4 and from 3) we have that It ) ɛ exp µ+γ)t >. Note that from Definition 5.1 we have that R λ) = µ + γ)µ + ) t,lt µ + )yt ζ) exp µ + + λ)ζdζ. 1 exp µ + + λ)lt

13 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 13 Given a small enough η > choose ɛ 1 and an integer k so that µ + γ)µ + ) t,lt µ + )yt ζ) exp µ + + λ 1 )ζdζ 1 + exp µ + + λ 1 )LT + exp µ + + λ 1 )2LT exp µ + + λ 1 )k 1)LT ) > R λ 1 )1 η), 16) and < ɛ 1 < min { ɛ 2 exp µ + γ)t + klt ), } 2µ + γ + λ 1 ) E 1 exp µ + )t T 4 + klt ). Then Et ) 2ɛ 1 µ + γ + λ 1 )/) and It ) 2ɛ 1. We assert that provided that It) remains continuously below the level ɛ then It +τ) ɛ 1 expλ 1 τ klt ) and Et +τ) µ + γ + λ 1 ɛ 1 expλ 1 τ klt ). For define τ such that { τ = ξ : It + τ) ɛ 1 expλ 1 τ klt ) and Et + τ) µ + γ + λ 1 ɛ 1 expλ 1 τ klt ) for τ, ξ By continuity τ > and if τ < then either It +τ ) = ɛ 1 expλ 1 τ klt ) or Et +τ ) = µ + γ + λ 1) ɛ 1 expλ 1 τ klt ). 17) We shall show that 17) leads to a contradiction. For τ klt we have that It + τ ) ɛ exp µ + γ)t + τ ), > ɛ 1 expλ 1 τ klt ), and Et + τ ) E 1 exp µ + )t T 4 + τ ), > µ + γ+ λ 1)ɛ 1 For τ > klt, and from 3) we have that It + τ ) = It ) exp µ + γ)τ + exp µ + γ)t + τ ) expλ 1 τ klt ). t+τ t } Eζ) expµ + γ)ζdζ, > ɛ 1 expλ 1 τ klt ). 18).

14 14 I. A. MONEIM AND D. GREENHALGH Also for τ > klt we have that, Et + τ ) = Et ) exp µ + )τ + exp µ + )t + τ ) t+τ t βζ)sζ)iζ) expµ + )ζdζ, Et ) exp µ + )τ + ɛ 1 exp µ + )t + τ ) ) t+τ R λ 1 ) 1)µ+γ)µ+)1 η) βζ) Ŝζ) 2 β sup t expµ + )ζ + λ 1 ζ t klt )dζ, > µ + γ + λ 1 ɛ 1 exp µ + )τ + ɛ 1 expλ 1 τ klt ) ) t+τ R λ 1 ) 1)µ+γ)µ+)1 η) βζ) Ŝζ) 2 β sup t > ɛ 1 expλ 1 τ klt ) > = τ βt + τ u) expµ + +λ 1 )ζ t τ )dζ, Ŝt + τ u) ) R λ 1 ) 1)µ+γ)µ+)1 η) 2 β exp µ + +λ 1 )udu, sup ) µ + γ R R λ 1 ) 1)µ+γ)1 η) λ 1 )1 η) 2 R = µ + γ + λ 1 λ 1 ) + 1)µ+γ)1 η) 2 ɛ 1 expλ 1 τ klt ), using Definition 5.2 and inequality 16), ) ɛ 1 expλ 1 τ klt ), µ + + λ 1 1 η)ɛ 1 expλ 1 τ klt ), µ + using equation 14). As η is arbitrarily small we can choose η small enough so that µ + + λ 1 )1 η)/µ + ) > 1 and Et + τ ) > µ + γ + λ 1) ɛ 1 expλ 1 τ klt ). 19) Hence 18) and 19) contradict 17) so we deduce that τ = and assuming that Iζ) always lies below ɛ, It + τ) ɛ 1 expλ 1 τ klt ) and Et + τ) µ + γ + λ 1)ɛ 1 expλ 1 τ klt ), for τ. In particular It) must rise again above its initial value ɛ by a time at most T = klt + T 7 + τ 1 where τ 1 = 1/λ 1 ) ln ɛ /ɛ 1 ) and this time depends only ɛ, ψ and the model parameters. Moreover for all t we have that It) η = ɛ exp µ + γ)klt + T 7 + τ 1 ). Next suppose that ɛ > I) > η 1 >. A similar argument shows that provided that η 1 >, It) rises above ɛ by a time at most T η 1 ) = klt + T 7 + τ 2 which

