The reproduction number for Covid-19 lies between 2 and 3 if we take no action to stop it spreading. With proper measures such as social distancing it may be reduced to below one. It is known that infections are more lethal for the elderly (and for persons who suffer from obesity, diabetes, high blood pressure) than for healthy young persons 1 . These two facts suggest that a social distancing policy which takes into account the difference in risk for the vulnerable and for the fit might be effective in reducing the overall mortality.
So consider a population consisting of a million vulnerable persons and two million fit persons. Assume that the mortality is ten times as high for the vulnerable as for the fit.
To be concrete, assume an Infection Fatality Rate (IFR) of 0.01 for the vulnerable and 0.001 for the fit. This yields an overall IFR of 0.004 = (0.01 + 2 * 0.001)/3.
Assume a two type model. A vulnerable person infects r v = 0.7 persons on average, a fit person r f = 1.3. Ten vulnerable persons and twenty fit, all infectious, on average will yield 7 + 26 = 33 new infections (corresponding to the overall reproduction number r 0 = 1.1). Start with a hundred infections among the vulnerable and two hundred among the fit. The population is compartmentalized. If the compartments are watertight, there is no contact between the vulnerable and the fit. The epidemic will die out among the vulnerable since r v = 0.7 < 1, but it will be more severe among the fit than in the corresponding homogeneous model with reproduction number r 0 = 1.1 since r f = 1.3 > 1.1. Computations show that the total number of deaths is lower than in a homogeneous population.
In a more realistic model the expected number r f,v of vulnerable persons infected by an infectious fit person and the number r v,f of fit persons infected by an infectious vulnerable person are positive. So suppose ten infectious fit persons infect on average eleven fit persons and two vulnerable persons, and ten infectious vulnerable persons infect five vulnerable persons and two fit persons. This "standard model" will play a prominent role in the discussion below. The infection rate between the two groups is low. Such a society may be said to shield the vulnerable. How effective is this shield? Simulations show that in the heterogeneous society there will be more deaths than in a homogeneous population
3 with reproduction number r 0 = 1.1. In the homogeneous population the death toll has a mean value of 2124; in the two type model the mean is ten per cent higher, 2347.
The result is not implausible. There is a considerable increase in the total number of infections among the fit. The positive cross infection r f,v = 0.2 from the fit to the vulnerable has the effect that the vulnerable are pulled along in this more severe epidemic, yielding a higher overall mortality. Compartmentalization increases the death toll. It is not clear how the transitions r f,v and r v,f are related. Intuitively for nursing homes one might interpret r f,v as measuring infections caused among the elderly by visits by the family, and r v,f as infections among the nurses and staff caused by illness among the elderly. Section 4 will look at this issue more closely.
The main purpose of the paper is to exhibit possible adverse effects of shielding in a heterogeneous population. This aim has been achieved already above by mentioning the results of the computations for the standard model. For the given values r v = 0.7 and r f = 1.3 above, and r v,f = 0.2, a large value of r f,v will reduce the total mortality. Based on that result the government might consider launching a campaign: "Tonight don't meet at the pub; visit your granny instead."
The benefits of social distancing measures which differentiate between the fit and the vulnerable are incontestable. This paper shows that policies have to be chosen with care to avoid adverse affects. Here we should mention a different beneficial effect of variations in the reproductive number across a population. Heterogeneity may reduce the herd immunity threshold 2,3 .
For an introduction to the mathematical background see 4,5 . The exposition below is self-contained. It relies on simulations. Readers with some experience in R are invited to use their skill to explore the effects of variations in the reproduction matrix.
Section 2 introduces the binomial Reed-Frost model. Section 3 presents the results.
These are discussed in Section 4. Section 5 contains our conclusion. The Appendix contains two sections: a non technical explanation of the decrease in mortality associated with large values of r f,v and a discussion of the role of eigenvalues and eigenvectors in the two-type Reed-Frost model.
In a homogeneous population of size n 0 with initially i 0 infections the probability of noninfection for any susceptible member of the population is q = q i 0 0 where q 0 = 1 − p 0 and p 0 = r 0 /n 0 is the probability of infection in a homogeneous population of size n 0 with reproduction number r 0 . The total number of new infections i among the n = n 0 − i 0 susceptibles is binomial-(n, p) with p = 1 − q. This yields a recursion starting with n = n 0 , i = i 0 . In R the recursion consists of three commands: n<-n-i; p<-1-q0^i; i<-rbinom(1,n,p);
The sequence of commands runs while i is positive. The total number infected is j = n 0 − n. This is the binomial or Reed-Frost model. In her exposition of this model in 1952 seconds on a ten year old iMac OS 10.11.6 with a 3.06 GHz Intel Core processor and 1067
MHz DDR3 memory modules in two out of four memory slots. The program determines the mean and sd of j, the total number infected. If the initial number of infections is small the epidemic may die out, but for i 0 = 300 initial infections each of the 10 5 simulations gives rise to a full blown epidemic.
The program for the multitype binomial model is similar. Start with a population n 0 = (10 6 , 2 * 10 6 ) of a million vulnerable and two million fit persons and i 0 = (100, 200) initial infections, a hundred among the vulnerable and two hundred among the fit. The probability that the infection will die out may be neglected. The reproduction number r 0 now becomes a matrix R which is akin to the transition matrix in a Markov chain where the kth row contains the probabilities p k,m of a transition from state k to m. We first look at a specific case, the standard model discussed in the introduction:
(2.1)
The first row states that ten infectious vulnerable persons will infect on average five vulnerable and two fit persons. The second row states that similarly ten infectious fit persons will infect on average two vulnerable and eleven fit persons.
For a vulnerable person the probability of not being infected is 
The program for a simulation of the epidemic in the multitype model then is:
The R program for the two type binomial Reed − Frost model. 
With r v = 0.7 and r f = 1.3 the reproduction matrix has the form The plot for r v = 0.8 and r f = 1.1 with r 0 = 1, Figure 3 , left, paints a darker picture.
For a = 0.2 (green curve) and c = 0.1 the overall death toll is twice as high as in the homogeneous case. Mortality among the fit is one third of the total rather than one sixth as in the homogeneous case. The reason for this behaviour? In the heterogeneous case there will be a full blown epidemic among the fit because r f = 1.05 > 1. The constant c = r f,v > 0 will pull the vulnerable into the epidemic. The overall reproduction number here has the critical value r 0 = 1. In the homogeneous model epidemics will be short and die out.
The right side of Figure 3 shows the subcritical case, r 0 = 0.95. The adverse effects of shielding the vulnerable here are already apparent for c = 0. The mortality curves are decreasing. The excess death toll is small, less than ten. It is of little interest since it reflects the size of the contiguous initial infection rather than of the population.
In all three figures the situation brightens for c in the upper half of the interval [0, 0.8].
The total mortality decreases dramatically as c → 0.8. The low mortality for R(0.2, 0.8) on first sight is a mystery. A large value r f,v = 0.8 indicates many infections from fit to vulnerable. Why should the mortality almost vanish? Restrict attention to the fit. First assume a = r v,f = 0. The top row of R then is (0.7, 0), but if we turn to the fit we see that the infections will die out since 0.5 < 1. There also are infections from the fit to the vulnerable, but these do not concern the fit since we have assumed that the vulnerable do not infect the fit, a = r v,f = 0. The infections among the vulnerable will also die out (since r v,v = 0.7 < 1), apart from the import from the fit, but the import will die down as the epidemic among the fit dies out. If a = r v,f is positive this will not alter the situation as long as a is small. Section 6 contains a more detailed analysis. 
One cannot argue with the result of a computation. One can argue about the interpretation. We restrict the discussion to two topics.
In the basic model the reproduction matrix is
A vulnerable infectious person infects on average 0.7 persons of whom a are fit; a fit infectious person infects on average 1.3 persons of whom c are vulnerable. Infection is due to contact. Social contacts of the healthy and young are more varied and more intense than for the old or sick. This difference has increased as the vulnerable have become more aware of their vulnerability. Human beings are social animals, but older people are perhaps better able to endure solitude and live with their thoughts and memories than the young. A factor 1.3/0.7 ≈ 2 may be excessive. In the example for the critical case, r 0 = 1, the factor is less, 1.05/0.9. The effect is similar.