15 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 15 depends only on η 1 and the model parameters, not on the initial conditions, where τ 2 = 1/λ 1 ) ln ɛ /ɛ 2 ) and { η 1 < ɛ 2 < min 2 exp µ + γ)t + klt ), } 2µ + γ + λ 1 ) E 1 exp µ + )t + klt T 4 ). Moreover by our previous argument It) remains above η for t T η 1 ). Hence if I) > η 1 then It) η for all t T η 1 ), where T η 1 ) depends only on η 1 and the model parameters. So we can look for non-zero periodic solutions for our system when R > 1. The following theorem gives the existence of a periodic solution of the system 1)-5) with period LT. Theorem 5.2. The system 1)-5) has a non-zero LT -periodic solution. Proof The set Z = R 4 with the norm z = S 2 + E 2 + I 2 + R 2 is a Banach space 3. Define the sets { } L = L 1 = and L 2 = { S, E, I, R) : S, E, I η, R, S + E + I + R = N S, E, I, R) : S, E, I > η 2, R, S + E + I + R = N { } S, E, I, R) : S, E, I, R, S + E + I + R = N we find that L and L 2 are compact, L 1 is open relative to L 2 i.e. L 2 -L 1 is closed) and L, L 1 and L 2 are convex sets. Define the mapping h : L 2 L 2 such that hz ) = zlt,, z ). hz ) is the solution of the initial value problem 1)-5) at time LT with z = S), E), I), R)) at time t =. The mapping h is continuous since the right hand of the system 1)-5) is differentiable. Then for any positive integer j the image of the set L 1 remains contained completely in the set L 2 so for j = 1, 2, 3,... we find that h j L 1 ) L 2. Now suppose that z L 1. Then for t T 1 2 η ), It) η. Hence if m LT T 1 2 η ) then we have h m L 1 ) L for all m m. Then by Horn s Fixed Point Theorem 8, the mapping h has a fixed point in the set L. Hence the system 1)-5) has a non-zero LT -periodic solution., },,

16 16 I. A. MONEIM AND D. GREENHALGH 6. Persistence results Stability of the DFE when R > 1. Suppose first that E) = I) =. Then from equations 1)-5) we find that Et) = It) = for all t. Arguing as in the proof of Theorem 4.1 we deduce that St), Rt)) Ŝt), ˆRt)) as t whatever the value of R > 1. Next consider the case where E) > or I) >. If I) = then it is straightforward to show from 3) that I t) > for t small and positive. So by changing the time origin if necessary we deduce from Theorem 5.1 that: Corollary 6.1. The DFS is unstable if R > Persistence of the disease when R > 1. Now we examine the persistence of the disease when R > 1. Here persistence means that the number of ectives is bounded away from zero. From Theorem 5.1 we have the following corollary. Definition 6.1. For a function f :, ) R + we define f = lim t ft). Definition 6.2. Uniform Strong Repeller Following 16 we say that the set q = {I = : St) N, Et) N, St) + Et) N}, is a uniform strong repeller for the set G = {S, I, E) : S N, < I N, E N, S + I + E N}, if I >. Definition 6.3. Uniform Persistence We say that the disease is uniformly persistent if each of St), Et), It) and Rt) is strictly bounded away from zero and moreover this bound depends only on the model parameters. Corollary 6.2. If R > 1 then the set q is a uniform strong repeller for the set G. Proof As I η the result is obvious. Our next step is to show that the disease is uniformly persistent when R > 1. From Theorem 5.1 It) is bounded away from zero for large times. We need to show that St), Et) and Rt) are similarly bounded away from zero for large times. Lemma 6.1. R η 3 > where η 3 depends only on the model parameters. Proof Given ɛ 1 > there exists t 1 such that It) η ɛ 1 for t t 1. Then for t t 1 from 4), dr µnp + γη ɛ 1 ) µ + δ)r. Integrating this inequality and arguing as in Lemma 4.1 we deduce that R µnp + γη ɛ 1 ). µ + δ) Letting η 3 = µnp + γη )/µ + δ) the result follows by letting ɛ 1. Lemma 6.2. S η 4 > where η 4 depends only on the model parameters.

17 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 17 Proof Given ɛ 1 > there exists t 2 such that Rt) 1 2 η 3 ɛ 1 for t t 2. Then for t t 2 from equation 1) we have that ) ds 1 µn1 p) + δ 2 η 3 ɛ 1 β max N + µ + r max ) S. Arguing as in Lemma 6.1 we deduce that, S η 4 = µn1 p) δη 3 β max N + µ + r max >. Lemma 6.3. The exposed population is bounded away from zero by a bound which depends only on the model parameters. Proof There exists t 3 such that for t t 3, St) S / 2) and It) I / 2). Hence for t t 3 from 2) de + µ + )E 1 2 βt)s I. 2) Picking n such that nlt t 3 then by multiplying 2) by expµ + )t and integrating En + 1)LT ) expµ + )n + 1)LT So E nlt ) exp µ + )nlt + 1 n + 1)LT 2 S I βt) exp µ + )t. nlt En + 1)LT ) 1 2 S I βt) expµ + )t LT ) = ɛ 1 >, using the fact that βt) is a non-zero, continuous, positive function so the integral is strictly positive. Moreover ɛ 1 depends only on the model parameters. But de µ + )E. So for n + 2)LT t n + 1)LT, Et) ɛ 1 exp µ + )t n + 1)LT ) ɛ 2 = ɛ 1 exp µ + )LT. Hence Et) ɛ 2 for all t n + 1)LT. Lemma 6.3 follows. From Lemmas and Theorem 5.1 we find that min S, E, I, R ) = c >. So if R > 1 then St), Et), It) and Rt) are strictly bounded away from zero and moreover this bound depends only on the model parameters. Hence the disease is uniformly persistent when R > Implications of Results. Our results for the SEIRS model with a periodic vaccination strategy are less complete than those for the corresponding SIRS model 1. We have not been able to show that the disease will die out if R c 1 and that it will take off if R c > 1. We have shown that the disease always dies out if R sup < 1 and that if R > 1 the DFE is unstable, the disease is uniformly persistent if initially present and an LT -periodic solution exists. As the disease is uniformly persistent if initially present when R > 1 we deduce that instability of the DFS when R > 1 can be extended from not being locally asymptotically stable, to being Lyapunov unstable. It is straightforward from the definitions of R sup and R to show that

18 18 I. A. MONEIM AND D. GREENHALGH R max = β maxŝmax µ + )µ + γ) Rsup R R min = β minŝmin µ + )µ + γ). Here β min = u,lt βu), Ŝ max = sup u,lt Ŝu) and Ŝmin = u,lt Ŝu). Hence if R max < 1 then the disease will die out, and if R min > 1 the DFS is unstable, the disease is uniformly persistent and an LT -periodic solution exists. We can also show that R sup R c R with both inequalities being strict if βt)ŝt) is non-constant on, LT. Define Zt) = yt u)fu)du, where yt) = βt)ŝt) and fu) = µ+) exp µ+)u/1 exp µ+)lt ), u LT. By the definitions of R, c R sup and R we need only to show that Zt) 1 yτ)dτ sup Zt), t,lt LT t,lt with strict inequality if yt) is non-constant. Now Zt) = = u u = LT y = LT y. yt u)fu)du, ys)fu)dsdu, letting s = t u, fu)du, Hence Zt) y = 1 t,lt LT Zt) being strict unless Zt) is a constant. Writing Zt) = Z t) = t t LT t t LT If Zt) is a constant then Z t) = so yt) = ys)ft s)ds, sup Zt), with both inequalities t,lt ys)f t s)ds yt)f) flt )). µ + ) flt ) f)) Zt) is also a constant. A sufficient condition for the DFS to be GAS is that R sup < 1, equivalently sup t,lt µ + )yt ζ) exp µ + )ζdζ 1 exp µ + )LT < µ + )µ + γ), 21)