In mathematics it is good practice to vary one variable at a time. The effect depends on the variables which are kept constant. If one entry of the reproduction matrix goes down and the other three are constant the epidemic will be less severe. In a partition the lockdown for the vulnerable becomes stricter while at the same time it is relaxed for the fit. This approach makes it natural to assume r f and r v to be constant. We then A healthy young person will cough with more force than a feeble old person. This may make the fit more infectious than the vulnerable. Thus there are indications that adults infect children but children hardly infect adults 7, 8, 9 . (There also is contrary evidence 10 .)
There are more reasons for a lack of symmetry. If the vulnerable are tested at regular intervals and visits are only allowed when the test result is negative this will not affect r f,v , but it will reduce the value of r v,f and hence increase the parameter t in Figure 4 .
A representative list of pairs, infector and infectee, together with age and medical condition, might help to determine the role of the values of t > 0.5 in figure 4 .
2) Do the results apply to real life?
The proportion of vulnerable to fit is 1:2. This is realistic for the Netherlands and In the more realistic model the infection curves in Figure 6 will have different shapes, but their relation to the dotted lines associated with the eigenvectors will be the same; the positions of the points describing the total number infected will change, but the shape of the curve which they form might well again be a question mark. This is speculation. Whether the suppositions hold can only be determined by doing the necessary simulations and observations.
The results presented in this article are indicative rather than descriptive. They suggest that our intuition fails us in understanding how the parameters of the reproduction matrix affect the outcome of an epidemic in a population divided into two classes, the vulnerable and the fit. It only takes a little effort to run the Reed-Frost model on a pc and simulate two type epidemics. Speed makes this model a viable alternative to our intuition. It may be a better guide to reality.
[1] Levin, A.T., Cochran, K.B. and Walsh, S.P. Assessing the age-specificity of infection fatality rates for COVID-19: Meta-analysis, and public policy implications. medRxiv, www.nber.org/papers/w27597 (2020).
[2] Britton, T., Ball, F. and Trapman, P. A mathematical model reveals the influence of Consider a billowing cloud of viruses in the region of the fit, a cloud which doubles in size every few days. In the region of the vulnerable the cloud of viruses shrinks and will fade away. Now assume the excess of viruses produced in the region of the fit every day is diverted to a region which is less hospitable and where the viruses will die out. The cloud in the region of the fit no longer grows. If one increases the part which is diverted by a fraction the cloud of viruses in the region of the fit will shrink at an exponential rate. This will be the case even if some of the viruses manage to find their way back to the region of the fit.
Can one arrange things such that in the region of the vulnerable the cloud will fade away in spite of the influx from the region of the fit? One may argue that the influx is only temporary since the cloud in the region of the fit will fade away at an exponential rate, and so will the fraction which is diverted to the region of the susceptible. The situation becomes less clear if a fraction of the cloud above the region of the susceptible manages to return to the region of the fit. In order to handle the situation where there is traffic between the two regions in both directions we have to be more specific. It helps to look at steady states. We give two examples. In both cases the top row of the matrix R is 
The new value i k+1 of the vector i is written as a linear transformation R k+1 of the old
, where R k+1 = RD k+1 is a modulation of the reproduction matrix.
The key to linear dynamical systems x k+1 = x k R is the left eigenvector associated with the largest eigenvalue. In the multitype epidemic the maximal eigenvalue of the reproduction matrix R determines the severity of the epidemic; the corresponding left eigenvector determines the proportion of vulnerable to fit among the infected.
These words should not be taken literally. The relation between the eigenvalues and eigenvectors of the reproduction matrix R and the course of the epidemic is not perfect as one sees on comparing the curves on the left side and the right side in Figure 2 . The coordinates associated with the left eigenvectors of R make R diagonal but destroy the diagonality of the modulator D k+1 in the recursion (7.8). decreases from slightly more than a hundred thousand to slightly more than a thousand. The death toll for R(0.2, c) in the Reed-Frost model and three approximations