19 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 19 and a necessary condition is that R > 1, equivalently t,lt µ + )yt ζ) exp µ + )ζdζ 1 exp µ + )LT In the case that β is a constant the first condition becomes sup t,lt and the second t,lt µ + )Ŝt ζ) exp µ + )ζdζ 1 exp µ + )LT µ + )Ŝt ζ) exp µ + )ζdζ 1 exp µ + )LT > < > µ + )µ + γ). 22) µ + )µ + γ), 23) β µ + )µ + γ). 24) β We might conjecture that a necessary and sufficient condition for the DFS to be GAS is R c 1, which is equivalent to βτ)ŝτ)dτ µ + )µ + γ)lt LT = S c. 25) βτ)dτ βτ)dτ Condition 25) says that the average number of susceptibles in the DFS weighted by βt) over the period of the vaccination function is less than or equal to a critical value S c. Recall that a similar condition for local stability of the DFS, Ŝt), for L = 1, T βτ)ŝτ)dτ T βτ)dτ < µ + γ)t T βτ)dτ = S c was obtained for a pulse vaccination function rt) = p n= δt nt ) by Shulgin et al. 15 in an SIR model. In the case that βt) is a constant and L = 1 condition 25) becomes 1 T µ + )µ + γ) Ŝτ)dτ, 26) T β and Shulgin s condition 1 T T Ŝτ)dτ < µ + γ). β 8. Summary and Discussion. We have extended the work of 5 and 1 from SIRS epidemic models to the more realistic and complicated SEIRS model. It is important to include an exposed or latent class into our models as many childhood diseases have a latent period. We have studied control of the dynamics of the ectious disease by using periodic vaccination of susceptibles of all ages, combined with immunisation of a given proportion of newborns. We did this both for a constant and for a seasonally varying disease transmission rate. Using a periodic vaccination strategy in such an SEIRS model seems to lead to periodicity in the disease dynamics. This work can be summarised as follows: In Section 1 a short introduction to common practically used vaccination strategies was given. Section 2 outlined the SEIRS model which we studied and gave the assumptions underpinning the model. Section 3 showed that there is a unique DFS for our SEIRS epidemic model and this solution is periodic with period equal to that of the vaccination function. We also gave a conjectured expression for R, the basic reproduction number of the disease, when the vaccination campaign rt) is used. Lower and upper bounds,

20 2 I. A. MONEIM AND D. GREENHALGH R and R sup respectively, for this expression were also defined. In this section we also studied the stability of the DFS of our model. We found that the DFS was GAS when R sup < 1 and in this case the ection will ultimately fade out of the population. In Section 4 fixed point theory was utilized to show the existence of non-trivial periodic solutions. Horn s Fixed Point Theorem showed that the model equations 1)-5) have a non-trivial LT -periodic solution if rt) has period LT and βt) has period T. Section 5 proved some persistence results when R > 1. The DFS is unstable both not locally asymptotically stable and unstable in the Lyapunov sense). When R > 1, if the disease is present initially it will persist and remain endemic in the population. We also showed that all of St), Et), It) and Rt) were uniformly bounded away from zero. Finally Section 6 presented a conjecture for a condition under which an immunisation program can prevent epidemics from occurring in the population. This is to keep a weighted mean value of the susceptible population at the DFS over the period of the vaccination function beneath a certain critical value. Our upper and lower bound results go some way towards proving this. A similar result for an SIRS model was found in 1. We also gave an equivalent condition for R sup < 1, that the maximum of a weighted average of the number of susceptibles at the DFS is less than a certain critical value, and a similar condition for R > 1, that the minimum of the same weighted average exceeds the same critical value. The results were obtained for an SEIRS epidemic model with the disease transmission rate βt) having period T or being a constant) and the vaccination function having period LT, where L 1. We conjectured that to control or eradicate the disease it was both necessary and sufficient to keep the mean value of the product of the disease transmission rate and the susceptible population at the DFS beneath a critical threshold value. If this is true then it is possible for sometimes only a few individuals to be vaccinated, provided only that the weighted mean value of the number of susceptibles at the DFS over the period of the vaccination function does not exceed the threshold value. The disease will still be eradicated. This contrasts with the strategy of constant vaccination where a critical fixed level of immunisation effort must always be applied to guarantee eradication, and this is an advantage of a periodic vaccination strategy over a constant one. For a highly ectious disease such as measles, using vaccination at birth only, approximately 91-94% of newborn individuals must be vaccinated to guarantee elimination of the disease 1. It is hard to achieve this level of vaccination coverage, particularly bearing in mind that measles vaccine efficacy is only around 95%, that some individuals may be difficult for the health services to locate and others may refuse vaccination for religious or other reasons. Using a continuous periodic vaccination strategy in conjunction with vaccination of a fixed proportion of newborn individuals, reduces the proportion of newborns who need to be immunised to a more realistic level. Moreover from 9) one can see easily that using such a mixed vaccination strategy uniformly reduces the level of fluctuation of susceptibles in the DFS compared with a purely periodic vaccination function p = ). Hence it is more optimal to use a combined vaccination approach in order to prevent major outbreaks of ectious disease occurring.

21 VACCINATION OF EPIDEMICS WITH SEASONALLY VARYING CONTACT RATE 21 REFERENCES 1 R. M. Anderson and R. M. May, Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford, Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson and Y. Danon, Pulse mass measles vaccination across age cohorts. Proc Natl Acad Sci USA 91993) T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations. Academic Press, New York, K. Dietz, The incidence of ectious diseases under the luence of seasonal fluctuations. In: J. Berger, W. Bühler, R. Repges and P. Tautu, eds.) Proceedings of a workshop on mathematical modelling in medicine. Mainz. Lecture Notes in Biomathematics ), Springer-Verlag, Berlin, pp D. Greenhalgh and I. A. Moneim, SIRS epidemic model and simulations using different types of seasonal contact rate. Systems Analysis Modelling Simulation 43:523) C. A. De Quadros, J. K. Andrus and J. M. Olivé, Eradication of poliomyelitis: progress. Am Pediatr Inf Dis J 11991) J. K. Hale, Ordinary differential equations. Wiley, New York. 8 W. A. Horn, Some fixed point theorems for compact maps and flows in Banach spaces. Trans Amer Math Soc ) W. P. London and J. A. Yorke, Recurrent outbreaks of measles, chickenpox and mumps, I. Amer J Epidemiol ) I. A. Moneim and D. Greenhalgh, Threshold and stability results for an SIRS epidemic model with a general periodic vaccination strategy. To appear. J Biol Systems 25). 11 D. Nokes and J. Swinton, The control of childhood viral ections by pulse vaccination. IMA J Math Appl Med Biol ), S. C. Pannuti, J. C. Moraes, V. A. Souza, M. C. C. Camargo and N. T. R. Hidalgo, Measles antibody prevalence after mass immunization in São-Paulo, Brazil. Bull WHO ) A. B. Sabin, Measles, killer of millions in developing countries: Strategies of elimination and continuing control. Eur J Epidemiol 71991), I. B. Schwartz and H. L. Smith, Infinite subharmonic bifurcations in an SEIR model. J Math Biol ), B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model. Bull Math Biol 61998), H. R. Thieme, Persistence under relaxed point-dissapitivity with application to an endemic model). SIAM J Math Anal ), P. J. Williams and H. F. Hull, Status of measles in The Gambia, Rev Inf Dis 51983) Received on Dec. 2, 24. Revised on March 9, 25. address: moneim97@hotmail.com address: david@stams.strath.ac.uk